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+{"text":"\\section{Introduction}\\label{sec:intro}\n\nThe class of force-directed graph drawing algorithms is large\nboth in terms of objectives and optimization algorithms~\\cite{k-fdda-13,b-fdgd-14}. \nExperimental~\\cite{bp-esdbgd-09} and anecdotal evidence suggest\nthat a most desirable objective is\nthe stress function of distance-based multidimensional scaling~\\cite{m-maed-66}.\nGiven a simple undirected graph $G=(V,E)$,\nthe layout $x=(\\mathbb{R}^2)^V$ of a straight-line drawing is considered suitable,\nif the weighted deviation\n\\begin{equation}\n  \\mathop{stress}(x)=\\sum_{i<j} d_{ij}^{-2}(\\|x_i-x_j\\|-d_{ij})^2\n  \\label{eq:stress}\n\\end{equation}\nof Euclidean distances $\\|x_i-x_j\\|$ in the layout\nfrom shortest-path distances $d_{ij}$ in the graph \nis small. \n\nThe stress function has been varied in numerous ways\nto accommodate additional objectives or constraints~\\cite{bm-qcsma-12,bp-mfrl-11,ghn-msmgl-13,dkm-cglsmgp-09,nob-uhmcsw-15}.\nSince stress minimization is computationally intractable, \nsimilarly many approaches have been proposed to save computation time~\\cite{imo-gmmdsgpu-09,mns-dlgmmso-18,okb-ssm-17}.\nThese methods are generally designed to improve an initial layout iteratively\nand thus yield local minima of the stress function\nthat cannot be improved further by moving single vertices.\n\nHere we are interested in assessing a recent proposal\nby Zheng, Pawar, and Goodman~\\cite{zpg-gdsgd-18}\nthat is based on stochastic gradient descent\nand claimed to outperform majorization approaches~\\cite{gkn-gdsm-05}.\n\nOur own computational experiments suggest\nthat the new approach does not lead to better layouts,\nbut that it is still preferable due to its simplicity\nand, crucially, indifference to initialization.\nWe do not address actual running times\nbecause any comparison would be relative to the choice of speed-up techniques \nand the overall similarity of the computation suggests\nthat the same algorithm engineering techniques could be used in either approach.\n\nThe remainder is organized as follows. \nIn Sect.~\\ref{sec:techniques},\nwe briefly describe the proposal of Zheng et al.\\ in the context of previous approaches.\nThe results of our experiments are presented and discussed in Sect.~\\ref{sec:experiments},\nand we conclude with some general implications in Sect.~\\ref{sec:concl}.\n\n\n\\section{Stress Minimization}\\label{sec:techniques}\n\nWe very briefly review some major developments\nin the use of multidimensional scaling in graph drawing.\nThis is not to provide the details of each method\nbut to contrast the approach based on stochastic gradient descent\nwith previous approaches.\n\n\\paragraph{Gradient descent.}\nWhile first uses of multidimensional scaling for graph drawing date back to the 1960s,\nit was popularized by Kamada and Kawai~\\cite{kk-adgug-89},\nwho also introduced a localized version of the gradient descent approach used until then.\nSince a necessary condition for a local minimum of the stress function is that\nall partial derivatives are zero, they iteratively pick a vertex for which\nthe vector of partial derivatives with respect to its two coordinates has maximum length.\nThen a two-dimensional Newton-Raphson method is applied to the stress function with all other vertices fixed.\nTheir layout is thus obtained by iteratively moving one vertex at a time\ntoward a position where the different stress terms cancel each other out.\n\n\\paragraph{Majorization.}\nGanser, Koren, and North~\\cite{gkn-gdsm-05} proposed to use majorization~\\cite{l-acamds-77} instead.\nHere, the complex stress function is replaced with a convex function\nthat is larger for each layout but the current, for which it is equal. \nMinimizing this function leads to a new layout\nthat is guaranteed to have lower stress,\nand the process is iterated until it converges to a local minimum.\n\nThe process can also be localized to move only a single vertex\nsuch that the majorizing function is reduced. \nThis yields an intuitive algorithm because the update \n$$\n  x_i\\gets \\frac{1}{\\sum\\limits_{j\\neq i}d_{ij}^{-2}}\n             \\sum_{j\\neq i} d_{ij}^{-2}\\cdot\\frac{x_j+d_{ij}(x_i-x_j)}{\\|x_i-x_j\\|}\n$$\nplaces vertex~$i$ directly into a position\nthat balances out the influences of all other vertices.\nOne iteration consists of an update of each vertex.\n\nBecause of its simplicity and guaranteed convergence,\nthis approach is considered the state of the art.\n\n\\paragraph{Stochastic gradient descent.}\nIn this method the gradient is replaced by an unbiased estimator.\nFor additive objective functions such as the stress function in Eq.~\\eqref{eq:stress},\nthe estimator may simply be a single term of the sum.\nSince stress has one term for every pair of vertices, \nthe contribution of this term can be reduced by\nmoving the two vertices either closer together or farther apart.\n\nA single update thus moves both vertices along the vector~$\\delta$  \nto extend or shrink the line segment $\\overline{x_i x_j}$ to match the target length~$d_{ij}$ more closely,\n$$\n \\begin{array}[c]{l}\n   x_i \\gets x_i-\\frac{\\mu(t)}{2}\\cdot\\delta\\\\[1ex]\n   x_j \\gets x_j+\\frac{\\mu(t)}{2}\\cdot\\delta\n \\end{array}\\qquad\\text{where}\\qquad\n \\delta = \\frac{\\|x_i-x_j\\|-d_{ij}}{\\|x_i-x_j\\|}\\cdot(x_i-x_j)~, \n$$\nand~$\\mu(t)=\\min\\{1,d_{ij}^{-2}\\eta(t)\\}$ is a weighted step width capped at~1. \nSince an individual move is almost certainly in conflict\nwith the desired distances of other pairs,\nthe method does not converge in general.\nInstead, the unweighted step width~$\\eta(t)$\nis made to exhibit an exponential decay over iteration time~$t$,\nand convergence is thus enforced.\n\n\\begin{figure}\n    \\mbox{\\includegraphics[width=0.25\\linewidth]{dwt_1005_sgd_random_00.png}%\n    \\includegraphics[width=0.25\\linewidth]{dwt_1005_sgd_random_01.png}%\n    \\includegraphics[width=0.25\\linewidth]{dwt_1005_sgd_random_06.png}%\n    \\includegraphics[width=0.25\\linewidth]{dwt_1005_sgd_random_15.png}}\n    \\caption{Example run of stochastic gradient descent on graph \\texttt{dwt\\_1005} with random initialization and intermediate layouts after 1, 6, and 15 iteration.}\n    \\label{fig:example}\n\\end{figure}\n\nOne iteration consists of an update of all pairs of vertices in random order.\nThe method is thus similar to localized majorization\nbut instead of aggregating the influence of all other vertices before moving one,\nthose influences are considered separately in random order. \nThe running time of one iteration is in $\\Theta(n^2)$\nfor both stochastic gradient descent and localized majorization, \nbut instead of over a linear number of linear-time vertex movements\nthe computation is spread out over a quadratic number of constant-time dyadic updates.\n\n\n\\section{Experiments}\\label{sec:experiments}\n\nOur experiments address the claim~\\cite{zpg-gdsgd-18}\nthat stochastic gradient descent~(SGD) outperforms majorization~(SMACOF).\nThe graphs used as benchmarks \nare from the University of Florida sparse matrix collection~\\cite{dh-ufsmc-11}.\n\n\\paragraph{On par, but not better.}\nThe claim of superior performance is based on experiments \nin which both approaches are initialized with a random layout\nas in the example in Fig.\\ref{fig:example}. \nIt was already concluded from earlier experiments, however,\nthat the performance of majorization depends on the initialization\nand that random initialization leads to poor local minima~\\cite{bp-esdbgd-09}.\n\n\\begin{figure}\n    \\centering\n    \\begin{tabular}{r|cccl}\n    initially & 1 iteration & 6 iterations & 15 iterations\\\\\\hline\n    \\raisebox{0.1\\linewidth}{random}\n    & \\includegraphics[width=0.27\\linewidth]{1138_bus_smacof_random_01}\n    & \\includegraphics[width=0.27\\linewidth]{1138_bus_smacof_random_06}\n    & \\includegraphics[width=0.27\\linewidth]{1138_bus_smacof_random_15} \n    & \\raisebox{0.1\\linewidth}{SMACOF}\\\\\n    \\raisebox{0.1\\linewidth}{random} \n    & \\includegraphics[width=0.27\\linewidth]{1138_bus_sgd_random_01}\n    & \\includegraphics[width=0.27\\linewidth]{1138_bus_sgd_random_06}\n    & \\includegraphics[width=0.27\\linewidth]{1138_bus_sgd_random_15} \n    & \\raisebox{0.1\\linewidth}{SGD}\\\\\n    \\raisebox{0.1\\linewidth}{CMDS} \n    & \\includegraphics[width=0.27\\linewidth]{1138_bus_smacof_cmds_01}\n    & \\includegraphics[width=0.27\\linewidth]{1138_bus_smacof_cmds_06}\n    & \\includegraphics[width=0.27\\linewidth]{1138_bus_smacof_cmds_15} \n    & \\raisebox{0.1\\linewidth}{SMACOF}\\\\\n    \\raisebox{0.1\\linewidth}{CMDS} \n    & \\includegraphics[width=0.27\\linewidth]{1138_bus_sgd_cmds_01}\n    & \\includegraphics[width=0.27\\linewidth]{1138_bus_sgd_cmds_06}\n    & \\includegraphics[width=0.27\\linewidth]{1138_bus_sgd_cmds_15} \n    & \\raisebox{0.1\\linewidth}{SGD}\n    \\end{tabular}\n    \\caption{An example graph (\\texttt{1138\\_bus}) after 1, 6, and 15 iterations.}\n    \\label{fig:layouts}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\includegraphics[width=\\textwidth]{stressValues.png}\n\\caption{Stress values for SGD and SMACOF on two example graphs. Random initialization is within a unit square whilst classical MDS is used at an appropriate scale. The plots show results of 10~runs for each algorithm, with circles representing single runs and lines interpolating through the means of all 10~runs. Initial stress omitted.}\n\\label{fig:stress}\n\\end{figure}\n\n\\begin{figure}[b!]\n\\includegraphics[width=\\textwidth]{RelativeDeviation.png}\n\\caption{Stress relative to baseline from SMACOF after CMDS.\nWith 10~runs for each instance, we find that \nrandom initialization results in significantly higher stress for SMACOF (left chart). \nThe stress obtained from SGD differs by about $\\pm1\\%$ (rescaled on the right).}\n\\label{fig:deviation}\n\\end{figure}\n\nWe therefore ran experiments comparing the reduction in stress \nwhen initializing at random or with classical MDS~(CMDS).\nClassical MDS results in layouts that are essentially unique and represent large distances well.\nMoreover, it can be approximated at comparatively negligible cost using PivotMDS~\\cite{bp-empms-07}.\nTwo typical examples of the results are shown in Fig.~\\ref{fig:stress},\nand for a better intuition, \nwe also show some of the corresponding layouts in Fig.~\\ref{fig:layouts}.\nWhile the result on all benchmark graphs confirm \nthat SGD indeed yields much lower stress than majorization \nwhen initialized with a random layout,\nthere is no noteworthy difference in the final stress \nwhen the initial layout takes care of the global arrangement.\nNotably, the result of SGD is largely independent of the initialization strategy. \n\nOur experiments on a much larger set of benchmark graphs support these conclusions.\nThe evaluations in Fig.~\\ref{fig:deviation} confirm quantitatively \nthat majorization with random initialization is a poor baseline \nbecause it results in significantly higher stress \ncompared to majorization after classical scaling. \nWhether SGD or the latter combination yield lower stress depends on the graph,\nbut relative differences are small, anyway.\n\n\\paragraph{Self-initializing.}\nThe seeming indifference of SGD to the initial layout prompted a second suite of experiments.\n\nWe hypothesized that the initially large displacements in SGD \nare responsible for the overall quality of the final outcome. \nIf this was the case, then the differences between SGD and SMACOF should \ndisappear when we initialize SMACOF with a small number of SGD iterations. \n\n\\begin{figure}[tb]\n\\includegraphics[width=\\textwidth]{SGD_Local_combination.png}\n\\caption{Stress reduction by SMACOF initialized with CMDS or a few iterations of SGD (left) \nand the relative deviation of the final stress from the baseline of SMACOF with CMDS (right)\non example graphs \\texttt{1138\\_bus} and \\texttt{dwt\\_1005}. \nThe initial iterations of SGD start from a random initialization in the unit square,\nand each instance was run 10~times.} \n\\label{fig:initialization}\n\\end{figure}\n\nAs illustrated in Fig.~\\ref{fig:initialization} this is indeed the case.\nEven a single step of SGD prevents majorization from sinking into a poor local minimum.\nAfter about seven iterations of SGD, majorization yields layouts \nthat are even slightly better than those obtained from initialization with CMDS.\nWe also note that in the next iterations, \nSMACOF reduces stress faster than SGD,\nbut the number of iterations to the final layout is roughly the same for both.\nThis number becomes smaller than for SMACOF initialized with CMDS,\noffsetting the higher cost of SGD iterations compared to PivotMDS. \n\nWe conclude that a, if not the, major advantage of the approach based on stochastic gradient descent\nis the reliable untangling of any initial layout during the first few iterations.\nNo separate initialization strategy is required.\n\n\\paragraph{Well designed.}\nWe performed a number of additional experiments \nthat generally confirm the recommendations given for stochastic gradient descent~\\cite{zpg-gdsgd-18},\nand indicate that little can be gained by straightforward attempts at improvement\nsuch as an initial focus on long distances or the integration of majorization steps.\n\n\n\\section{Conclusions}\\label{sec:concl}\n\nWe have presented computational experiments comparing two approaches \nfor graph drawing by multidimensional scaling of shortest-path distances.\n\nContrary to claims by the authors, we do not find that stochastic gradient descent, \nwhich was recently proposed as an alternative to majorization,\nleads to better layouts~\\cite{zpg-gdsgd-18}.\nWe find no significant differences in stress,\nprovided that majorization is initialized appropriately. \n\nThe true advantage of stochastic gradient descent\nappears to lie in its indifference to initialization. \nIt is striking that this very simple and uniform algorithm\nyields results that are on par with the state of the art.\n\nWe did not compare running times in this short paper because\nboth approaches largely perform the same operations in different order\nand speed-up techniques such as subsampling and spatial aggregation abound.\nSince many of these apply similarly to both approaches, \nwe expect differences to be too subtle for any general claims.\nSince pairs in a maximal matching can be updated without interference,\nstochastic gradient descent appears to be more amenable to parallelization, though.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nFor   a compact and Hausdorff  space $X$,  let $C(X, I)$ denote the semigroup of all\ncontinuous functions on $X$ taking values in the closed unit interval $I = [0,1]$. Motivated by a result of L. Moln\\'ar (see \\cite{Mo}), J. Marovt studied the form of multiplicative bijections on $C(X, I)$\nin a special case, namely, when $X$ satisfies the first axiom of countability (see \\cite{M}). He proved that\nif $\\varphi$ is such a map, then there exist a  homeomorphism $\\mu : X \\longrightarrow X$ and a continuous map $p: X \\longrightarrow (0, +\\infty)$ such that\n$$\\varphi (f) (x) = f(\\mu (x))^{p(x)}$$\nfor every $f \\in C (X, I)$ and $x \\in X$.\nWe call maps of this form {\\em standard}.  Marovt conjectured  that every multiplicative bijection \non $C (X, I)$ was necessarily standard, and that the assumption on $X$ of being first countable could be dropped.\n\n\n\n\nIn this paper we prove that Marovt's conjecture does not hold. In fact we give a characterization of spaces $X$ for which there exists a multiplicative bijection of $C(X,I)$ that is {\\em not}  standard. Roughly speaking, we prove that this is  the case if and only if $X$ is the Stone-\\v{C}ech compactification of a proper subset (see Theorem~\\ref{sete}).\n\nWe study a more general case, as is that of multiplicative bijections between semigroups of functions. Of course, a multiplicative bijection between arbitrary semigroups of $C(X, I)$ need not be of the above form, even if they separate points. A simple example\nis the following: take $X$ having only one point, so each semigroup of $C(X, I)$ can be identified with a\nsemigroup of $I$. Consider for instance $s=1\/2$, $t= 1\/3$, and $\\mathcal{A} := \\{s^n t^m : n, m \\in \\mathbb{N}\\}$. It is now clear that the map sending each $s^n t^m$ into $s^m t^n$ is multiplicative and bijective, but cannot be described as above. \n\nNevertheless, our technique can be used for semigroups other than $C(X, I)$. In fact, it works if we just  require  these\nsemigroups $\\mathcal{A} \\subset C(X, I)$ to satisfy the following three properties (where, as usual,\nif $f \\in C(X, I)$,  $\\mathrm{coz} \\hspace{.02in}  f$ denotes the set $\\{x \\in X: f(x) \\neq 0 \\}$, and \n $\\mathrm{supp} \\hspace{.02in}  f$ denotes its closure).\n\n\n\\begin{description}\n\\item[Property 1] Given any $x \\in X$ and any neighborhood $U$ of $x$,\nthere exists $f \\in \\mathcal{A}$ such that $f(x) \\equiv 1$ on a neighborhood of $x$ and $\\mathrm{supp} \\hspace{.02in}  f \\subset U$.\n\\item[Property 2] If $f, g \\in \\mathcal{A}$ satisfy $f (x) \\le  g (x)$ for every $x \\in X$,  \nand $\\mathrm{supp} \\hspace{.02in}  f \\subset \\mathrm{coz} \\hspace{.02in}  g $, \nthen there exists $k \\in \\mathcal{A}$ such that\n$f=gk$.\n\\item[Property 3] For each $x \\in X$, the set $(0, 1) \\cap \\{f(x) : f \\in \\mathcal{A}\\} $ is nonempty.\n\\end{description}\n\n\nIt is easy to see that the image by  a standard multiplicative bijection of a  semigroup satisfying Properties 1, 2, and 3 is a semigroup that also satisfies them. One of such semigroups is, for instance, that of all continuously differentiable functions on $I$ (also taking values in $I$). The general form of multiplicative bijections between this kind of semigroups will be given in Theorem~\\ref{derr-laredo}. Finally we also give an example where Marovt's result does not hold for multiplicative bijections defined between general semigroups of this type, even when the ground spaces are first countable (see\nExample~\\ref{carras-si-cero}). \n\n\\smallskip\n\n\nSuppose that $\\varphi: \\cx \\ra \\cy$ is multiplicative and bijective, where\n $\\mathcal{A} \\subset C(X, I)$ and $\\mathcal{B} \\subset C(Y, I)$ are \nsemigroups satisfying Properties 1, 2, and 3.  We say that $y \\in Y$ is a {\\em standard point} for $\\varphi$ \n   if there exist a number $p \\in (0, + \\infty)$ and a point $x \\in X$ such that\n $\\varphi (f) (y) = f(x)^{p}$ for every $f \\in \\mathcal{A}$. \nWe denote by $\\mathcal{R} (\\varphi)$ the set of standard points for $\\varphi$.\n \n Throughout we assume that $X$ and $Y$ are compact and Hausdorff spaces. Given a (completely regular) space $Z$,\n we denote by $\\beta Z$ its Stone-\\v{C}ech compactification.\n \n \\smallskip\n \n Remark finally that our results are essentially different from those given in \\cite{Mi} for multiplicative bijections between  spaces \n$C(X, \\mathbb{R})$ of real-valued continuous functions on $X$. Even so, there is a similarity in that the multiplicative structure of the spaces of functions determines the space\n $X$ up to homeomorphism. More recent results in this line, concerning multiplicative maps on $C(X, \\mathbb{R})$, are given for instance in \\cite{LS1}  and \\cite{LS2}.\n \n \n \\section{Main results}\n \n We  first give a theorem that provides  a description of multiplicative bijections between semigroups \n satisfying the above properties. Recall that a subset $Z$ of $Y$ is a {\\em cozero-set} if  there exists a real-valued\n continuous function $f$ on $Y$ such that $Z = \\{y \\in Y: f(y) \\neq 0\\}$.\n \n \\begin{thm}\\label{derr-laredo}\n Suppose that $\\varphi: \\cx \\ra \\cy$ is multiplicative and bijective, where\n $\\mathcal{A} \\subset C(X, I)$ and $\\mathcal{B} \\subset C(Y, I)$ are \nsemigroups satisfying Properties 1, 2, and 3. Then \n\\begin{enumerate}\n\\item\\label{astamarte} there exist a \n homeomorphism $\\mu$ from $Y$ onto $X$\n and a continuous map $p: \\mathcal{R} (\\varphi) \\longrightarrow (0, + \\infty)$ such that\n$$\\varphi(f) (y) = f(\\mu(y))^{p(y)}$$\nfor every $f \\in \\mathcal{A}$ and $y \\in \\mathcal{R} (\\varphi)$; \n\\item\\label{santamarta} the set $\\mathcal{R} (\\varphi)$ is a  \ndense cozero-set in $Y$, and the map $p$ can be extended  to a continuous map from $Y$ into $[0, + \\infty]$ taking\nvalues in $\\{0 , + \\infty\\}$ at every point of $Y \\setminus \\mathcal{R} (\\varphi)$. \n\\end{enumerate}\n\\end{thm}\n\n\\begin{rem}\\label{negro}\nProperty 3 is necessary to ensure that Theorem~\\ref{derr-laredo} holds. For instance if we take $X=Y=[-1,1]$ and \n$\\mathcal{A} := \\{f \\in C(X,I) : f(0) \\in \\{0,1\\}\\}$, it is clear that $\\mathcal{A}$ satisfies both Properties 1 and 2, and that the map $\\varphi: \\mathcal{A} \\longrightarrow \\mathcal{A}$ defined, for every $f \\in \\mathcal{A}$, as $\\varphi (f) (x) := f(x)$ if $x \\le 0$, and $\\varphi (f) (x) := f(x)^2$ if $x >0$, is a multiplicative bijection. Nevertheless, $\\mathcal{R} (\\varphi) =X$, so each point is standard for $\\varphi$, but we cannot find a continuous map $p$ as given in Theorem~\\ref{derr-laredo}.\n\\end{rem}\n\n We next state the theorem characterizing all spaces on which  can be defined a multiplicative bijection that is not standard. Recall that a topological space $Z$ is said to be {\\em pseudocompact} if every real-valued continuous map on $Z$ is bounded.\n \n \\begin{thm}\\label{sete}\n There exists a bijective and multiplicative  map  $\\varphi: \\dx \\ra \\dy$ \n that is not standard if and only if $X$ and $Y$ are homeomorphic and there exists a \n subset $Z$ of $Y$, not pseudocompact, such that $Y = \\beta Z$.\n \\end{thm}\n \n Obviously Theorem~\\ref{sete} gives an answer in the negative to Marovt's conjecture. It is enough now to take any completely regular space $Z$ (thus ensuring that its Stone-\\v{C}ech compactification exists) that is not pseudocompact (as for instance $\\mathbb{N}$, $\\mathbb{R}$, or any unbounded subset of a normed space), and we have that there are always multiplicative bijections on $C (\\beta Z, I)$ that\n are not standard. \n\n\n\n\n\n\\section{Some other results and proofs}\n\n The following is a key lemma to prove Theorem~\\ref{derr-laredo}.\n\n\\begin{lem}\\label{azero}\nSuppose that\n$u: \\mathscr{A}_1 \\longrightarrow \\mathscr{A}_2$ is an order preserving  multiplicative bijection, \nwhere $\\mathscr{A}_1$ and $ \\mathscr{A}_2$ are semigroups contained in $(0,1)$. Then there exists \n$p \\in (0 , + \\infty)$ such that\n$u(\\gamma) = \\gamma^p$ for every $\\gamma \\in \\mathscr{A}_1$.\n\\end{lem}\n\n\\begin{proof}\nSuppose on the contrary that there exist $\\alpha, \\beta \\in \\mathscr{A}_1$ such that $u( \\alpha) = \\alpha^p$ and $u(\\beta ) =\\beta^q$, where $0<p<q$. By taking integer powers of $\\alpha$ if necessary, we may assume without loss of generality that $\\alpha < \\beta$. Define $a := - \\log \\alpha$ and $b:= - \\log \\beta$. Obviously $0 < b <a$, and we can find natural numbers $n, m$ such that \n$$ \\frac{b}{a} < \\frac{n}{m} <  \\frac{q}{p} \\frac{b}{a}.$$\nNow it is clear that $m b < na$ and $npa <mqb$. These inequalities lead easily to $\\alpha^n < \\beta^m$ and $\\alpha^{np} > \\beta^{mq}$, that is, $u(\\alpha^n) > u (\\beta^m)$, against the fact that $u$ is order preserving.\n\\end{proof}\n\n\n\n\\begin{proof}[Proof of Theorem~\\ref{derr-laredo}]\n(\\ref{astamarte})\nFor each $x  \\in X$,  let $\\mathscr{U}_x$ be the set of all $f \\in \\mathcal{A}$ for which there exists a neighborhood of $x$ where $f \\equiv 1 $. \n\nLet us see that $\\mathscr{C}_x := \\{y \\in Y: \\varphi (f) (y) =1 \\hspace{.03in} \\forall f \\in \\mathscr{U}_x\\}$ is nonempty.\nFirst notice that, given any $f \\in \\mathscr{U}_x$, the set $\\varphi (f)^{-1} (\\{1\\})$ is compact.\nOn the other hand, given $f_1, \\ldots, f_n \\in \\mathscr{C}_x$, we can find $f_0 \\in \\mathcal{A}$, $f_0 \\neq 0$, such that\n$\\mathrm{supp} \\hspace{.02in}  f_0 \\subset \\{z \\in X: \\prod_{i=1}^n f_i (z) =1\\}$. This obviously implies that\n$f_0 f_i =f_0$ for each $i$, so $\\varphi(f_0) \\varphi(f_i) = \\varphi (f_0) \\neq 0$. We deduce that \n$\\bigcap_{i=1}^n \\varphi (f_i)^{-1} (\\{1\\}) \\neq \\emptyset$. Then $\\{ \\varphi (f)^{-1} (\\{1\\}) : f \\in \\mathscr{U}_x \\}$ satisfies\nthe finite intersection property, and we conclude that\n$\\mathscr{C}_x $ is nonempty.\n\n\n\nOn the other hand,  $\\varphi^{-1}$ is also multiplicative, so given any $y \\in \\mathscr{C}_x$, we  have that the set $\\mathscr{C}_y$ (defined in a similar way as $\\mathscr{C}_x$) is nonempty. Let us see that $\\mathscr{C}_y =\\{x\\}$. \nSuppose that  there exists  $z \\in \\mathscr{C}_y$, $z \\neq x$. Then we can find $f_z \\in \\mathscr{U}_x$ such that $z \\notin \\mathrm{supp} \\hspace{.02in}  f_z $.  Since $z \\in \\mathscr{C}_y$\n  and $\\varphi (f_z) (y) =1$,\nthen we can find $k \\in \\mathcal{B}$ with $\\mathrm{supp} \\hspace{.02in} k \\subset \\mathrm{coz} \\hspace{.02in} \\varphi (f_z)$\nand $\\varphi^{-1} (k) (z) \\neq 0$. Clearly, \nif we now  take $g \\in \\mathscr{U}_z$ with $f_z g=0$,  then $g \\varphi^{-1} (k) \\neq 0$, but $\\varphi (g) k =0$, which is impossible.\n\n\n\n\n\nThe above process lets us define a map $\\mu : Y \\longrightarrow X$, which turns out to be bijective, such that $\\mu (y)$ is the only point in $\\mathscr{C}_y$, and $y$ is the only point\nin $\\mathscr{C}_{\\mu (y)}$, for each $y \\in Y$.\n\n\n\nWe prove next that $\\mu$ is continuous at every point of $Y$ (and is consequently a homeomorphism). Take any $y \\in Y$, and let $U$ be an open \nneighborhood of $\\mu (y)$. We will see that, if $f \\in \\mathscr{U}_{\\mu (y)}$ and $\\mathrm{supp} \\hspace{.02in}   f \\subset U $, \nthen $\\mu (\\mathrm{coz} \\hspace{.02in} \\varphi (f))$ is contained in $U$. Otherwise\nthere exists $z \\in Y$ such that $\\varphi (f) (z) \\neq 0$ and $\\mu (z) \\notin \\mathrm{supp} \\hspace{.02in}   f$,\nso we can take $k \\in \\mathscr{U}_{\\mu (z)}$ such that $kf =0$. Obviously \n$\\varphi (k) (z) \\varphi (f) (z) \\neq 0$, which \nis absurd.\n\nFinally, we have that, by definition, if $y \\in \\mathcal{R} (\\varphi)$, then there exist $p(y) \\in (0, + \\infty)$\nand $x \\in X$ such that $\\varphi(f) (y) = f (x)^{p(y)}$ for every $f \\in \\mathcal{A}$. It is easy to check that $x = \\mu (y)$.\nAs for the map $p: \\mathcal{R} (\\varphi) \\longrightarrow (0, + \\infty)$, we have that for each $y \\in \\mathcal{R} (\\varphi)$, \n $$p(y) = \\frac{\\log \\varphi (f) (y)}{\\log f \\left( \\mu \\left( y \\right) \\right)}$$for every $f \\in \\mathcal{A}$ with \n$f \\left( \\mu \\left( y \\right) \\right) \\neq 0,1$. This implies  that $p$ is continuous at $y$, and consequently on $\\mathcal{R} (\\varphi)$.\n\n\\medskip\n\n(\\ref{santamarta}) \nFor each $y \\in Y$,  let $\\mathcal{B}_y := \\{g (y) : g \\in \\mathcal{B}\\} \\cap (0,1)$, and define  $\\mathcal{A}_{x}$  for each $x \\in X$ in a similar way. Consider also \n the set $\\mathcal{R}_1 (\\varphi)$ of all $y \\in Y$ such that\n$\\varphi (f) (y) \\neq 0, 1$ whenever $f \\in \\mathcal{A}$ satisfies $f (\\mu (y)) \\neq 0, 1$. We need the following claim.\n\n\\smallskip\n\n{\\bf Claim.} {\\em Let $y \\in \\mathcal{R}_1 (\\varphi)$. If $f, g \\in \\mathcal{A}$ satisfy \n$g(\\mu (y)) \\le f(\\mu (y))$,  then $\\varphi (g) (y) \\le \\varphi (f) (y)  $. Moreover $\\mu (y) $ belongs to $\\mathcal{R}_1 \\left( \\varphi^{-1} \\right)$.}\n\\smallskip\n\nSuppose first that $g(\\mu (y)) < f(\\mu (y))$, and  take   a neighborhood $U$ of $\\mu (y)$ with $ g(x) < f(x)$ for every $x \\in U$. We pick $f_0, g_0 \\in \\mathscr{U}_{\\mu (y)}$\nsuch that $\\mathrm{supp} \\hspace{.02in}  f_0 \\subset U$, and \nsuch that $\\mathrm{supp} \\hspace{.02in}  g_0 \\subset \\{ x \\in X : f_0 (x) =1\\}$, respectively. Since $\\mathcal{A}$ \nsatisfies Property 2, then   \nthere exists $k \\in \\mathcal{A}$ such that $(ff_0)k=  gg_0$. Also $k(\\mu(y)) \\in (0,1)$, and consequently $$\\varphi (g) (y) = \\varphi (gg_0) (y) = \\varphi (ff_0) (y) \\varphi (k) (y) < \\varphi (ff_0) (y) = \\varphi (f) (y).$$\nWe now prove that $\\mu (y)$ belongs to $\\mathcal{R}_1 (\\varphi^{-1})$. Let $h \\in \\mathcal{B}$ be such that $ h (y) \\neq 0,1$.\nSuppose that $\\varphi^{-1} (h) (\\mu (y)) = 0$ and take any $l \\in \\mathcal{A}$ with $l (\\mu (y)) \\neq 0, 1$, and \n$n \\in \\mathbb{N}$ such that $\\left( \\varphi(l) (y) \\right)^n < h(y)$. We  then have that $ \\varphi^{-1} (h)(\\mu (y)) < l^n (\\mu (y))$ and\n$h (y) > \\varphi(l^n) (y)$, what goes against what we have proved above. We deduce that $\\varphi^{-1} (h) (\\mu (y)) \\neq 0$. In a similar way we can deduce that $\\varphi^{-1} (h) (\\mu (y)) \\neq 1$. Thus, $\\mu (y)$ belongs\nto $\\mathcal{R}_1 (\\varphi)$.\n\nNow, working with $\\varphi^{-1}$, it is clear that if $g(\\mu (y)) \\le f(\\mu (y))$, then we cannot get $\\varphi (g) (y) > \\varphi (f) (y)  $.\nThe claim is proved.\n\n\n\\smallskip\n\n\n\n\nFor any $y \\in \\mathcal{R}_1 (\\varphi)$, we may define a map $\\varphi_y : \\mathcal{A}_{\\mu(y)} \\longrightarrow \\mathcal{B}_y$ in\n the following way. Given $\\alpha \\in \\mathcal{A}_{\\mu (y)}$, there exists $f \\in \\mathcal{A}$ such that\n $f(\\mu(y)) = \\alpha$. Then define $\\varphi_y (\\alpha) := \\varphi (f) (y)$. It is clear by the above claim that \n $\\varphi_y$ is well defined, and obviously it is multiplicative, order preserving, and bijective. \nAlso, we have that  $\\varphi (f) (y) =1 $ whenever $f(\\mu (y))= 1$, and $\\varphi (f) (y) =0 $ whenever $f(\\mu (y))= 0$,\n$f \\in \\mathcal{A}$.\nConsequently, by \nLemma~\\ref{azero},  we have that $\\mathcal{R}_1 (\\varphi ) \\subset \\mathcal{R} (\\varphi)$. The other inclusion is immediate, so\n$\\mathcal{R}_1 (\\varphi) = \\mathcal{R} (\\varphi) $.\n\n\n\n\n\n\\medskip\n\n \nWe  prove that $\\mathcal{R} (\\varphi) $\nis dense in $Y$.\nLet $W_0 \\subset Y$ be open (and nonempty). Pick \n$y \\in W_0$ and assume that  $y \\notin \\mathcal{R} (\\varphi) = \\mathcal{R}_1 (\\varphi)$, so \n there exists\n$f_0 \\in \\mathcal{A}$ such that $f_0 (\\mu (y)) \\neq 0, 1$ and $ \\varphi (f_0) (y) \\in \\{0,  1\\}$. Let $V:= \\{x \\in X : 0<f_0 (x) <1\\}$. \nWe next see that $y$ does not belong to the interior of the set $W_1 := \\{ z \\in W_0 : \\varphi (f_0) (z) =0\\}$. Otherwise, we can find \n$g_0 \\in \\mathscr{U}_{y}$ with  $g_0 \\varphi (f_0) =0$. This implies that $\\varphi^{-1} (g_0) f_0 =0$, against the fact that $\\varphi^{-1} (g_0) (\\mu (y)) =1$ and $f_0 (\\mu (y)) \\neq 0$. On the other hand,  \n$y$ does not belong to the interior of the set $W_2 := \\{ z \\in W_0 : \\varphi (f_0) (z) =1\\}$, because $\\varphi (f_0) \\notin \\mathscr{U}_y$. Consequently, due to the form of $W_1$ and $W_2$, we have that $y$ belongs to the closure of $Y \\setminus \\left( W_1 \\cup W_2 \\right)$. This means that \n$$W:= \\mu^{-1} \\left( V \\right) \\cap \\left( W_0 \\setminus \\left( W_1 \\cup W_2 \\right) \\right) \\neq \\emptyset.$$\nAs in the proof of the claim, it is  clear that given any point in $Y$, and $f , g \\in \\mathcal{A}$ with $g(\\mu (y)) < f(\\mu (y))$, then $\\varphi (g) (y) \\le  \\varphi (f) (y)$. Now it is straightforward to check that every point in $W$ belongs to $\\mathcal{R}_1 (\\varphi) = \\mathcal{R} (\\varphi)$.\n\n\n\n\\medskip\n\nWe finally see that $\\mathcal{R} (\\varphi)$ is a cozero-set.  \nSuppose  that $y \\in Y  \\setminus  \\mathcal{R} (\\varphi)$. Take any net $\\left( z_{\\alpha} \\right)_{\\alpha \\in \\Lambda}$  in $\\mathcal{R} (\\varphi)$\n converging to $y$.  \nClearly,  since $y \\notin \\mathcal{R}_1 (\\varphi)$, then there exists $f \\in \\mathcal{A}$ with $f (\\mu (y)) \\neq 0, 1$ and \neither $\\varphi (f) (y) =0 $ or $1$.\nAlso  \n $$p(z_{\\alpha}) = \\frac{\\log \\varphi (f) (z_{\\alpha})}{\\log f \\left( \\mu \\left( z_{\\alpha} \\right) \\right)}$$\n for every $\\alpha \\in \\Lambda$. \n  The conclusion that $\\lim_{\\alpha}  p (z_{\\alpha}) \\in \\{0, + \\infty\\}$ follows easily.   Finally,  it is clear that\n$\\mathcal{R} (\\varphi)$ coincides with the  cozero-set of the continuous function  $Y \\longrightarrow \\mathbb{R}$ given by \n  $$  y \\mapsto p(y) \\left( p \\left( y \\right) ^2 +1 \\right)^{-1} $$\n if    $y \\in \\mathcal{R} (\\varphi )$, \nand $y \\mapsto 0$ otherwise.  \n\\end{proof}    \n\n\\begin{rem}\\label{cerotres}\nIf in Theorem~\\ref{derr-laredo} we assume that the semigroups satisfy just Properties 1 and 2, a similar proof\nensures the existence of the homeomorphism $\\mu$ between $Y$ and $X$, even if the theorem does not hold (see Remark~\\ref{negro}). \n\\end{rem}\n\n\n\n\n\n\nIn what follows, when we want to specify the homeomorphism $\\mu$ and the map $p$ corresponding to a multiplicative and bijective map $\\varphi$, as given in Theorem~\\ref{derr-laredo}, we will write $\\varphi [\\mu, p]$.\nIt is obvious that if $\\varphi = \\varphi [\\mu, p]$, then $\\varphi^{-1} = [\\mu^{-1}, q]$, \nwhere $q = 1\/\\left( p\\circ \\mu^{-1} \\right)$.\n\n\n\\begin{cor}\\label{paragogo} \nLet  $\\varphi: \\cx \\ra \\cy$ be multiplicative and bijective,\nwhere\n $\\mathcal{A} \\subset C(X, I)$ and $\\mathcal{B} \\subset C(Y, I)$ are \nsemigroups satisfying Properties 1, 2, and 3. If $\\mathcal{R} (\\varphi)$ \nis pseudocompact, then $\\mathcal{R} (\\varphi) = Y$.\n\\end{cor}\n\n\\begin{proof}\nIt is obvious that if $\\varphi = \\varphi [\\mu, p]$ and $\\varphi^{-1} = \\varphi^{-1}[ \\mu^{-1}, q]$, then both $p$ and $q$ are bounded, and consequently neither $+ \\infty$ nor \n$ 0$ are  limit points of $p (\\mathcal{R} (\\varphi ))$. The conclusion follows from Theorem~\\ref{derr-laredo}.\n\\end{proof}\n\n\n\n\n\n\\begin{prop}\\label{dismas}\nLet  $\\varphi: \\dx \\ra \\dy$ be multiplicative and bijective. Then $Y = \\beta (\\mathcal{R} (\\varphi))$.\n\\end{prop}\n\n\\begin{proof}\nWe follow the notation given  after\n Remark~\\ref{cerotres}. \nSuppose that $\\varphi = \\varphi [\\mu, p]$. \n It is clear that if $\\mathbf{1}$ is the map \nconstantly equal to $1$ on $X$, and we consider   the multiplicative bijection \n$\\psi = \\psi[\\mu^{-1}, \\mathbf{1}] : C (Y, I) \\longrightarrow C (X, I)$,\n then the composition $\\varphi \\circ \\psi : C (Y, I) \\longrightarrow C (Y, I)  $ is $\\varphi \\circ \\psi [\\mathbf{id}_Y, p] $ (where $\\mathbf{id}_Y$  is the identity map on $Y$). Also,  $\\mathcal{R}_1 (\\varphi) =  \\mathcal{R}_1 (\\varphi \\circ \\psi)$, \nso $\\mathcal{R} (\\varphi) =  \\mathcal{R} (\\varphi \\circ \\psi)$ (see Proof of Theorem~\\ref{derr-laredo}). Consequently we can assume without loss of \ngenerality that $X=Y$  and $\\mu = \\mathbf{id}_Y$. \n\nNotice  that if we consider the continuous extensions of $p$ and $q$ to maps from $Y$  to $[0, + \\infty]$,\n as seen in Theorem~\\ref{derr-laredo}, then $p^{-1} (\\{0\\}) =  q^{-1} (\\{ + \\infty\\}) $\n and $p^{-1} (\\{+ \\infty\\}) =  q^{-1} (\\{ 0\\}) $. Thus, assuming that $\\beta (\\mathcal{R} (\\varphi)) \\neq Y$, at least one of the sets $ p^{-1} (\\{0\\})$,  $q^{-1} (\\{ 0\\})$\n is nonempty. We conclude that there is a continuous map \n from $Y \\setminus p^{-1} (\\{0\\})$, or from $Y \\setminus  q^{-1} (\\{ 0\\})$, to $I$ which does not admit\n a continuous extension to $Y$.\n \n Taking into account that $\\mathcal{R} (\\varphi) = \\mathcal{R} \\left( \\varphi^{-1} \\right)$,  we \nassume without loss of generality that \n  there exists a continuous function $g_0 :Y \\setminus p^{-1} (\\{0\\})  \\longrightarrow I$ that \ncannot be continuously extended to $Y$. We consider the map $h \\in C (Y, I)$ whose image by $\\varphi$ is the constant function $1\/e$. It is easily seen that,\nsince $p(y) \\log h (y) = \\log \\varphi (h) (y)$ for every $y \\in \\mathcal{R} (\\varphi)$, then \n $$h (y)= \\exp \\left( - \\frac{1}{p(y)} \\right)$$\nif $y \\in Y \\setminus p^{-1} (\\{0\\})$, and $h \\equiv 0$ on $ p^{-1} (\\{0\\})$. \n\n\n\n\nSince $p^{-1} (\\{0\\})$ is a zero-set, then there exists a sequence $\\left( K_n \\right)$ of compact subsets of $Y$ with\n$K_{n+1}$ contained in the interior of $K_n$ for each $n \\in \\mathbb{N}$, and such that \n$p^{-1} (\\{0\\}) = \\bigcap_{n=1}^{\\infty}      K_n$. For each $n \\in \\mathbb{N}$, we take $g_n \\in C (Y, I)$ satisfying $g_n \\equiv g_0$ on\n $Y \\setminus K_n$. We denote by $f_n$ its counterimage by $\\varphi$.\n \n\nNotice that if we define $f_0 (z) = f_n (z)$ whenever $z \\notin K_n$, then \n$f_0 : \nY \\setminus p^{-1} (\\{0\\})  \\longrightarrow I$ is continuous. It is also obvious that $f_0 h$ (defined \nas $0$ on $p^{-1} (\\{0\\})$) belongs to $C(Y, I)$. On the other hand, we have that $f_0 h \\equiv f_n h$ outside $K_n$ for each\n$n \\in \\mathbb{N}$, and consequently $$\\varphi (f_0 h) \\equiv \\varphi (f_n h) \\equiv g_n \\varphi (h) \\equiv g_0 \\varphi (h)$$\noutside each $K_n$. This implies that $\\varphi (f_0 h) \\equiv g_0 \\varphi (h) $ on $ Y \\setminus p^{-1} (\\{0\\}) $. Now\n it is easy to see that, since $\\varphi (h) \\equiv 1\/e$ on $p^{-1} (\\{0\\}) $,  then $\\varphi (f_0 h)$ is not defined at some points of $ p^{-1} (\\{0\\}) $, which is absurd.\n\\end{proof}\n\n\n\\begin{rem}\\label{rosales}\nWe deduce in particular that if the set  $Y \\setminus \\mathcal{R} (\\varphi)$ is nonempty, then its cardinality is at least $2^{\\mathfrak{c}}$. Also, if it  is endowed with the restricted topology, then no point in \n$Y \\setminus \\mathcal{R} (\\varphi)$ is a $G_{\\delta}$ (see \\cite[Chapter 9]{GJ}).\n\\end{rem}\n\n\\begin{rem}\\label{azulbelga}\nBy Remark~\\ref{rosales}, we have that each point of $Y$ having a countable base of neighborhoods belongs\nto $\\mathcal{R} (\\varphi)$ for every $\\varphi$. In particular, if $Y$ is first countable, then $\\mathcal{R} (\\varphi) = Y$, which is essentially Marovt's result. But there can be spaces which are not  first countable, and   for which every point is standard. As an easy example,\nconsider a bijective and multiplicative map $\\varphi : C([0, \\omega_1], I) \\longrightarrow C([0, \\omega_1], I)$, where \n$\\omega_1$ denotes the first noncountable ordinal. Since each point of $[0, \\omega_1)$ has a countable base of\nneighborhoods, we deduce that $[0, \\omega_1] \\setminus \\mathcal{R} (\\varphi) \\subset \\{\\omega_1\\}$, and again by Remark~\\ref{rosales}, we cannot have $[0, \\omega_1] \\setminus \\mathcal{R} (\\varphi) = \\{\\omega_1\\}$. We conclude that $\\mathcal{R} (\\varphi) = [0, \\omega_1]$.\n\\end{rem} \n\n\\begin{rem}\\label{nikdezir}\nThe conclusion given in Remark~\\ref{azulbelga} is not true for more general semigroups, that is, not every point having a countable base of neighborhoods necessarily belongs to $\\mathcal{R} (\\varphi)$ (see Example~\\ref{carras-si-cero}).\n\\end{rem}\n\n\nUsing Theorem~\\ref{derr-laredo}, it is easy to see that every isolated point is standard for every multiplicative bijection. More generally, the next result allows us to identify some points that belong to $\\mathcal{R} (\\varphi)$ for every $\\varphi$.\n    \n\\begin{cor}\\label{afeite}\nLet $y \\in Y$ be such that $\\beta \\left( Y \\setminus \\{y\\} \\right) \\neq Y$. \nThen $y \\in \\mathcal{R} (\\varphi)$ for every multiplicative bijection $\\varphi: \\dx \\ra \\dy$.\n\\end{cor}\n\n\\begin{proof}\nSuppose that $y \\notin \\mathcal{R} (\\varphi)$. Thus, taking into account  Proposition~\\ref{dismas} and the fact that $\\mathcal{R} (\\varphi) \\subset  Y \\setminus \\{y\\}$, we deduce that $\\beta \\left( Y \\setminus \\{y\\} \\right) = Y$, and we \nare finished.\n\\end{proof}\n\n\\begin{rem}\nObviously, the converse of Corollary~\\ref{afeite} is in general not true. For an easy example, consider for instance\n the point $\\omega_1$ in Remark~\\ref{azulbelga}, and take into account that  $\\beta [0, \\omega_1) = [0, \\omega_1]$.\n\\end{rem}\n\n\n\\begin{rem}\nMarovt's result cannot  be given in general for the kind of semigroups we are dealing with, even if the ground spaces are assumed to be first countable. That is, it is possible that $\\mathcal{R} (\\varphi) \\neq X$ for a multiplicative bijection $\\varphi : \\mathcal{A} \\longrightarrow \\mathcal{B}$, even in the case when $\\mathcal{A}, \\mathcal{B} \\subset C (X, I)$ satisfy Properties 1, 2, and 3, and  the \nspace $X$ is first countable. We next include an example of this fact.\n\\end{rem}\n\n\n\n\\begin{ex}\\label{carras-si-cero}\nFor a topological space $Z$, denote by $C(Z, \\mathbb{R}_{\\ge 0})$ the set of all continuous maps from $Z$ to $[0, +\\infty)$. Let $\\mathbb{N} \\cup \\{ \\infty \\}  $ be the one-point compactification of $\\mathbb{N}$ and  let \n$\\varphi : C( \\mathbb{N} \\cup \\{ \\infty \\}  , \\mathbb{R}_{\\ge 0} ) \\longrightarrow C ( \\mathbb{N} , \\mathbb{R}_{\\ge 0} )$ be defined, for each $f \\in C( \\mathbb{N} \\cup \\{ \\infty \\}  , \\mathbb{R}_{\\ge 0}) $, as $\\varphi (f) (n) := f(n)^n$ for every $n \\in \\mathbb{N}$.\n It is clear that $\\varphi$ is multiplicative and injective. \nSince it also preserves order, then the image of each $f \\in C(\\mathbb{N} \\cup \\{ \\infty \\}  , I)$ takes values in $I$.\n\n\nConsider now  the subset $\\mathcal{A}_1$ of $C(\\mathbb{N} \\cup \\{ \\infty \\} , I )$\nof all characteristic  functions which are continuous, and let $f_0 \\in C( \\mathbb{N} \\cup \\{ \\infty \\}  , I)$   be the counterimage by $\\varphi$ of the constant function equal to $1\/2$. Define $\\mathcal{A}$ as the set of all functions of the form\n$f = \\alpha g f_0^m$ for some $\\alpha \\ge 0$, $g \\in \\mathcal{A}_1$, and $m \\in \\mathbb{Z}$, such that $f(n) \\in I$ for every $n \\in \\mathbb{N}$.\n\n It is easy to see that if $f \\in \\mathcal {A}$, then $\\lim_{n \\longrightarrow \\infty} \\varphi (f) (n)$ exists, so in a natural way\n we may define a map $\\varphi : \\mathcal{A} \\longrightarrow C( \\mathbb{N} \\cup \\{ \\infty \\} , I) $.\nPut  $\\mathcal{B} := \\varphi (\\mathcal{A})$. It is easy to see that \n$\\mathcal{A}$ and $\\mathcal{B}$ satisfy Properties 1, 2, and 3. On the other hand, every point in $\\mathbb{N}$ is\n isolated, and consequently $\\mathbb{N}$ is contained in $\\mathcal{R} (\\varphi)$. Nevertheless, $\\infty$ does not \nbelong to $\\mathcal{R} (\\varphi)$.\n\\end{ex}\n\n\n\n\\begin{proof}[Proof of Theorem~\\ref{sete}]\nSuppose  that there exists a multiplicative bijection $\\varphi: \\dx \\ra \\dy$ that is not standard. First, by \nTheorem~\\ref{derr-laredo}, $X$ and $Y$ must be homeomorphic. Also, calling $Z := \\mathcal{R} (\\varphi)$, we have \nby Corollary~\\ref{paragogo} that $Z$ is not pseudocompact, and \nby Proposition~\\ref{dismas} that\n$Y = \\beta Z$.\n\nConversely, suppose that $X$ and $Y$ are homeomorphic, and that there exists a proper subset $Z$ of $Y$ which is not pseudocompact, and such that\n$Y= \\beta Z$. It is clear that without loss of generality we may assume that $X=Y$. \nSince $Z$ is not pseudocompact,  then there exists an unbounded continuous function $u: Z \\longrightarrow [1 , + \\infty)$. We define a  map $\\varphi: C (Z, I) \\longrightarrow C (Z, I)$ as $\\varphi (g) (z) := g (z)^{u(z)}$ for each $g \\in C (Z, I)$ and $z \\in Z$. \n\nIt is easy to check that $\\varphi$ is multiplicative and injective.\nLet us see that $\\varphi$ is surjective. To this end, we take any  $k \\in C (Z, I)$, and consider the map\n $L_k : Z \\longrightarrow [- \\infty, 0]$\ndefined as $L_k (z) := (\\log k(z)) \/ u(z)$ if $k (z) \\neq 0$, and $L_k (z) := -\\infty$ if $k(z) =0$. It is easy to \ncheck that $L_k$ is continuous and that $\\varphi (\\exp L_k) = k$ (assuming $\\exp (- \\infty) =0$).\n\nObviously $\\varphi$ can be seen as a multiplicative bijection on $C(\\beta Z, I)$. On the other hand,  if \nfor some homeomorphism $\\mu: \\beta Z \\longrightarrow \\beta Z$ and a continuous map \n$v: \\beta Z \\longrightarrow (0, + \\infty)$,  \n$\\varphi (f) (x) = f(\\mu (x))^{v (x)}$  for every $f$, then $\\mu$ must be the identity on $\\beta Z$, and $v(z) = u(z)$ for every $z \\in Z$. \nThis obviously implies that $v$ must attain the value $+ \\infty$ on some points of $\\beta Z \\setminus Z$, which is \nabsurd.\n\\end{proof}\n\n\n\n\\section{Acknowledgements}\nThe author wishes to thank  the referee for his\/her  valuable remarks and, also,  Ana M. R\\'odenas for drawing his attention to this subject.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nM33 (NGC~598) is a spiral galaxy whose closeness (840 kpc, Freedman et al. 1991; optical size 53'\n$\\times$53', Holmberg 1958) and inclination (i=53 $^\\circ$) allow detailed studies of its stellar \npopulations and ionized nebulae.\nPlanetary Nebulae (PNe) and H~{\\sc ii}\\ regions represent two very different stages in the lifetime of a galaxy, \nand their comparison provides insight on the chemical evolution of the host galaxy. \nPNe are indeed the final ejecta of evolved low- and intermediate-mass stars with mass \nbetween 1 and 8 M$_{\\odot}$, which must have formed between 3$\\times$10$^7$ yr and 10 Gyr ago, e.g., \nMaraston \\cite{maraston05}, while H~{\\sc ii}\\ regions belong to a very young population.\n\nOne of the most debated questions about the chemical evolution of galaxies is the metallicity gradient \nbehaviour with time.  Chemical evolution models predict different temporal behaviors of the metallicity gradient, depending on the assumptions one makes on gas inflow and outflow rates, and the star and cloud formation efficiencies.  Observations are needed to constrain these assumptions, but so far they have been insufficient, especially for the older populations. \nThe idea behind the observations leading to our work is to study the chemical and physical properties of a large number of PNe and H~{\\sc ii}\\ regions in M33, using the same set of observations, the same data reduction and analysis techniques, and identical abundance determination methods, to avoid all biases due to the stellar vs. nebular analysis. The aim is to derive abundances of the $\\alpha$-elements for as many PNe \nand H~{\\sc ii}\\ regions as possible, and to study the variation of the metallicity gradient and the average \nabundances in M33. \n\n \nIn this paper, several aspects of the study of M33 will be described. \nIn Section \\ref{sect_mmt} the observations of a sample of PNe and H~{\\sc ii}\\ regions with MMT will be presented. \nThese data, together with a large literature data-set, allow us to recompute the PN and H~{\\sc ii}\\ region \nmetallicity radial gradients.\nIn Section \\ref{sect_distr} the spatial distribution of the metallicity will be shown. \nThe off-center of the metallicity maps obtained from H~{\\sc ii}\\ regions and from PNe  will be discussed.\nIn Section \\ref{sect_model} the chemical evolution model of M33 (Magrini et al. 2007a, hereafter M07) will be revised, \nincluding a star formation process, parametrized by a Schmidt law, consistent with the observations. \nThe consequences, in particular the \ntime evolution of the metallicity gradient, will be analyzed. \nFinally, our conclusions and summary will be given in Section \\ref{sect_conclu}. \n \n\n\\section{The MMT observations}\n\\label{sect_mmt}\nIn semester 2007B we observed $\\sim$150 ionized nebulae in M33 in multi-object spectroscopic mode.\nWe used the Hectospec fiber-fed spectrograph (Fabricant et al. 2005) on the Multi Mirror Telescope \n(MMT), with a 270 mm$^{-1}$ grating at a dispersion of 1.2 \\AA\\ pixel$^{-1}$. \nThe instrument deploys 300 fibers over a 1$^\\circ$\\ diameter field of view; the fiber diameter is 1.5\\arcsec\\ (6 pc using a distance of 840 kpc to M33).  \nWe obtained spectra of 102 PNe and 48 HII regions with resulting total spectral coverage from ~3600 to 9100 \\AA. \nThe details on the data reduction and analysis are in Magrini et al. (2009, hereafter M09). \nMost of the PNe observed in M33 belong to its disk, and they are non-Type I, implying a population \nmainly composed by PNe with old progenitors, i.e. M$<$3 M$_{\\odot}$, and ages $>$0.3 Gyr.\n\nThe MMT observations allowed us to obtain a good determination of the O\/H gradient (together with \nNe\/H, S\/H, and Ar\/H) both from H~{\\sc ii}\\ regions and PNe. In the case of H~{\\sc ii}\\ regions, our cumulative sample includes: \ni)  H~{\\sc ii}\\ regions by Magrini et al. \\cite{magrini07b} which comprises their own determinations and all previous abundance determinations  with available $t_{e}$, recomputed uniformly; ii) the sample by  Rosolowsky \\& Simon \n(2008); iii) the new MMT sample. For PNe, we use the sample of PNe presented by M09. \nThe resulting gradients are the following, for PNe and H~{\\sc ii}\\ regions respectively, 12 + {\\rm log(O\/H)} =  \n$$   -0.031 (\\pm 0.013) ~  {\\rm R_{GC}} + 8.44 (\\pm\n0.06),\\eqno(1)$$ \n$$   -0.032 (\\pm 0.009) ~  {\\rm R_{GC}} + 8.42 (\\pm\n0.04),\\eqno(2)$$ where R$_{GC}$ is the galactocentric distance in kpc. \nThe two gradient are identical within the errors, both in the their slopes and central values. \n\\section{The metallicity in M33 and its evolution}\n\\label{sect_distr}\n\nThe large amount of chemical abundance data from H~{\\sc ii}\\ regions and PNe collected \nto date in M33 allow us to reconstruct  its metallicity map.\nIn this section, both the 2D  and the radial distribution are analyzed\ntaking advantages also of all previous abundance determinations with measured $t_{e}$\\ and of our new\nresults.   \n\n\n\\subsection{The 2D distribution of metals}\n\nThe usual way to study the metallicity distribution in disk galaxies is \nto average it azimuthally, assuming that: i)  the centre of the galaxy coincides\nwith the peak of the metallicity distribution; ii) at a given radius, the metallicity is the \nsame in each side of the galaxy. \nThe large number of metallicity measurements in M33, both from H~{\\sc ii}\\ regions and from PNe, \nallow us to reconstruct not only their radial gradient, but also their spatial distribution \nprojected onto the disk.  \nIn Figure 1, the two-dimensional metallicity distributions for M33 from H~{\\sc ii}\\ regions and \nfrom PNe are shown. \nThe highest metallicity H~{\\sc ii}\\ regions and PNe are not located at the center of the\ngalaxy, but rather lie at a radius of 1-2 kpc, in the southern arm.  \nSimon \\& Rosolowsky (2008) noticed this behavior for H~{\\sc ii}\\ regions and suggested as \nexplanation that the material enriched by the most recent generation\nof star formation in the arm has not yet been azimuthally  mixed through the\ngalaxy.  Whilst it might be true for H~{\\sc ii}\\ regions, it cannot be the reason for the off-center \nmetallicity distribution of PNe since they belong to an older population. \nColin \\& Athanassoula \\cite{colin81} noticed several evidences of asymmetries in the inner regions \nof M33, such as the distribution of HI atomic gas, of H~{\\sc ii}\\ regions, of  high luminosity stars. They\nproposed a kinematical model with a displaced bulge having a retrograde motion around the center. \nAlso detailed analysis of the innermost regions of M33 by Corbelli \\& Walterbos \\cite{corbelli07} \nfound possible asymmetries in the stellar and gas velocity pattern which might  be related \nto the displacement of a small bulge.\nThe off-center metallicity distribution might thus be related to the lack of a dominant gravitational \nsource in the center of this galaxy with a consequent motion at different epochs of the peak of the highest star \nformation region around the M33 visual center.  \n \n \\begin{figure}\n  \\resizebox{\\hsize}{!}{\\includegraphics[clip=true,angle=270]{metmap_hii_bw1.ps}}\n  \\resizebox{\\hsize}{!}{\\includegraphics[clip=true,angle=270]{metmap_pn_bw1.ps}}\n  \\caption{\\footnotesize{The metallicity maps: PNe (top) and H~{\\sc ii}\\ regions (bottom). \n  The 12 + log(O\/H) scale is shown on the y axe. The centre of M33 corresponds to the 0,0 position.      }}\n  \\label{oxy}%\n   \\end{figure}\n\n\\subsection{The time-evolution of the abundance gradient}\n\\label{sect_model}\n\nM07 built a chemical evolution model of M33, called {\\em accretion} model,  \nable to reproduce its main features, \nincluding the radial trends at  present-time of molecular gas, atomic gas,  stars,  SFR, and the \ntime evolution of the metallicity gradient using the available  constraints at that time. \nIn that model, the disk of M33 was formed by continuous accretion of primordial \ngas from the intergalactic medium. \nHowever, new observations have been made recently available, rendering necessary \na revision of the model. \nIn particular, the global slope  of the radial O\/H gradient  has been confirmed much \nshallower than retained in the past (cf. Rosolowsky \\& Simon 2008) \nand its evolution much slower (cf.  M09).\n\nThe general assumption of multiphase chemical evolution models, \nsuch as the  M33 one (M07), \nis to describe the formation and disruption of diffuse gas, clouds, and stars,   \nby means of physical processes, e.g., Ferrini et al. \\cite{ferrini92}.  \nIn particular, the SF is represented \nwith two processes: the interaction of molecular clouds  with the \nradiation field of massive stars and the  collisions between two molecular clouds.\nThe dominant process is due to cloud collisions.  \nIn this kind of model the relationship between the star formation rate (SFR) and the surface density of gas (molecular or total) is, thus, a by-product of the model, and cannot be assumed 'a priori'.\n\nHowever, the law relating the surface densities of SF and cold gas is one of the most fundamental laws \ndescribing the galaxy behaviour. Virtually the entire range of global star formation rates in galaxies \ncan be reproduced by a Schmidt power law relation. \nIn the particular case of M33, the relation between the SFR, measured from the FUV emission, \nand molecular gas has a well-defined slope (Verley et al., in prep.) corresponding to\n\\begin{equation}\n\\Sigma_{SFR} = A  \\Sigma^{1.2}_{mol gas}.\n\\end{equation}\nA pure cloud-cloud collision process for the star formation is not able to reproduce it.  \nFor this reason, we have taken into account other parameterizations of  the SF process. \nIn particular, a process which is dominated by cloud collisions close to the center, while in the \nintermediate and peripheral regions is proportional to the fraction of clouds with a 1.5 exponent, \nis able to reproduce the observations. It reproduces  the \nhigher cloud surface density in the inner regions, rendering the conversion into star less effective, \nwhile in the outer regions it takes into account a more efficacious SF.   \nThe resulting Schmidt law would have an average exponent all over the radial range (or the molecular gas surface density range) of 1.1, thus consistent with the observations. \n\nIn addition, the introduction of the Schmidt law allowed us  to better reproduce \nthe O\/H gradient and its evolution \n(see Figure 2): a gradient almost flat both at present-time both at the epoch of the formation of the PNe progenitors, \nwith a little evolution of its absolute value and slope. \nNote from Figure 2 the steeper gradient predicted by the previous model, where the SF process \nwas dominated by cloud-cloud collision all over the radial range. \n  \n  \\begin{figure}\n   \\centering\n   \\resizebox{\\hsize}{!}{\\includegraphics[clip=true]{M33_oxy_sl.ps}}  \n      \\caption{\\footnotesize{The time-evolution of the O\/H radial gradient. Filled symbols  (blue circles and magenta squares) are \n      the oxygen abundances of H~{\\sc ii}\\ regions and PNe, respectively, averaged in bins 1 kpc wide.\n      Model with the Schmidt law: present time (red continuous line), 1 Gyr ago (long-short dashed line), \n      5 Gyr ago (dot-dashed line).  Model with the cloud-cloud collision process (M07): present time (green  dashed line)}. \n       }\n              \\label{sl}%\n    \\end{figure}\n\n\n\\section{Summary and conclusions}\n\\label{sect_conclu}\nThe chemical evolution of M33 is studied by means of new spectroscopic observations \nof PNe and H~{\\sc ii}\\ regions.\nTheir 2D metallicity maps have been found both off-centered, with a peak in the southern\narm, at 1-2 kpc from the center. This might be related to the absence  of a dominant gravitational \nsource in the center. \nThe slow evolution of the metallicity gradient from the present time to the \nbirth of the PNe progenitors is explained with an {\\em accretion} model \nwhere the SF process is driven by the Schmidt law.  \n\n\n\n \n\\begin{acknowledgements}\nI warmly thank Edvige Corbelli, Daniele Galli, \nLetizia Stanghellini, and Eva Villaver for their collaboration in this work. \n\\end{acknowledgements}\n\n\\bibliographystyle{aa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nField perturbations of a curved background spacetime obey a wave equation which dictates that they propagate ``mainly\" along null geodesics. However, in general, there is also a part of the field, called the {\\it tail}, which propagates slower than light (even in the case of a massless field). In other words, the  strong Huygens principle is generally violated in curved spacetimes~\\cite{McLenaghan:1974,CzaporMcLenaghan}.\nThis tail  is important for different reasons. For example, obtaining the tail term is useful for calculating the self-force~\\cite{Poisson:2011nh} on a point particle via the method of matched expansions~\\cite{Anderson:Wiseman:2005}, where the retarded Green function of the wave equation needs to be regularized~\\cite{CDOW13,PhysRevD.89.084021}.\nAlso, the tail may give rise to an interesting relevant contribution  to the communication between quantum particle detectors~\\cite{blasco2015violation,Jonsson:2020npo}.\n\nMathematically, the tail term may be defined in local neighbourhoods of spacetime points via the Hadamard form~\\cite{Hadamard} of the retarded Green function. Specifically, and henceforth focusing on the case of a scalar field for simplicity, the tail is the bitensor $V(x,x')$  in the term in the Hadamard form which has support only {\\it inside} the (past) light cone of the field point. This tail bitensor satisfies the {\\it homogeneous} wave equation, constrained  by its value on the light cone, which satisfies a transport equation (along null geodesics) with a given initial condition~\\cite{Poisson:2011nh}.\n\nThe Hadamard tail $V(x,x')$ has been obtained in closed form in only a very few settings of high symmetry - specifically, and to the best of our knowledge, only when the background spacetime is flat~\\cite{M&F} or conformally-flat\\footnote{In these cases, the field considered is non-conformal, so that there exists a nontrivial tail.}, such as simple (spatially-flat) cosmological model spacetimes~\\cite{Burko:2002ge,haas2005mass}, including (a patch of) de Sitter~\\cite{Friedlander}.\nA simplifying feature of flat and conformally-flat spacetimes is that null geodesics emanating from a point do not cross.\nFurthermore, the maximal symmetry of both flat  and de Sitter spacetimes, in particular, means that it is possible to rewrite the  partial differential wave equations in these spacetimes as {\\it ordinary} differential equations (where the independent variable is the geodesic distance), which are much easier to solve.\nHowever, in other spacetimes which are less symmetric  and where null geodesics emanating from a point do cross, such as black hole spacetimes, no closed form expression for $V(x,x')$ is known and, instead, one needs to resort to numerical or approximation analytical techniques. \nFocusing on black hole spacetimes, some of these techniques have been fairly successful in calculating $V(x,x')$ in Schwarzschild~\\cite{CDOWb,Ottewill:2009uj,Decanini:Folacci:2005a,CDOW13,PhysRevD.89.084021} but face much more significant difficulties in the case of Kerr. It is thus important to develop alternative methods for calculating the tail term.\n\nIn this paper we pursue the endeavour of calculating $V(x,x')$ by directly (numerically) integrating the (homogeneous) wave equation with given Characteristic Initial Data (CID) on the light cone.\nSpecifically, we apply this method to the case of a scalar field propagating on Pleba{\\'n}ski-Hacyan spacetime  (PH), $\\mathbb{M}_2\\times{\\mathbb{S}^2}$~\\cite{Griffiths&Podolsky}.\nFrom a physical point of view, this spacetime serves as a black hole toy-model and it  captures the important feature that null geodesics emanating from a point do cross (in fact, similarly to Schwarzschild, there exist caustics where an ${\\mathbb{S}^2}$-envelope of null geodesics focus).\nIn its turn, from a technical point of view, the fact that PH is not a maximally-symmetric spacetime but is the direct product of two (two-dimensional) maximally-symmetric spacetimes (namely, $\\mathbb{M}_2$ and ${\\mathbb{S}^2}$) means that its wave equation, while it is not an {\\it ordinary} differential equation as in flat or de Sitter spacetimes, it is reduced to a {\\it two}-dimensional PDE; furthermore, the value of $V(x,x')$ on the light cone is known in closed form~\\cite{Casals:2012px}. Our calculation of $V(x,x')$ in the specific case of a massles field with a coupling constant value of $\\xi=1\/8$ and  $x$ and $x'$ on a static path, agrees with~\\cite{Casals:2012px}, where $V$ is calculated via a completely different method which involves infinite sums and integrals.\nThis provides a check of our method and calculation and serves as a proof-of-concept for this method.\nWe also obtain new results for $V$ in PH: for {\\it any} pair of spacetime points where it is defined for $\\xi=1\/8$ as well as for $\\xi=0$, $1\/6$, $1\/4$ and $1\/2$.\n\nFor solving the homogeneous \n(two-dimensional)\nwave equation in PH by evolving CID we use a finite-difference scheme.  Refs.~\\cite{mark2017recipe,Lousto:1997wf} proposed and implemented a CID scheme for obtaining the multipolar modes of the field (or retarded Green function) in Schwarzschild spacetime. We adapted this scheme to calculate the full $V(x,x')$ in PH spacetime. We note that some peculiarities of the PDE satisfied by $V$ in PH will also be present in the PDE satisfied by $V$ in Schwarzschild, so that our adaptation of the scheme will probably be useful for any future investigation in the latter spacetime. Furthermore, we developed the scheme to higher order than in~\\cite{mark2017recipe,Lousto:1997wf} by providing additional data on the light cone. \n\nThe rest of this paper is organized as follows. In Sec.~\\ref{sec:Hadamard} we introduce the Hadamard form and give the explicit forms of the scalar wave equation and some Hadamard quantities  in PH.\nIn Sec.~\\ref{sec:CID} we present the Characteristifc Initial Value problem in PH, our finite difference scheme for solving it and the results of our calculations.\nWe conclude in Sec.~\\ref{sec:Discussion} with a brief discussion.\n\nWe choose units such that $G=c=1$.\n\n\n\\section{Wave equation, Hadamard Form and Tail}\\label{sec:Hadamard}\n\n\n\\subsection{A general spacetime}\n\nA scalar field perturbation of a background spacetime satisfies a wave equation.\nSpecifically, its retarded Green function satisfies:\n\\begin{equation}\\label{eq:GF eq}\n\\left(\\Box-m^2-\\xi R\\right)G_{\\textrm{ret}}(x,{x^\\prime})=-4\\pi\\delta_4(x,{x^\\prime}),\n\\end{equation}\nwhere $m$ is the mass of the field, $R$ is the Ricci scalar, $\\xi$ is a coupling constant and \n$x$ and ${x^\\prime}$ are, respectively, the field  and  base spacetime points.\n\nThe Hadamard form provides an analytic expression for the singularities of the retarded Green function when $x$ is in a local (normal\\footnote{A normal\nneighbourhood of ${x^\\prime}$ is a region containing ${x^\\prime}$ such that\nevery $x$ in that region is connected to ${x^\\prime}$ by a unique\ngeodesic which lies within the region.}) neighbourhood of  ${x^\\prime}$~\\cite{DeWitt:1960,Friedlander,Poisson:2011nh}:\n\\begin{align} & G_{\\textrm{ret}}(x,{x^\\prime})=\n\\label{grhad} \\\\ & [U(x,{x^\\prime})\\delta(\\sigma(x,{x^\\prime}))+V(x,{x^\\prime})\\theta(-\\sigma(x,{x^\\prime}))]\\theta_+(x,{x^\\prime}),\n\\nonumber\n\\end{align}\nwhere $\\delta$ and $\\theta$ are, respectively, the  Dirac delta and Heaviside distributions,\n\\[ \\theta_+(x,{x^\\prime})\\equiv \\left\\{\\begin{array}{ll} 1 & \\hbox{if $x$ lies to the future of ${x^\\prime}$};\\\\ 0 & \\hbox{otherwise},\\end{array}\\right. \\]\nand $U$ and $V$ are   biscalars which are smooth in that local neighbourhood. \nHere, $\\sigma$ is Synge's world-function, i.e., one-half of the\nsquare of the geodesic distance along the unique geodesic\nconnecting ${x^\\prime}$  and $x$.\nThus, clearly, the term with $U$ in Eq.~\\eqref{grhad} has support only on the light cone whereas the term with $V$ has support {\\it inside} the light cone: this is the tail term which is the focus of this paper.\n\nThe Hadamard tail $V$ satisfies the {\\it homogeneous} wave equation:\n\\begin{equation}\\label{eq:V wave eq}\n\\left(\\Box-m^2-\\xi R\\right)V(x,{x^\\prime})=0,\n\\end{equation}\nconstrained by its value on the light cone:\n\\begin{equation}\\label{eq:transp eq V}\n\\hat{V}_{,\\alpha}\\sigma^{\\alpha}+\\frac{1}{2}\\left(\\sigma^{\\alpha}{}_{\\alpha}-2\\right)\\hat{V}=\\frac{1}{2}\\left(\\Box-m^2-\\xi R\\right)\\left.U\\right|_{\\sigma=0},\n\\end{equation}\nwhere $\\hat{V} \\equiv V|_{\\sigma=0}$ and  $\\sigma^{\\alpha}{}_{\\alpha}\\equiv\\nabla_\\alpha\\nabla^\\alpha\\sigma$.\nEq.\\eqref{eq:transp eq V} is in fact a transport equation along a light cone-generating null geodesic. It is to be solved together with the initial condition corresponding to the value of $V$ at coincidence (i.e., at ${x^\\prime}=x$):\n\\begin{equation}\\label{eq:IC V}\nV(x,x)=\\frac{1}{12}\\left(1-6\\xi\\right)R(x)-\\frac{1}{2}m^2,\n\\end{equation}\nwhich follows partly from the fact that  $V$ is smooth at coincidence.\n\nOne method for trying to calculate $V(x,{x^\\prime})$ is to  express it as an asymptotic series:\n\\begin{equation} \nV(x,x')=\\sum_{n=0}^\\infty \\nu_n(x,x')\\sigma^n\\label{eqn:VsigmaExpansion},\n\\end{equation}\nwhere the coefficients $\\nu_n(x,x')$ satisfy\ncertain recurrence relations~\\cite{DeWitt:1960,Decanini:Folacci:2005a}.\nRef.\\cite{Ottewill:2009uj} provided a complete procedure for calculating $\\nu_n$ by solving transport equations. Unfortunately, however, as the coefficient order $n$ increases, these transport equations become increasingly hard to solve (even numerically and for low $n$).\nFurthermore, although the series in Eq.~\\eqref{eq:IC V} converges uniformly in subregions of normal neighbourhoods~\\cite{DeWitt:1960,Friedlander}, it is not actually guaranteed to converge in the whole maximal normal neighbourhood of a point.\nA more practical method for calculating $V(x,x')$ in spherically-symmetric spacetimes is to expand this bitensor in small {\\it coordinate} distance between $x$ and $x'$~\\cite{CDOWb}.\nAlthough this method has proven to be very useful in Schwarzschild spacetime~\\cite{CDOW13,PhysRevD.89.084021}, it is naturally adapted to spherical symmetry and so it still needs to be developed in Kerr spacetime.\n\nIn this paper, we shall use the first two orders in the series in Eq.~\\eqref{eq:IC V} in order to calculate the characteristic initial data for a numerical scheme for solving the full wave Eq.~\\eqref{eq:V wave eq} for $V(x,x')$ for {\\it any} pair of points in PH spacetime.\n\n\n\\subsection{PH spacetime}\\label{sec:PH}\nPH spacetime is the direct product of two-dimensional Minkowski spacetime $\\mathbb{M}_2$ and the two-sphere ${\\mathbb{S}^2}$~\\cite{Griffiths&Podolsky}. This becomes manifest when writing its line element as\\footnote{We make the units choice that the radius of the two-spheres is equal to one.}\n\\begin{equation} ds^2=-dt^2+dy^2+d\\Omega^2,\\label{PHlel}\\end{equation} where $$d\\Omega^2=d\\theta^2+\\sin^2\\theta d\\varphi^2,$$\nwith $(t,y)\\in \\mathbb{R}^2$ global inertial coordinates in  $\\mathbb{M}_2$, and $\\theta\\in [0,\\pi]$ and $\\varphi\\in (-\\pi,\\pi]$\nthe standard angular coordinates in ${\\mathbb{S}^2}$. \nThe  Ricci scalar is $R=2$.\n\nPH being the direct product $\\mathbb{M}_2\\times {\\mathbb{S}^2}$, its  world function $\\sigma$ is readily given~\\cite{Casals:2012px} as the sum of the world functions in $\\mathbb{M}_2$ and  ${\\mathbb{S}^2}$, respectively $\\sigma_{\\mathbb{M}_2}$  and $\\sigma_{{\\mathbb{S}^2}}$: $\\sigma(x,{x^\\prime}) = \\sigma_{\\mathbb{M}_2}+\\sigma_{{\\mathbb{S}^2}}$.\nIn their turn, these world functions are given, in normal neighbourhoods, by\n\\begin{equation} \\sigma_{\\mathbb{M}_2}=-\\frac12\\eta^2\\equiv -\\frac12(t-{t^\\prime})^2+\\frac12(y-{y^\\prime})^2 \\label{sigbardef}\\end{equation}\nand\n\\begin{align} \n\\sigma_{{\\mathbb{S}^2}}=\\,&\\frac{\\gamma^2}{2},\\\\\n\\cos\\gamma\\equiv\\,& \\cos\\theta\\cos{\\theta^\\prime}+\\sin\\theta\\sin{\\theta^\\prime}\\cos(\\varphi-{\\varphi^\\prime}).\\nonumber \\label{gamdef}\\end{align}\nWe thus have\n\\begin{equation} \n\\sigma=\n-\\frac12\\eta^2 + \\frac12\\gamma^2=-\\frac12(t-{t^\\prime})^2+\\frac12(y-{y^\\prime})^2+ \\frac12\\gamma^2.\n\\end{equation}\nClearly, $\\eta\\in \\mathbb{R}$ is the geodesic distance in the whole of $\\mathbb{M}_2$. In its turn,  $\\gamma\\in [0,\\pi]$ is the geodesic (or angle) {\\it separation} in  ${\\mathbb{S}^2}$, while it also is the geodesic distance in normal neighbourhoods of ${\\mathbb{S}^2}$ (see~\\cite{Casals:2019heg,casals2016global} for this subtle but important distinction between geodesic separation and geodesic distance in the context of Schwarzschild spacetime).\nNull geodesics (for which $\\sigma=0$ in normal neighbourhoods) focus at the first caustic points: $\\eta=\\gamma=\\pi$.\nAfter crossing the first caustic, the envelope of null geodesics emanating from a base point forms the (future) boundary of the maximal normal neighbourhood of the base point; this boundary is given by $\\eta=2\\pi-\\gamma\\in [\\pi,2\\pi]$  (see the left panel of Fig.1 in \\cite{Casals:2012px}).\nSince $V(x,x')$ is only defined in normal neighbourhoods, this will also be part of the boundary of the grid in our numerical scheme.\nThe other part  is given by the null hypersurface corresponding to the envelope of future-directed {\\it direct} null geodesics, i.e., by $\\eta=\\gamma\\in [0,\\pi)$ (so that it is $\\sigma=0$ with $\\eta\\geq 0$).\nThat is, the future boundary $\\eta=2\\pi-\\gamma\\in [\\pi,2\\pi]$  of the maximal normal neighourhood of the base point together with the boundary $\\eta=\\gamma\\in [0,\\pi)$ of the causal future of the base point form the boundary of our numerical grid.\n\nIt is straight-forward to obtain the d'Alembertian in PH in the above coordinates:\n\\begin{align}\n&\n\\Box=\\Box_{\\mathbb{M}_2}+\\Box_{{\\mathbb{S}^2}},\\\\\n&\n\\Box_{\\mathbb{M}_2}=-\\frac{\\partial^2}{\\partial t^2}+\\frac{\\partial^2}{\\partial y^2},\n\\nonumber\\\\\n&\n\\Box_{{\\mathbb{S}^2}}=\\frac{1}{\\sin\\theta}\\frac{\\partial}{\\partial \\theta}\\sin\\theta\\frac{\\partial}{\\partial \\theta}+\\frac{1}{\\sin^2\\theta}\\frac{\\partial^2}{\\partial\\varphi^2}.\\nonumber\n\\end{align}\n\nNow, given that both  $\\mathbb{M}_2$ and  ${\\mathbb{S}^2}$ are maximally-symmetric manifolds, it is easy to see that the above operators $\\Box_{\\mathbb{M}_2}$ and $\\Box_{\\mathbb{S}^2}$ become {\\it ordinary} differential operators when rewritten in terms of the corresponding geodesic distances.\nExplicitly,\n\\begin{equation}\n\\Box_{\\mathbb{M}_2}=-\\frac{\\partial^2}{\\partial \\eta^2}-\\frac{1}{\\eta}\\frac{\\partial^2}{\\partial \\eta}\n\\end{equation}\nand\n\\begin{align}\n\\Box_{\\mathbb{S}^2}=\\frac{\\partial^2}{\\partial \\gamma^2}+\\cot\\gamma \\frac{\\partial}{\\partial \\gamma}=\n\\frac{1}{\\sin\\gamma}\\frac{\\partial}{\\partial \\gamma}\\left(\\sin\\gamma\\frac{\\partial}{\\partial \\gamma}\\right).\n\\end{align}\n\nThe wave equation~\\eqref{eq:V wave eq} thus becomes\n\\begin{equation}\\label{eq:wave eq PH}\n\\left[\n\\frac{\\partial^2}{\\partial \\gamma^2}+\\cot\\gamma \\frac{\\partial}{\\partial \\gamma}\n-\\frac{\\partial^2}{\\partial \\eta^2}-\\frac{1}{\\eta}\\frac{\\partial}{\\partial \\eta}\n-\\zeta\\right]\\! \\! V(x,{x^\\prime})\\!=0\n\\end{equation}\nwhere $\\zeta\\equiv m^2+\\xi R=m^2+2\\xi$.\nThus, we have reduced the  wave equation, which is generally a four-dimensional PDE, to a two-dimensional PDE in PH.\n\nIn~\\cite{Casals:2012px}, it was found that, in PH, it is\n$V=V(\\eta,\\gamma)$ and $\\nu_k=\\nu_k(\\gamma)$ and \nclosed form expressions for some Hadamard quantities were obtained. Specifically, it was found that\n\\begin{align} \\label{udef}\n&\nU(x,x')= U(\\gamma)=\n\\left|\\frac{\\gamma}{\\sin\\gamma}\\right|^{1\/2},\n\\end{align}\nand, by solving Eqs.~\\eqref{eq:transp eq V}\nand \\eqref{eq:IC V}, that\n\\begin{align}\n\\hat{V} =\\nu_0(\\gamma)=\\frac18U(\\gamma)\\left(1-4\\zeta\n+\\frac{1}{\\gamma^2}-\\frac{\\cot\\gamma}{\\gamma}\\right).\\label{v0def}\n\\end{align}\nThe higher orders $\\nu_k$, $k>0$, can in principle be obtained from $\\nu_{k-1}$ via a recurrence relation.\nFor the specific case $\\zeta=1\/4$ it was obtained that\n\\begin{align} \\label{eq:hat V1 M2xS2}\n& \\nu_1=\n\\\\ &\nU(\\gamma)\\frac{2\\gamma^2-3 \\csc ^2(\\gamma ) \\left[6 \\gamma ^2+2 \\gamma \\sin (2 \\gamma )+5 \\cos (2 \\gamma )-5\\right]}{256 \\gamma ^4},\n\\nonumber\n\\end{align}\nfor which regularity of $\\nu_1$ at $\\gamma=0$ was required.\n\nWe note that $\\hat{V}$ is regular for all $\\gamma\\in [0,\\pi)$ but it diverges (like $(\\pi-\\gamma)^{-3\/2}$) at the antipodal points $\\gamma=\\pi$. These antipodal points, however, lie outside maximal normal neighbourhoods: \nAs is manifest, ${\\mathbb{S}^2}$-envelopes of null geodesics focus along $\\gamma=\\pi$ (a line of caustics), as in Schwarzschild spacetime. It has been observed~\\cite{Dolan:2011fh,Zenginoglu:2012xe,casals2016global,Ori1,Harte:2012uw,Casals:2012px,CDOWa} that, when this happens, the retarded Green function diverges when ${x^\\prime}$ and $x$  are connected by a null geodesic (even beyond normal neighbourhoods) displaying the following {\\it global} fourfold (leading) singularity structure\\footnote{Here, $\\sigma$ refers to a well-defined extension of the world\nfunction outside normal neighbourhoods~\\cite{casals2016global,Casals:2019heg}. We also note that this structure does not hold at caustics~\\cite{casals2016global} and that the subleading order (in Schwarzschild and outside caustics) is given in~\\cite{casals2016global}.}: $\\delta(\\sigma) \\to \\text{PV}\\left(1\/\\sigma\\right)\\to -\\delta(\\sigma) \\to -\\text{PV}\\left(1\/\\sigma\\right)\\to \\delta(\\sigma)\\dots$, where $\\text{PV}$ denotes the principal value distribution.\nSince $V$ is equal to $G_{\\textrm{ret}}$ in a region of causal separation which lies inside a normal neighbourhood,\n$V$ must diverge like $G_{\\textrm{ret}}$, i.e. as $\\text{PV}\\left(1\/\\sigma\\right)$, when approaching the end of the normal neighbourhood in this direction.\nAs we shall see in Sec.~\\ref{sec:results V}, the divergence of $\\hat{V}$ at $\\gamma=\\pi$ propagates in this manner throughout the end of the maximal normal neighbourhood of the base point.\n\nThe upshot is that, in PH, we have reduced \nthe original wave equation  to a two-dimensional PDE (see Eq.~\\eqref{eq:wave eq PH}), and that the Hadamard tail $\\hat{V}$ on the light cone is known in closed form  (see Eq.~\\eqref{v0def}).\nIn the next section we will use these advantageous features  to help us numerically solve the wave equation \\eqref{eq:wave eq PH} and thus to calculate $V$ inside the light cone.\n\n\n\\section{Solving the Characteristic Initial Value Problem for the Hadamard Tail}\\label{sec:CID}\n\nIn this work, we will directly solve the wave equation \\eqref{eq:wave eq PH} as a characteristic initial value problem.\nMore concretely, we shall develop a numerical scheme which will evolve initial data on the light cone, i.e., on\n$$\n    \\sigma=-\\frac{1}{2}\\eta^2+\\frac{1}{2}\\gamma^2=0.\n$$\nThis CID is given by Eqs.\\eqref{eqn:VsigmaExpansion}--\\eqref{eq:hat V1 M2xS2}.\nThis section provides the details of the  scheme and the results. We split this section into three subsections: we first rewrite the wave equation in variables suitable to the CID problem; we then describe the numerical scheme; finally, we show our results for $V$.\n\n\\subsection{Wave equation as a Characteristic Initial Value problem} \n\nLet us  introduce the variables \n\\begin{equation}\\label{eq:uv}\nu\\equiv \\eta-\\gamma, \\quad v\\equiv \\eta+\\gamma,\n\\end{equation}\nwhich are naturally adapted to the Characteristic Initial Value problem, since $\\sigma=-uv\/2$.\nIn these variables, the d'Alembertian in PH (see Eq.~\\eqref{eq:wave eq PH}) becomes\n\\begin{equation}\n    \\Box=-4\\frac{\\partial^2}{\\partial u\\partial v}-Q\\frac{\\partial}{\\partial v}-S\\frac{\\partial}{\\partial u},\n\\end{equation}\nwhere\n\\begin{align*}\n    Q\\equiv \\,&\\frac{2}{v+u}-\\cot\\frac{v-u}{2},\\\\\n    S\\equiv\\,&\\frac{2}{v+u}+\\cot\\frac{v-u}{2}.\n\\end{align*}\n\nTherefore, Eq.~\\eqref{eq:wave eq PH} turns into\n\\begin{equation}\n    \\left(4\\frac{\\partial^2}{\\partial u\\partial v}+Q\\frac{\\partial}{\\partial v}+S\\frac{\\partial}{\\partial u}+\n    \\zeta\n    \\right)V(x,{x^\\prime})=0.\\label{eqn:VUVEqn}\n\\end{equation}\nWe note the appearance of {\\it first}-order derivatives with respect to $u$ and $v$, arising from the first-order derivatives with respect to $\\eta$ and $\\gamma$ in Eq.~\\eqref{eq:wave eq PH}.\nWe also note that, even though $u$ and $v$ range over the reals to cover the whole spacetime, the domain over which we solve Eq.~\\eqref{eqn:VUVEqn} is limited by the range of $\\gamma\\in [0,\\pi]$ and the region where $V$ is defined.\nIn the next subsection we detail how these constraints are reflected on the domains for $u$ and $v$.\n\nLet us now turn to the CID. On the $u-v$ plane, the light cone is located along the $u=0$ and $v=0$ lines. Thus, $\\hat{V}$ in terms of $u$ and $v$, is given by\n\\begin{equation}\\label{eqn:VCIDBC}\n    \\begin{aligned}\n    \\left.V\\right|_{u=0}=\\,&\\nu_0\\left(\\frac{v}{2}\\right),\\\\\n    \\left.V\\right|_{v=0}=\\,&\\nu_0\\left(-\\frac{u}{2}\\right),\n    \\end{aligned}\n\\end{equation}\nwhere $\\nu_0=\\nu_0(\\gamma)$ is given in Eq.~\\eqref{v0def}.\n\n\nThe wave Eq.~\\eqref{eqn:VUVEqn}, together with the CID \\eqref{eqn:VCIDBC} constitutes our Characteristic Initial Value problem.\nWe next present  \n the finite difference method that\n we shall use to   solve it.\n\n\\subsection{Numerical Scheme}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics{Figures\/CIDGridM2xS2.pdf}\n    \\caption{\n    Grid distribution for a finite difference scheme for solving a two-dimensional PDE where $u$ and $v$ denote the independent variables and $2h$ is the stepsize.\n    }\n    \\label{fig:CIDGridM2xS2}\n\\end{figure}\n\nThere have already been some implementations of CID schemes for solving the two-dimensional PDE (not containing first order derivatives, unlike \\eqref{eqn:VUVEqn}) which is obeyed by the (smooth factor\\footnote{The other factor contains  non-smooth Heaviside distributions.} in the) $\\ell$-multipolar modes of the retarded Green function in Schwarzschild spacetime (see, e.g., Eq.~(C2) in~\\cite{Jonsson:2020npo}).\nLet us briefly discuss these schemes.\nIn Ref.~\\cite{mark2017recipe}, the authors implemented a scheme (previously proposed by Lousto and Price \\cite{Lousto:1997wf}) of order $h^4$, where $2h$ is the stepsize of the grid (see Fig.~\\ref{fig:CIDGridM2xS2}). In order to calculate the value of the field at a point, their scheme required field data on the immediately ``previous\" grid points -- e.g., on the points S, E and W in order to obtain the value of the field at the point $N$ in Fig.~\\ref{fig:CIDGridM2xS2}.\nIn~\\cite{Jonsson:2020npo}, together with collaborators, we  extended the scheme in~\\cite{mark2017recipe} to order $h^6$ at the expense of requiring the value of the field and of its first-order derivatives at the same grid points as in the lower-order scheme of~\\cite{mark2017recipe}.\nMore recently, Ref.~\\cite{PhysRevD.103.124022} came up with another scheme which they implemented to order $h^6$ (although in principle can be generalized to any higher order) at the expense of requiring the value of the field at more points than in Refs.~\\cite{mark2017recipe,Jonsson:2020npo}.\n\nIn our current work in PH, the PDE \\eqref{eqn:VUVEqn} is also two-dimensional but, unlike in the case of Schwarzschild just reviewed,  and as we have emphasized, it contains first order derivatives and it is satisfied by the full field.\nWe note that first order derivatives also appear in Schwarzschild spacetime  in the PDE satisfied by the full field as well as in the Teukolsky PDE satisfied by multipolar field modes for fields of higher spin~\\cite{Teukolsky:1973ha}.\nFor solving  Eq.~\\eqref{eqn:VUVEqn}, we choose to essentially follow the  fourth order scheme of Ref.~\\cite{Jonsson:2020npo} and adapt it to our specific PDE.\n\nAnother difference between our setup and that in Refs.~\\cite{mark2017recipe,Jonsson:2020npo,PhysRevD.103.124022} is that, since Eq.~\\eqref{eqn:VUVEqn} is obeyed by the Hadamard tail $V$, we only solve it inside the maximal normal neighbourhod. Therefore, the independent variables should only range over the finite intervals $v\\in [0,2\\pi)$ and $u\\in [0,v]$  (dictated by the range $\\gamma\\in [0,\\pi)$ inside the maximal normal neighbourhood). This is unlike the problem in Refs.~\\cite{mark2017recipe,Jonsson:2020npo,PhysRevD.103.124022}, which is for the retarded Green function, and so with independent variables, which are null coordinates, in principle ranging over the whole real line.\nIn order to be able to map better our problem to that in Refs.~\\cite{mark2017recipe,Jonsson:2020npo,PhysRevD.103.124022} and the CID problem in Schwarzschild in general,  we shall henceforth consider, without loss of generality, that the spacetime points $x$ and ${x^\\prime}$ have $\\theta=\\theta'=\\pi\/2$ and $\\varphi'=0$ and, further, that $\\gamma\\equiv \\varphi\\in(-\\pi,+\\pi)$ denotes the azimuthal angle of $x$  (instead of the angular separation, which is in $[0,\\pi]$, as until now).\nThe variables $u$ and $v$ continue to be defined as in \\eqref{eq:uv} but now with  $\\gamma\\in (-\\pi,+\\pi)$ being an azimuthal angle. This means that the region of interest for calculating $V(x,x')$, which is the part of the region of causal separation which lies inside the maximal normal neighbourhood, is bounded by\n$\\eta=2\\pi- \\gamma$ together with $\\eta=\\gamma$ if $\\gamma\\in [0,\\pi]$\\footnote{\\label{ftn:gamma}Here we allow $\\gamma$ to take on the value $\\pi$ (respectively $-\\pi$) so as to refer to the {\\it boundary} of the region of interest.}, as explained in Sec.~\\ref{sec:PH}, and by\n$\\eta=2\\pi+ \\gamma$ together with $\\eta=-\\gamma$ if $\\gamma\\in [-\\pi,0]$\\textsuperscript{\\ref{ftn:gamma}}\nafter extending the range of $\\gamma$ \ninto the negative line.\nIn the CID variables, this means that the region of interest covered by the numerical grid, is given by $u,v\\in [0,2\\pi)$.\n\n\nWe next describe our scheme and its implementation for obtaining $V$ for various values of $\\zeta$.\nWe start by describing a lower order, $\\order{h^4}$ of accuracy, version of the scheme. We do so in order to more clearly highlight the key distinction in solving the PDE \\eqref{eqn:VUVEqn}, containing first-order derivatives, as opposed to the PDE for  multipolar  modes of Refs.~\\cite{mark2017recipe,Jonsson:2020npo,PhysRevD.103.124022}, not containing any first-order derivatives.\nWe then describe the method at the higher order $\\order{h^6}$ of accuracy. \n\n\\subsubsection{CID scheme setup}\n\nIn order to solve the wave Eq.~\\eqref{eqn:VUVEqn} with the CID in Eq.~\\eqref{eqn:VCIDBC}, we first establish a uniform grid of points on the $u-v$ plane, which we show in Fig.~\\ref{fig:CIDGridM2xS2}. The spacing between grid points is $2h$.  The next step is to integrate Eq.~\\eqref{eqn:VUVEqn} on each of the squares in the grid. For the $S$-$E$-$N$-$W$ square in Fig.~\\ref{fig:CIDGridM2xS2} we have\n\n\\begin{widetext}\n    \\begin{align}\n        4\\int\\limits_{SENW} \\frac{\\partial^2V}{\\partial v\\partial u} \\,\\textrm{d}v\\,\\textrm{d}u+\\int\\limits_{SENW} Q\\frac{\\partial V}{\\partial v} \\,\\textrm{d}v\\,\\textrm{d}u\n        +\\int\\limits_{SENW} S\\frac{\\partial V}{\\partial u} \\,\\textrm{d}v\\,\\textrm{d}u+\n       \n        \\zeta\n        \\int\\limits_{SENW}V \\,\\textrm{d}v\\,\\textrm{d}u=0.\\label{eqn:squareIntegralV}\n    \\end{align}\n\\end{widetext}\nThe first integral in the left hand side of Eq.~\\eqref{eqn:squareIntegralV} can be readily evaluated exactly as\n\\begin{equation}\n    \\int\\limits_{SENW} \\frac{\\partial^2V}{\\partial v\\partial u} \\,\\textrm{d}v\\,\\textrm{d}u=V_N-V_E-V_W+V_S,\n\\end{equation}\nwhere the subindices $N$, $E$, $W$ and $S$ in $V$ refer to the point on the grid where $V$ is to be evaluated. For the remaining three integrals, we Taylor expand the integrands about the central point $O=(u_O,v_O)$ in the square. The Taylor series expansion of the product of two smooth functions $F(v,u)$ and $G(v,u)$ about $O$ is\n\\begin{widetext}\n    \\begin{equation}\n        F(v,u)G(v,u)=\\sum_{\\substack{0\\leq m,n\\leq K\\\\m+n\\leq K}}\\frac{1}{m!\\, n!}\\left(\\frac{\\partial^{m+n}}{\\partial v^m\\partial u^n}(F\\,G)\\right)_O(v-v_0)^m(u-u_0)^n+\\mathcal{O}(h^{K+1}),\n        \\label{eqn:FGTaylorSeries}\n    \\end{equation}\n\\end{widetext}\nwhere  $K$ determines the order in the expansion. We expand in this manner the last three integrands in Eq.~\\eqref{eqn:squareIntegralV} (replacing $F$ and $G$ in Eq.~\\eqref{eqn:FGTaylorSeries} by the appropriate functions) to the desired order.\n\n\n\\subsubsection{CID scheme to $\\mathcal{O}(h^4)$}\n\nUsing \\eqref{eqn:FGTaylorSeries} to order $h^2$, the last three\nintegrals in \\eqref{eqn:squareIntegralV} are given by\n\\begin{align}\\label{eqn:squareInts}\n    \\int\\limits_{SENW} Q\\frac{\\partial V}{\\partial v} \\,\\textrm{d}v\\,\\textrm{d}u=\\,&4h^2Q_O\\left(\\frac{\\partial V}{\\partial v}\\right)_O+\\mathcal{O}(h^4)\\\\\n    \\int\\limits_{SENW} S\\frac{\\partial V}{\\partial u} \\,\\textrm{d}v\\,\\textrm{d}u=\\,&4h^2S_O\\left(\\frac{\\partial V}{\\partial u}\\right)_O+\\mathcal{O}(h^4),\\\\\n    \\int\\limits_{SENW}V \\,\\textrm{d}v\\,\\textrm{d}u=\\,&4h^2V_O+\\mathcal{O}(h^4),\n\\end{align}\nwhere,  again, the subindex $O$ indicates that the corresponding quantity is evaluated at the point $O$. In this result we only considered the first two leading orders in the Taylor series to obtain the integrals to order $h^3$. However, it can be shown that the contribution to the integral from the next-to-leading order term in the Taylor series vanishes.\n\nIn order to calculate $V$ and its derivatives at the point $O$, we evaluate its Taylor expansion at the points $E,W$ and $S$ to order $h$. This allows us to construct a system of three equations where the unknown variables are $V_O$, $\\left(\\pd{V}{u}\\right)_O$ and $\\left(\\pd{V}{v}\\right)_O$. We find:\n\\begin{align}\n    V_O=\\,&\\frac{V_E+V_W}{2}+\\mathcal{O}(h^2),\\\\\n    \\left(\\frac{\\partial V}{\\partial u}\\right)_O=\\,&\\frac{V_W-V_S}{2h}+\\mathcal{O}(h),\\\\\n    \\left(\\frac{\\partial V}{\\partial v}\\right)_O=&\\,\\frac{V_E-V_S}{2h}+\\mathcal{O}(h).\\label{eqn:dVdvO}\n\\end{align}\n\nIn this way, by using Eqs.~\\eqref{eqn:squareInts}-\\eqref{eqn:dVdvO} and Eq.~\\eqref{eqn:squareIntegralV}, we find that the sought-after value of $V$ at the point $N$ is given by\n\\begin{widetext}\n    \\begin{equation}\\label{eqn:VNExpression}\n        V_N=-V_S-\\left(\\frac{V_E+V_W-2V_S}{u_O+v_O}+\\frac{1}{2}(V_E-V_W)\\cot{\\frac{v_O-u_O}{2}}\\right)h+\\left(1\n       \n        -\\frac{\\zeta}{2}\n        h^2\\right)(V_E+V_W)+\\order{h^4},\n    \\end{equation}\n\\end{widetext}\nIf we compare this expression for $V_N$ with its equivalent in Refs.~\\cite{mark2017recipe,Jonsson:2020npo,PhysRevD.103.124022}, we immediately note the additional term linear in $h$ in Eq.~\\eqref{eqn:VNExpression}. This term is related with the two {\\it first} order derivatives in Eq.~\\eqref{eqn:VUVEqn}, absent in~\\cite{mark2017recipe,Jonsson:2020npo,PhysRevD.103.124022}.\n\nAnother key difference appears when $u_O=v_O$ (i.e., along $\\gamma=0$). The spherical symmetry of $\\mathbb{M}_2\\times{\\mathbb{S}^2}$ implies that $V(\\eta,\\gamma)=V(\\eta,-\\gamma)$ (or $V_E=V_W$) for any square with a central point such that $u_O=v_O$. More precisely, for all squares  with $u_O=v_O$, $V_N$ is calculated as\n\\begin{align}\\label{eqn:VNDiag}\n    V_N=\\,&-V_S-2h\\left(\\frac{V_E-V_S}{u_O+v_O}\\right)+ \\nonumber\\\\\n    &\n    \\left(2-\\zeta h^2\n    \\right)V_E+\\order{h^4}.\n\\end{align}\nThis symmetry reduces by a half the amount of data to calculate. \n\nWith the above expression for $V_N$, we can reformulate the CID problem in the following way. We first split the grid in Fig.~\\ref{fig:CIDGridM2xS2} into two triangles, one on each side of the $u=v$ line (i.e., $\\gamma=0$). For the bottom triangle, $V_N$ in any square with $u_O=v_O$ only depends on $V_E$ and $V_S$ (see Eq.~\\eqref{eqn:VNDiag}). This implies that, for the bottom triangle, and after having imposed the symmetry along $\\gamma=0$, we only require $V|_{u=0}$ as initial data.  As already pointed out, the values at points in the top triangle can just be obtained using the $V(\\eta,\\gamma)=V(\\eta,-\\gamma)$ symmetry.\nAlternatively, one could choose to apply the scheme  to points in the top triangle, in which case we would have the opposite situation: the only required  initial data in that case would be $V|_{v=0}$.\n\n\nBy knowing the first term, $\\nu_0(\\gamma)$, in Eq.~\\eqref{eqn:VsigmaExpansion}, we were able to calculate $V$ to order $\\order{h^4}$ using this scheme. \nWe next develop a higher order scheme by considering additional initial data on the light cone with the help of the next term, $\\nu_1(\\gamma)$, in Eq.~\\eqref{eqn:VsigmaExpansion}.\n\n\\subsubsection{CID scheme to $\\mathcal{O}(h^6)$}\\label{eq:Oh6}\nIn order to implement a higher order CID scheme, we simply have to calculate the integrals in Eq.~\\eqref{eqn:squareIntegralV} (except the first one) to the desired order. We achieve this by increasing the order in the Taylor expansion of each integrand. In the previous subsection, in order to obtain $V$ to $\\order{h^4}$, we Taylor expanded the integrands in Eq.~\\eqref{eqn:squareIntegralV} to $\\order{h^2}$. Thus, for a CID scheme to $\\order{h^6}$ we Taylor expand it to $\\order{h^4}$. This increase in the order requires  additional data since there are more Taylor coefficients to be calculated. These additional data are easily obtained by differentiating Eq.~\\eqref{eqn:VsigmaExpansion} once with respect to $u$ and $v$ and evaluating it at $\\sigma=-\\frac{1}{2}uv=0$. This procedure leads to the following initial data:\n\\begin{align}\\label{eqn:addVCIDBC}\n    \\left.\\frac{\\partial V}{\\partial u}\\right|_{u=0}=\\,&-\\frac{1}{2}{\\nu_0}'\\left(\\frac{v}{2}\\right)-\\frac{1}{2}\\nu_1\\left(\\frac{v}{2}\\right)\\,v,\\\\\n    \\left.\\frac{\\partial V}{\\partial u}\\right|_{v=0}=\\,&-\\frac{1}{2}{\\nu_0}'\\left(-\\frac{u}{2}\\right),\\\\\n    \\left.\\frac{\\partial V}{\\partial v}\\right|_{u=0}=\\,&\\frac{1}{2}{\\nu_0}'\\left(\\frac{v}{2}\\right),\\\\\\label{eqn:addVCIDBCEnd}\n    \\left.\\frac{\\partial V}{\\partial v}\\right|_{v=0}=\\,&\\frac{1}{2}{\\nu_0}'\\left(-\\frac{u}{2}\\right)-\\frac{1}{2}\\nu_1\\left(-\\frac{u}{2}\\right)\\,u.\n\\end{align}\nTogether with Eq.~\\eqref{eqn:VCIDBC}, we have just established how to obtain the necessary initial data to calculate $V_N$ using the scheme here. \n\nWe now proceed to calculate the integrals in Eq.~\\eqref{eqn:squareIntegralV}, with  the last three integrands  now expanded to $\\order{h^4}$. This yields\n\\begin{widetext}\n    \\begin{align}\\notag\n        V_N=\\,&V_E+V_W-V_S-\\left[Q_O\\evalO{\\pd{V}{v}}+S_O\\evalO{\\pd{V}{u}}+\n        \\zeta\\,\n        V_O\\right]h^2-\\\\\\notag\n        &\\frac{1}{6}\\left[\n        \\zeta\\,\n        \\evalO{\\pd{{}^2V}{u^2}+\\pd{{}^2V}{v^2}}+2\\evalO{\\pd{Q}{v}}\\evalO{\\pd{^2V}{v^2}}+2\\evalO{\\pd{S}{u}}\\evalO{\\pd{^2V}{u^2}}\\right.+\\\\\\notag\n        &\\evalO{\\pd{V}{v}}\\evalO{\\pd{^2Q}{u^2}+\\pd{^2Q}{v^2}}+2\\evalO{\\pd{^2V}{u\\partial v}}\\evalO{\\pd{Q}{u}+\\pd{S}{v}}+\\evalO{\\pd{V}{u}}\\evalO{\\pd{^2S}{u^2}+\\pd{^2S}{v^2}}+\\\\\\label{eqn:higherVNExpresion}\n        &\\left.S_O\\evalO{\\pd{{}^3V}{u\\partial v^2}+\\pd{^3V}{u^3}}+Q_O\\evalO{\\pd{{}^3V}{u^2\\partial v}+\\pd{^3V}{v^3}}\\right]h^4+\\order{h^6}.\n    \\end{align}\n\\end{widetext}\nIn order to calculate $V$ and its corresponding derivatives at the point $O$, we construct a system of 12 equations by evaluating Eq.~\\eqref{eqn:FGTaylorSeries} (with $F\\cdot G=V$) and its first order derivatives at the points $N$, $E$, $W$ and $S$. Ten of these 12 equations are then used to express $V_O$ and its 9 derivatives appearing in Eq.~\\eqref{eqn:higherVNExpresion} in terms of $V_E$, $V_W$, $V_S$ and their derivatives:\n\\begin{widetext}\n    \\begin{align}\n        4V_O&=2V_E+2V_W+h\\left(\\frac{\\partial V}{\\partial u}-\\frac{\\partial V}{\\partial v}\\right)_E-h\\left(\\frac{\\partial V}{\\partial u}-\\frac{\\partial V}{\\partial v}\\right)_W+\\order{h^4},\\\\\n        8h\\left(\\frac{\\partial V}{\\partial u}\\right)_O&=-5V_S-V_E+5V_W+V_N\n        -2h\\left(\\frac{\\partial V}{\\partial u}\n        +\\frac{\\partial V}{\\partial v}\\right)_S\n        -2h\\left(\\frac{\\partial V}{\\partial u}\n        -\\frac{\\partial V}{\\partial v}\\right)_W+\\order{h^4},\\\\\n        8h\\left(\\frac{\\partial V}{\\partial v}\\right)_O&=-5V_S+5V_E+V_W-V_N\n        -2h\\left(\\frac{\\partial V}{\\partial u}\n        +\\frac{\\partial V}{\\partial v}\\right)_S\n        +2h\\left(\\frac{\\partial V}{\\partial u}\n        -\\frac{\\partial V}{\\partial v}\\right)_E+\\order{h^4},\\\\\n        4h^2\\left(\\frac{\\partial^2V}{\\partial u^2}\\right)_O&=V_S-V_E-V_W+V_N+2h\\left(\\frac{\\partial V}{\\partial u}\\right)_E-2h\\left(\\frac{\\partial V}{\\partial u}\\right)_W+\\order{h^4},\\\\\\label{eqn:GdGofEWNSLast}\n        4h^2\\left(\\frac{\\partial^2V}{\\partial v^2}\\right)_O&=V_S-V_E-V_W+V_N-2h\\left(\\frac{\\partial V}{\\partial v}\\right)_E+2h\\left(\\frac{\\partial V}{\\partial v}\\right)_W+\\order{h^4},\\\\\n        4h^2\\left(\\frac{\\partial^2V}{\\partial v\\partial u}\\right)_O&=V_N+V_S-V_E-V_W+\\order{h^4},\\\\\n        \\frac{2}{3}h^3\\left(\\frac{\\partial^3V}{\\partial v^3}\\right)_O&=V_S-V_E+h\\left(\\frac{\\partial V}{\\partial v}\\right)_S+h\\left(\\frac{\\partial V}{\\partial v}\\right)_E+\\order{h^4},\\\\ \\frac{2}{3}h^3\\left(\\frac{\\partial^3V}{\\partial u^3}\\right)_O&=V_S-V_W+h\\left(\\frac{\\partial V}{\\partial u}\\right)_S+h\\left(\\frac{\\partial V}{\\partial u}\\right)_W+\\order{h^4},\\\\\n        4h^3\\left(\\frac{\\partial^2V}{\\partial v^2\\partial u}\\right)_O&=V_N+V_S-V_E-V_W+2h\\left(\\frac{\\partial V}{\\partial v}\\right)_S-2h\\left(\\frac{\\partial V}{\\partial v}\\right)_W+\\order{h^4},\\\\\\label{eqn:GdGofEWNSNLast}\n        4h^3\\left(\\frac{\\partial^2V}{\\partial v\\partial u^2}\\right)_O&=V_N+V_S-V_E-V_W+2h\\left(\\frac{\\partial V}{\\partial u}\\right)_S-2h\\left(\\frac{\\partial V}{\\partial u}\\right)_E+\\order{h^4}.\n    \\end{align}\n\\end{widetext}\n\nThe remaining two equations are used to calculate the first order derivatives of $V$ at the point $N$:\n\\begin{widetext}\n    \\begin{align}\n        \\left(\\pd{V}{u}\\right)_N=\\,&\\frac{V_S-V_E-V_W+V_N}{h}-\\left(\\pd{V}{u}\\right)_E+\\left(\\pd{V}{u}\\right)_W+\\left(\\pd{V}{u}\\right)_S+\\order{h^3},\\\\\n        \\left(\\pd{V}{v}\\right)_N=\\,&\\frac{V_S-V_E-V_W+V_N}{h}+\\left(\\pd{V}{v}\\right)_E-\\left(\\pd{V}{v}\\right)_W+\\left(\\pd{V}{v}\\right)_S+\\order{h^3}.\\nonumber\n    \\end{align}\n\\end{widetext}\nAs can be seen in the above equations, these derivatives do depend on $V_N$.\n\nSimilarly to the $\\order{h^4}$ scheme of the previous subsection, for squares in the grid  with $u_O=v_O$, we have to consider the symmetries of $\\mathbb{M}_2\\times{\\mathbb{S}^2}$ again. Besides the $V_E=V_W$ symmetry condition, the higher order scheme will also make use of additional symmetry conditions for the derivatives of $V$. Specifically, for Eq.~\\eqref{eqn:higherVNExpresion} we make use of the identities \n\\begin{align}\\label{eq:ids high O}\n    \\left(\\pd{V}{v}-\\pd{V}{u}\\right)_E=\\,&\\left(\\pd{V}{v}-\\pd{V}{u}\\right)_W,\\\\\\label{eqn:Vsym3}\n    \\left(\\pd{V}{u}\\right)_E-\\left(\\pd{V}{u}\\right)_S=\\,&\\left(\\pd{V}{v}\\right)_E-\\left(\\pd{V}{v}\\right)_S,\n\\end{align}\nwhich respectively arise from  the following  conditions:\n\\begin{equation}\\label{eq:symms high O}\n    \\pd{^2V}{u\\partial v}=\\pd{^2V}{v\\partial u}\n    \\end{equation}\n and\n \\begin{equation}\n    \\left(\\pd{V}{\\gamma}\\right)_{\\gamma=0}=0.\n\\end{equation}\n\nSummarizing, in order to use our CID scheme to $\\order{h^6}$, one should: start with CID given in Eqs.~\\eqref{eqn:VCIDBC} and \\eqref{eqn:addVCIDBC}-\\eqref{eqn:addVCIDBCEnd}, then use Eq.~\\eqref{eqn:higherVNExpresion} to obtain $V$. \nIn fact, this would be the procedure for finding numerically the values at both upper and lower triangles using CID data along both $u=0$ and $v=0$. In practise, however, it is more efficient to just use half the CID data in Eqs.~\\eqref{eqn:VCIDBC} and \\eqref{eqn:addVCIDBC}-\\eqref{eqn:addVCIDBCEnd}, either just along $u=0$ or along $v=0$, and then use $V_E=V_W$ together with the identities in Eqs.~\\eqref{eq:ids high O}-\\eqref{eqn:Vsym3} to evolve the data in one triangle only.\n\nAfter having the prescription for the CID scheme to two different orders, we implemented it using the computer algebra software {\\it Mathematica}. In the following subsection we show our results, compare them against previous results obtained using different approaches to calculate $V$ (see \\cite{Casals:2012px}) and provide new results.\n\n\\subsection{Results for $V$}\\label{sec:results V}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[scale=0.65]{Plots\/GretComparisonPlot.pdf}\\\\~\\\\\n    \\includegraphics[width=0.45\\textwidth]{Plots\/schemesRelErrorPlot.pdf}\n    \n    \\caption{Top plot: Retarded Green function as an $\\ell$-mode sum (blue) and $V$ using the higher order CID scheme (dashed red), a small coordinate-separation expansion (green) and approximated by $\\nu_0+\\nu_1\\sigma$ (dashed gray), for $\\mathbb{M}_2\\times{\\mathbb{S}^2}$ with \n    $\\zeta=1\/4$,\n    $\\Delta y=y-y'=0$ and $\\gamma=\\pi\/2$ as functions of $\\eta=\\Delta t$.\n    Bottom plot: Relative error between the $\\mathcal{O}(h^4)$ and $\\mathcal{O}(h^6)$ schemes to calculate $V$ with $h=0.00261799$.}\n    \\label{fig:GretComparison}\n\\end{figure}\n\nIn this section we show our results for the Hadamard biscalar $V(x,x')$.\nIn the top plot of Fig.~\\ref{fig:GretComparison} we consider the case of $x$ and $x'$ on a static path with $y=y'$ and $\\gamma=\\pi\/2$ for  $\\zeta=1\/4$.\nIn it, we compare the following: $V$ obtained using the higher order CID scheme of Sec.~\\ref{eq:Oh6} with $h=0.00261799$ (dashed red); the retarded Green function $G_{\\textrm{ret}}$ calculated with the multipolar $\\ell$-mode sum (up to $\\ell=800$) expression given in Eq.~(134)~\\cite{Casals:2012px}\\footnote{We note that in the last expression in Eq.~(134)~\\cite{Casals:2012px} there is a  missing factor $\\theta(-\\sigma_{\\mathbb{M}_2})$.} (blue); the crude approximation $\\nu_0+\\nu_1\\sigma$ to $V$ (dashed gray) from Eq.~\\eqref{eqn:VsigmaExpansion}; $V$ calculated using a small coordinate distance expansion (see Ref.~\\cite{CDOWb}) (green). \nThe first divergence at $\\eta=\\pi\/2$ of $G_{\\textrm{ret}}$  corresponds to the direct null geodesic divergence $\\delta(\\sigma)$ as per the Hadamard form Eq.\\eqref{grhad}.\nThis divergence signals the start of causal separation.\nAs explained in Sec.\\ref{sec:PH}, $V$  and $G_{\\textrm{ret}}$ should agree in the region in-between this divergence, and the next divergence at $\\eta=3\\pi\/2$, which signals the end of the maximal normal neighbourhood and corresponds to a null geodesic having crossed a caustic at $\\gamma=\\pi$. Thus, this latter divergence should be of type $\\text{PV}\\left(1\/\\sigma\\right)$, in agreement with the plot.\n\n\nThe top plot of Fig.~\\ref{fig:GretComparison}  also shows that the CID scheme has good agreement with the $\\ell$-mode-sum calculated $G_\\textrm{ret}$. Indeed, in the bottom plot of Fig.~\\ref{fig:GretComparison} we show that the relative error between the two CID schemes, to $\\order{h^4}$ and to $\\order{h^6}$, with the same stepsize $h=0.00261799$, is at least of order $10^{-4}$. Let us check this value for consistency. Let $e_2$ and $e_4$ be the cumulative errors for the schemes of $\\mathcal{O}(h^4)$ and $\\mathcal{O}(h^6)$, respectively. For the $n$th evolved point in the grid, these errors are given by $e_2=\\order{n(2h)^4}$ and $e_4=\\order{n(2h)^6}$. In the case of Fig.~\\ref{fig:GretComparison}, for a point close to the end of the normal neighbourhood we have $n=\\order{10^5}$. This gives $e_2=\\order{10^{-4}}$ and $e_4=\\order{10^{-9}}$, which  can be taken as relative errors  since $V=\\order{1}$ close the end of the normal neighbourhood. As expected based on this, $e_2$ does agree with the relative error between the two schemes shown in the bottom plot of Fig.~\\ref{fig:GretComparison}.\n\nIn Fig.~\\ref{fig:V3D} we show the  plot of $V$ (to $\\order{h^6}$) for $\\zeta=1\/4$\nfor {\\it any} pair of spacetime\npoints (as long as they lie in normal neighbourhoods, so that $V$ is defined). The red line corresponds to the static worldline ($y=y'$ and $\\gamma=\\pi\/2$) of Fig.~\\ref{fig:GretComparison}. Evolving  CID  has allowed us to calculate $V$  everywhere where it is defined.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[scale=0.5]{Plots\/V3DPlot.pdf}\n    \\caption{Plot of $V$ for\n    all pairs of points in normal neighbourhoods with\n    $\\zeta=1\/4$. The $u=2\\pi$ and $v=2\\pi$ lines correspond to the end of the normal neighbourhood where the (leading) singularity of $G_\\textrm{ret}$, and so of $V$, is of type $\\text{PV}\\left(1\/\\sigma\\right)$ (when away from caustics). The red line is along the static worldline considered in Fig.~\\ref{fig:GretComparison}.}\n    \\label{fig:V3D}\n\\end{figure}\n\nWe also calculated $V$ to $\\order{h^4}$ for various   values of $\\zeta\\neq 1\/4$. In the top plot of Fig.~\\ref{fig:VXi0468} we show these results for the same worldline as in Fig.~\\ref{fig:GretComparison}. For this particular worldline (which has $\\gamma=\\pi\/2$), as $\\zeta$ increases, the magnitude of $V$ decreases. \nIn the bottom plot of Fig.~\\ref{fig:VXi0468} we again plot $V$ for all possible pairs of spacetime points, but now  for $\\zeta=1$. \nWe can see in it that there is a more marked change in the form, with respect to Fig.~\\ref{fig:V3D} for $\\zeta=1\/4$,  near the caustic $\\gamma=\\pm\\pi$.\nWe note that we also calculated $V$ for all pairs of points for the other values of $\\zeta$ that we used in the top plot of Fig.\\ref{fig:VXi0468} but we do not display the full results since their behaviour was not so different from Fig.~\\ref{fig:V3D} for $\\zeta=1\/4$. \n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.45\\textwidth]{Plots\/VXi068Plot.pdf}\\\\~\\\\\n    \\includegraphics[width=0.45\\textwidth]{Plots\/VXi068Plot3D2.pdf}\n    \\caption{Top: Plot of $V$  for different values of $\\zeta$ and for the same scenario (and stepsize) as in Fig.~\\ref{fig:GretComparison}. Bottom: 3D plot of $V$ for $\\zeta=1$.}\n    \\label{fig:VXi0468}\n\\end{figure}\n\n\n\\section{Discussion}\\label{sec:Discussion}\n\nIn this work we have presented and implemented a new method for calculating the tail term $V(x,x')$ in wave propagation on curved background spacetimes where null geodesics cross: by integrating the homogeneous wave equation using Characteristic Initial Data on the light cone.\nWe have provided a proof-of-concept for this method  by applying it to PH, $\\mathbb{M}_2\\times{\\mathbb{S}^2}$.\nFurthermore, we have calculated $V$ for new cases: at {\\it all} spacetime points where it is defined, for various values of $\\zeta\\equiv m^2+2\\xi$.\nThe calculation of $V$ (and of the retarded Green function) at all pairs of points is useful, in particular, for a potential application to the self-consistent orbital evolution of a particle via the self-force.\n\nThe calculation in $\\mathbb{M}_2\\times{\\mathbb{S}^2}$ is technically easier than in black hole spacetimes: First, the Characteristic Initial Data $\\left.V(x,x')\\right|_{\\sigma=0}$  is known analytically and, second, the wave equation is reduced to a {\\it two}-dimensional PDE. As for the first point, we note that $V(x,x')$ was numerically calculated  along null geodesics in Schwarzschild in~\\cite{Ottewill:2009uj} by solving transport equations, and thus its Characteristic Initial Data in Schwarzschild is readily available, while~\\cite{Ottewill:2009uj} provides a prescription for its calculation in Kerr. \nAs for the second point, the wave equation in Schwarzschild would acquire an extra dimension, thus becoming a {\\it three}-dimensional PDE, for which there exist numerical techniques. \nFurthermore, the three-dimensional PDE in Schwarzschild contains first-order derivatives (at least through the angular part),  which the scheme presented here in the context of PH has dealt with.\nIn Kerr, the PDE would become {\\it four}-dimensional, entailing a greater numerical challenge.\nWe intend to undertake the calculation of $V(x,x')$ in these black hole spacetimes in the future.\n\n\n\n\\section{{Acknowledgments.}}\n\nWe are grateful to Adrian Ottewill and Barry Wardell for useful discussions.\nM.C.\\ acknowledges partial financial support by CNPq (Brazil), process number 314824\/2020-0. D. Q. A. acknowledges support from FAPERJ (process number 200.804\/2019) and CNPq (process number 140951\/2017-2).\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\subsection{Tree-likeness}\n\nTrees are graphs with some very distinctive and fundamental properties and it is legitimate to ask to what degree those properties can be transferred to more general structures that are tree-like in some sense \\cite[p. 253]{Diestel}.\nRoughly speaking, tree-likeness stands for  something related to low\n dimensionality, low complexity, efficient information deduction (from local to global),  information-lossless  decomposition (from global into simple pieces)\n  and nice shape for efficient implementation of  divide-and-conquer strategy. For the very basic\n interconnection structures like a graph or a  hypergraph,\n tree-likeness is\n  naturally reflected by the strength of interconnection, namely its\n  connectivity\/homotopy type or cyclicity\/acyclicity, or just the degree of derivation\n  from\n  some characterizing conditions of a tree\/hypertree and its various associated structures and generalizations.\n\nIn vast applications,  one finds that the borderline between\ntractable and intractable cases\n may be the tree-like degree of\n the structure to be\ndealt with \\cite{CYY}.  A support to this   from the fixed-parameter\ncomplexity  point of view is the observation that on various\ntree-structures we can design very good algorithms for many purposes\nand these algorithms can somehow be lifted to tree-like structures\n\\cite{ALS,Downey,DMC,KFL}. It is thus very useful to get information\non approximating general structures by tractable structures, namely\ntree-like structures. On the other hand, one not only finds it\nnatural that tree-like structures appear extensively in many fields,\nsay biology \\cite{Dress96}, structured programs \\cite{Thorup}  and\ndatabase theory \\cite{Fagin}, as graphical representations of\nvarious types of hierarchical relationships, but also\n notice surprisingly that many practical structures we\nencounter are just tree-like, say the internet\n\\cite{ABKMRT,Kleinberg,ST} and chemical compounds  \\cite{YAM}. This\nprompts in many areas the very active study of tree-like structures.\nEspecially, lots of ways to define\/measure a tree-like structure\nhave been proposed in the literature from many different\nconsiderations, just to name a few, say tree-width \\cite{RS,RS86},\ntree-length \\cite{DG,UY}, combinatorial dimension\n\\cite{Dress96,Dress84}, $\\epsilon$-three-points condition\n\\cite{Dinitz},\n $\\epsilon$-four-points condition \\cite{ABKMRT},  asymptotic connectivity\n \\cite{Bahls},\n tree-partition-width \\cite{Bod,Wood}, tree-degree \\cite{CM},\nMcKee-Scheinerman chordality \\cite{McS}, $s$-elimination dimension\n\\cite{DKT}, linkage (degeneracy) \\cite{DKT,KT,LW}, sparsity order\n\\cite{Laurent},\n persistence \\cite{Downey}, cycle rank \\cite{CYY,Lee}, various\ndegrees of acyclicity\/cyclicity \\cite{Duke,Fagin},  boxicity\n\\cite{Roberts}, doubling dimension  \\cite{Gupta},  Domino treewidth\n\\cite{Bod}, hypertree-width \\cite{GLS}, coverwidth \\cite{Chen},\nspread-cut-width \\cite{CJG}, Kelly-width \\cite{Hunter},\n and many other width parameters\n  \\cite{DMC,HOSG}.   It is clear that many relationships among\n  these concepts should be expected as they are all formulated in different ways to represent\n     different aspects of our vague but intuitive idea of tree-likeness.  To clarify these relationship helps to bridge the study\n  in different fields focusing on different tree-likeness measures and helps to improve our understanding\n  of the  universal  tree-like world.\n\nAs a small step in pursuing further understanding of tree-likeness,\nwe take up in this paper the modest task of comparing two parameters\nof tree-likeness,\n  namely (Gromov) hyperbolicity and chordality of a graph.\n    Our main result is that  $k$-chordal graphs must be $\\frac{\\lfloor\n\\frac{k}{2}\\rfloor}{2}$-hyperbolic when $k\\geq 4$  (Theorem\n\\ref{main}). Besides that, we determine a complete set of\nunavoidable\n   isometric subgraphs of $5$-chordal  graphs attaining hyperbolicity $1$  (Theorem \\ref{main1}),   as a minor\n    attempt to   respond to the general  question,\n   ``what is the structure of graphs with relative small hyperbolicity'' \\cite[p. 62]{BKM01}, and the even more general\n   question,\n ``what is the structure of a very tree-like graph''.\n\nThe plan of the paper is as follows.  In Sections \\ref{chordal}\n and  \\ref{hyper} we introduce the two tree-likeness parameters,\n chordality and hyperbolicity, respectively.\n   Section \\ref{theorem} is devoted to a general discussion of the\n   relationship between chordality and hyperbolicity, including a\n   presentation of our main results (Theorems \\ref{main}  and\n   \\ref{main1}).\nSome  consequences of our main results will be listed in Section\n \\ref{oct}.\n In\nSection  \\ref{parameter}, we study the relationship between several\nother tree-likeness parameters and  the main objects of this paper,\nthat is to say, chordality and hyperbolicity, and make use of these\nrelationship to connect chordality and hyperbolicity. The\nrelationship between chordality and hyperbolicity thus obtained by\nnow  is not as strong as Theorem \\ref{main}. But the discussion may\nbe of some independent interest.\n We present a complete and\nself-contained proof of Theorems\n \\ref{main} and \\ref{main1}   in Section    \\ref{proofs} in two stages: some preliminary facts are prepared in Section\n \\ref{lemmas} and the final proof appears in Section \\ref{Proof}.     Following \\cite{BKM01,KM02}, the key to our work is to\n  examine the extremal local configurations as described by\n  Assumptions I and II (see Section \\ref{lemmas}).\nSeveral key   lemmas in Section \\ref{lemmas} are basically copied\nfrom \\cite{BKM01,KM02}.   It is  often that these lemmas are to be\nfound as pieces of a long proof of a big statement in\n\\cite{BKM01,KM02} and so the validity of these technical lemmas\nunder some weaker assumptions needs to be carefully checked. We\ninclude the complete proofs of them, more or less as they were\npresented in \\cite{BKM01,KM02}, not only for the convenience of the\nreader but also to convince the reader that they do hold in our\nsetting.\n\n\\subsection{Chordality}\\label{chordal}\n\n\nWe only consider  simple, unweighted, connected, but not necessarily\nfinite  graphs.   Any graph $G$  together with      the usual\nshortest-path metric on it, $d_G: \\ V(G)\\times V(G)\\mapsto\n\\{0,1,2,\\ldots\\}$,  gives rise to a metric space. We often suppress\nthe subscript and write  $d(x,y)$ instead of   $d_G(x,y)$ when the\ngraph is known by context.\n  Moreover, we\nmay   use the shorthand $xy$ for $d(x,y)$ to further simplify  the\nnotation. Note that a pair of vertices $x$ and $y$ form an edge if\nand only if $xy=1.$            For $S,T\\subseteq V(G)$, we write\n$d(S,T)$ for $\\min _{x\\in S,y\\in T}d(x,y)$.  We often omit the\nbrackets and adopt the convention that  $x$ stands  for the\nsingleton set $\\{x\\}$ when no confusion can be caused.\n\n\nLet  $G$  be a graph.  A  {\\em walk of length $n$} in $G$ is a\nsequence of vertices\n $x_0,x_1,x_2,\\ldots ,x_n$ such that $x_{i-1}x_{i}=1$ for  $i=1,\\ldots ,n$. If these $n+1$ vertices are\n  pairwise different, we call the sequence\n a {\\em path of length $n$}.  A {\\em pseudo-cycle} of length $n$ in $G$ is a\ncyclic sequence of $n$   vertices $x_1,\\ldots ,x_n\\in V(G)$ such\nthat $x_ix_{j}=1$ whenever $j=i+1 \\ (\\bmod \\ n)$; we will reserve\nthe notation $[x_1x_2\\cdots x_n]$ for this pseudo-cycle.      We\ncall this pseudo-cycle an {\\em $n$-cycle}, or a {\\em cycle of length\n$n$}, if $x_1,\\ldots,x_n$ are $n$ different vertices.\n   A {\\em chord} of a path or cycle is an\nedge joining nonconsecutive vertices on  the path or cycle.   An\n{\\em odd chord} of a cycle of even length is a chord connecting\ndifferent vertices the distance between which in the cycle is odd.\n A cycle without chord is called an\n{\\em induced cycle}, or a {\\em chordless cycle}.  For any $n\\geq 3,$\nthe {\\em $n$-cycle graph} is the  graph with $n$ vertices\n          which   has a chordless   $n$-cycle and we denote this graph by $C_n$.   A subgraph $H$ of a\ngraph $G$ is \\textit{isometric} if for any $u,v\\in V(H)$   it holds\n$d_H(u,v)=d_G(u,v)$. A $4$-cycle of a graph $G$ is an\n  {\\em isometric $4$-cycle}\n  provided   the subgraph of   $G$   induced by the vertices of this\ncycle  is isometric and the subgraph has only those four edges which\nare displayed in the cycle. Indeed, this amounts to saying that this\ncycle  is an induced\/chordless cycle; c.f.  Lemma  \\ref{EASY}.\n\n\n\nWe say that a graph is {\\em $k$-chordal} if it does not contain any\ninduced $n$-cycle for $n>k.$ Clearly,    trees are nothing but\n$2$-chordal graphs.  A $3$-chordal graph is usually termed as a {\\em\nchordal\n graph} and a $4$-chordal graph is often called a  {\\em\nhole-free\n graph}.    The class of   $k$-chordal graphs is also\ndiscussed under the name $k$-bounded-hole graphs \\cite{Gavril}.\n\n The  {\\em chordality}   of   a   graph  $G$ is   the smallest integer $k\\geq 2$  such that $G$  is  $k$-chordal\n \\cite{BT1}.           Following   \\cite{BT1},  we use the notation  $\\mathbbm{l}\\mathbbm{c}(G)$ for this parameter as it is merely the length of\n the longest chordless cycle in $G$ when $G$ is not a tree.         Note that our use of the concept of chordality is\n  basically the same as that used in \\cite{CLS,CR} but is very different\n from  the usage  of this term in  \\cite{McS}.\n\n\n\n\nThe recognition of  $k$-chordal graphs is coNP-complete for\n$k=\\Theta (n^{\\epsilon})$ for any constant $\\epsilon >0$\n\\cite{Uehara}. Especially, to determine the chordality of the\nhypercube is attracting much attention  under the name of the\nsnake-in-the-box problem   due to its connection with some\nerror-checking codes problem \\cite{Klee}. Just like the famous\nsnake-in-the-box problem, it looks hard to determine the exact value\nof the chordality of general grid graphs -- it is only easy to see\nthat $\\mathbbm{l}\\mathbbm{c}(G_{m,n})$   should be roughly\nproportional to $nm$ when $\\min (n,m)>2.$\n Nevertheless, just like\nmany other tree-likeness parameters, quite a few  natural graph\nclasses are known to have small chordality \\cite{BLS}. We review\n  some $5$-chordal  ($4$-chordal)  graphs  in the remainder of this subsection.\n\n\n\n\nAn {\\em asteroidal triple} ($AT$) of a graph $G$ is a a set of three\nvertices of $G$  such that for any pair of them there is a path\nconnecting the two vertices whose distance to the remaining vertex\nis at least two.  A graph is {\\em AT-free} if no three vertices form\nan $AT$    \\cite[p. 114]{BLS}. Obviously, all $AT$-free graphs are\n$5$-chordal.  A graph is an {\\em interval graph} exactly when it is\nboth chordal and $AT$-free \\cite[Theorem 7.2.6]{BLS}. $AT$-free\ngraphs also include {\\em cocomparability graphs} \\cite[Theorem\n7.2.7]{BLS}; moreover, all {\\em bounded\n tolerance graphs}\nare cocomparability\n graphs    \\cite{GMT}  \\cite[Theorem 2.8]{MA}  and a graph is a   {\\em permutation graph} if and only if\n  itself and its complement are cocomparability graphs \\cite[Theorem 4.7.1]{BLS}.\nAn important subclass of cocomparability graphs is the class of\n {\\em threshold graphs}, which are those graphs without any induced\nsubgraph isomorphic to the $4$-cycle, the complement of the\n$4$-cycle or the path of length $3$  \\cite[p. 23]{MA}.\n\n\n\n\n A graph is {\\em weakly chordal } \\cite{GMT,Hayward}  when both itself and\n its complement are $4$-chordal.\n  Note that all tolerance graphs  \\cite{MA}  are domination graphs \\cite{Rusu}   and all domination graphs\n are weakly chordal  \\cite{DHMO}.\n A graph is {\\em strongly chordal} if it is chordal and if every even\n cycle of length at least $6$ in this graph has an odd chord  \\cite[p.  21]{GMT}.\n    A\ngraph is {\\em distance-hereditary} if each of its induced paths, and\nhence each of its connected induced subgraphs, is isometric\n\\cite{Howorka}.       We call a graph a {\\em cograph} provided it\ndoes not contain any induced  path of length $3$ \\cite[Theorem\n11.3.3]{BLS}. It is easy to see that each cograph is\ndistance-hereditary and all distance-hereditary\n graphs form a proper subclass\nof  $4$-chordal graphs. It is also known that cocomparability graphs\nare all $4$-chordal \\cite{BT1, Gallai}.\n\n\n\n\n\n\n\\subsection{Hyperbolicity}\\label{hyper}\n\n\n\\subsubsection{Definition  and background}\n\nFor any vertices  $x,y,u,v$ of a graph $G,$ put $\\delta\n_G(x,y,u,v)$, which we often abbreviate to $\\delta (x,y,u,v)$, to be\n the difference between the largest and the\nsecond largest of the following three terms:\n$$\\frac{uv+xy}{2},\\, \\frac{ux+vy}{2},\\, \\text{and} \\  \\frac{uy+vx}{2}.$$ Clearly, $\\delta\n(x,y,u,v)=0$ if $x,y,u,v$ are not four different vertices. A graph\n$G$, viewed as a metric space as mentioned above, is {\\em\n$\\delta$-hyperbolic}  (or tree-like with defect at most $\\delta$)\nprovided for any vertices $x,y,u,v$ in $G$ it holds $\\delta\n(x,y,u,v)\\leq \\delta$ and the  (Gromov) {\\em hyperbolicity} of $G$,\ndenoted $\\delta^* (G)$, is the minimum half integer $\\delta$ such\nthat $G$ is $\\delta$-hyperbolic\n\\cite{Bow91,BH,CDEHV,CDEHVX,DD,Gromov}. Note that it may happen\n$\\delta ^*(G)=\\infty$.   But for a finite graph $G$,  $\\delta^* (G)$\nis clearly finite and polynomial time computable.\n\n\n\n\nNote that in some earlier literature the concept of Gromov\nhyperbolicity is used\n a little bit different from what we adopt here;\n what we call\n$\\delta$-hyperbolic here is called $2\\delta$-hyperbolic in\n\\cite{ABKMRT,BC,BC08,BKM01,CE,DHHKMW,Dress96,GL,KM02,MS} and hence\nthe hyperbolicity of a graph is always an integer according to their\ndefinition. We also refer to \\cite{Alonso,Bow91,BH,  Vai} for some\nequivalent and very accessible definitions of Gromov hyperbolicity\nwhich involve some other comparable parameters.\n\n\nThe concept of hyperbolicity comes from the work of Gromov in\ngeometric group theory  which encapsulates many of the global\nfeatures of the geometry of complete, simply connected manifolds of\nnegative curvature \\cite[p. 398]{BH}.    This concept   not only\nturns out to be strikingly useful in coarse geometry but also\nbecomes more and more important in many applied fields like\nnetworking and phylogenetics\n\\cite{CDEHV1,CDEHV,CDEHVX,CE,DraganX,Dress84,DHHKMW,DHM,Dress96,GL,JLB,JLHB,Kleinberg,ST}.\nThe hyperbolicity of a graph is a way to measure the additive\ndistortion with which every four-points sub-metric of the given\ngraph metric embeds into a tree metric \\cite{ABKMRT}. Indeed, it is\nnot hard to check that the hyperbolicity of a tree  is zero -- the\ncorresponding condition for this  is known as the\n  four-point condition (4PC) and is a characterization of\n  general tree-like metric spaces\n\\cite{Dress84,Dress96,Imrich}. Moreover, the fact that hyperbolicity\nis a tree-likeness parameter\n      is   reflected in the easy fact that the hyperbolicity of a graph is the maximum  hyperbolicity of its 2-connected components --\n      This observation implies the classical result    that\n$0$-hyperbolic graphs are exactly block graphs, namely those  graphs\nin which every $2$-connected subgraph is complete, which are also\nknown to be   those diamond-free chordal graphs\n\\cite{BM,DMS,Howorka}.\n  More results on bounding hyperbolicity of graphs and characterizing low hyperbolicity graphs  can be found in\n\\cite{BC,BC08,BKM01,CDEHV1,CDEHV,DG,KM02}; we will only  report  in\nSection \\ref{MR} some  work most closely related to ours and refer\nthe readers to corresponding references for many other interesting\nunaddressed work.\n\n\n\n\n\n\n\n\nFor any vertex $u\\in V(G)$, the {\\em Gromov product}, also known as\n  the {\\em overlap function}, of any two vertices $x$ and $y$ of $G$ with respect to $u$ is equal\nto $\\frac{1}{2}(xu + yu - xy)$  and is denoted by $(x\\cdot y)_u$\n\\cite[p. 410]{BH}.  As an important context in phylogenetics\n\\cite{DHHKMW,DHM,Farris},  for any real number $\\rho$, the {\\em\nFarris transform} based at $u$, denoted    $D_{\\rho ,u}$, is the\ntransformation which sends $d_G$ to the map\n$$D_{\\rho , u}(d_G): V(G)\\times V(G)\\rightarrow \\mathbb{R}:  \\ (x,y)\\mapsto \\rho -(x\\cdot\ny)_u.$$\n   We\nsay that $G$ is {\\em $\\delta$-hyperbolic with respect to $u\\in\nV(G)$} if the following inequality \\begin{equation}(x\\cdot y)_u\\geq\n\\min ((x\\cdot v)_u,(y\\cdot v)_u)-\\delta \\label{EQ}\\end{equation}\nholds for any vertices $x,y,v$ of $G.$ It is easy to check that the\ninequality \\eqref{EQ} can be rewritten as\n$$xy+uv\\leq \\max (xu+yv, xv+yu)+2\\delta$$ and so we see that $G$ is\n$\\delta$-hyperbolic if and only if  $G$ is $\\delta$-hyperbolic with\nrespect to every vertex of $G.$\n By a simple but nice argument, Gromov shows that  $G$ is   $2\\delta$-hyperbolic provided\n it is  $\\delta$-hyperbolic with respect to\n any given vertex \\cite[Proposition 2.2]{Alonso} \\cite[1.1B]{Gromov}.\n\n\n\nThe {\\em tree-length}   \\cite{Dour, DG, Lo, UY} of a graph $G$,\ndenoted  $\\mathbbm{t}\\mathbbm{l}(G)$, is the minimum integer $k$\nsuch that there is a chordal graph $G'$ satisfying $V(G)=V(G')$,\n$E(G)\\subseteq E(G')$ and  $\\max (d_G(u,v):\\ d_{G'}(u,v)=1)= k.$  We\nuse the convention that the tree-length of a graph without any edge\nis 1.\n It is straightforward\nfrom the definition that   chordal graphs are exactly the graphs of\ntree-length $1$.   It is also known that  $AT$-free graphs  and\ndistance-hereditary graphs have tree-length at most $2$ \\cite[p.\n367]{Dour}; a way to see this is to use the forthcoming result\nrelating chordality and tree-length as well as    the fact that\n$AT$-free graphs are $5$-chordal and distance-hereditary graphs are\n$4$-chordal.\n\n\n\n\n\\begin{te}\\cite[Lemma 6]{GKKPP}   \\cite[Theorem 3.3]{GKKPP1} If $G$ is a $k$-chordal graph, then $\\mathbbm{t}\\mathbbm{l}(G)\\leq \\lfloor \\frac{k}{2}\\rfloor.$\n\\label{thm9}\n\\end{te}\n\n\n\\begin{proof}[Outline] To obtain a minimal triangulation of $G,$ it suffices to select a maximal set of pairwise parallel\nminimal separators of $G$ and add edges to make each of them a\nclique \\cite[Theorem 4.6]{Parra}. It is easy to check that each such\nnew edge connects two points of distance at most  $\\lfloor\n\\frac{k}{2}\\rfloor$ apart in $G.$\n\\end{proof}\n\nThe following is an interesting extension  of the classical result\nthat trees are $0$-hyperbolic and its proof can be given in a way\ngeneralizing     the well-known proof of the latter fact.\n\n\n\n\\begin{te}\n\\cite[Proposition 13]{CDEHV}  A graph  $G$ is $k$-hyperbolic\nprovided its tree-length is no greater than  $k.$\n          \\label{thm10}\n\\end{te}\n\n\n\n\n\n\n\n\n\nIt is noteworthy that a converse of Theorem \\ref{thm10} has also\nbeen established, which means that hyperbolicity and tree-length are\ncomparable parameters of tree-likeness.\n\n\n\\begin{te}\\cite[Proposition 14]{CDEHV}   The inequality $\\mathbbm{t}\\mathbbm{l}(G)\\leq 12k+8k\\log_2n + 17$ holds for any\n $k$-hyperbolic graph $G$    with $n$ vertices.\n\\end{te}\n\n\n\n\\subsubsection{Three examples}\n\n\n\n\nLet us try our hand at three examples to get a feeling of the\nconcept of hyperbolicity.   The first example says that  graphs with\nsmall diameter, hence those so-called small-world networks, must\nhave low hyperbolicity. Note that additionally similar simple\nresults will be reported as    Lemmas \\ref{first} and \\ref{lem14}.\n\n\\begin{example}  \\cite[p. 683]{KM02} The hyperbolicity of a graph $G$  with diameter $D$ is at most\n$\\lfloor \\frac{D}{2}\\rfloor$. \\label{diam}\n\\end{example}\n\\begin{proof} Take $x,y,u,v\\in V(G)$. Our goal  is to show that   $\\delta (x,y,u,v)\\leq\n\\frac{D}{2}$. Without loss of generality, assume that\n\\begin{equation}  \\label{Leipzig}  xy+uv\\geq xu+yv\\geq xv+yu\n\\end{equation} and hence \\begin{equation}\\delta\n(x,y,u,v)=\\frac{1}{2}((xy+uv)-(xu+yv)). \\label{coffee}\n\\end{equation}\n In the first place, we have\n$$\nxu+yu\\geq xy, ux+vx\\geq uv,   xv+yv\\geq xy, vy+uy\\geq uv.\n$$\nSumming up these inequalities yields $(xu+yv)+(xv+yu)\\geq  xy+uv$,\nwhich, according to Eq. \\eqref{Leipzig}, implies that\n$$xu+yv\\geq \\frac{1}{2}(xy+uv).$$\nThis along with Eq. \\eqref{coffee}  gives  $\\delta (x,y,u,v)\\leq\n\\frac{1}{4}(xy+uv)\\leq \\frac{D}{2}.$\n Moreover, if  $\\delta (x,y,u,v)=\n\\frac{D}{2}$, then we have\n \\begin{equation}xu+yv=D,\n xv+yv=xy=D, xu+xv=uv=D.\n \\label{JS}\n \\end{equation}\nBy adding the equalities in Eq. \\eqref{JS} together, we see that\n$3D=2(xu+xv+yv)$ and so $D$ must be even.\n\\end{proof}\n\n\n\n\\begin{figure}\n\\hspace{-33mm}\n\\unitlength 1mm\n\\linethickness{0.4pt}\n\\ifx\\plotpoint\\undefined\\newsavebox{\\plotpoint}\\fi \\hspace{20mm}\n\\begin{center}\n\\begin{picture}(10,10)\n\\put(10,10){\\circle{20}}\\put(10,17){\\circle*{1}}\\put(10,18){\\makebox(0,0)[b]{$x$}}\n\\put(17,10){\\circle*{1}}\\put(18,10){\\makebox(0,0)[l]{$v$}}\n\\put(10,3){\\circle*{1}}\\put(10,2){\\makebox(0,0)[t]{$y$}}\n\\put(3,10){\\circle*{1}}\\put(2,10){\\makebox(0,0)[r]{$u$}}\n\\put(16,15){\\makebox(0,0)[l]{$xv$}}\\put(15,3){\\makebox(0,0)[l]{$vy$}}\n\\put(4,4){\\makebox(0,0)[r]{$yu$}}\\put(4,15){\\makebox(0,0)[r]{$ux$}}\n \\end{picture}\n\\end{center}\n\\caption{Four points in an $n$-cycle.}\\label{n-cycle}\n\\end{figure}\n\n\nThe bound asserted by  Example \\ref{diam} is clearly not tight when\n$D=1.$ But, as can be seen from the next example,   the bound given\nin  Example  \\ref{diam} in terms of the diameter $D$ is best\npossible for every $D\\geq 2$. Note that this forthcoming example can\nalso be seen directly via Example \\ref{diam}, as indicated in\n\\cite[p. 683]{KM02}.\n\n\\begin{example} \\cite[p. 683]{KM02}   For any $n\\geq 3,$ the  chordality  of the $n$-cycle  is $n$ while\nthe hyperbolicity of the $n$-cycle is \\begin{equation}\n\\delta^*(C_n)=\\begin{cases} \\lfloor\n\\frac{n}{4}\\rfloor-\\frac{1}{2},&\\text{if $n\\equiv 1\\ (\\!\\!\\!\\!\\!\\!\\mod  4)$;}\\\\\n\\lfloor \\frac{n}{4}\\rfloor,&\\text{else.}\n\\end{cases}\n\\label{USTC}\n\\end{equation}\nNote that the diameter of  $C_n$ is  $\\lfloor \\frac{n}{2}\\rfloor$\nand\n$$ \\delta^*(C_n)=\\begin{cases}\n\\frac{\\lfloor \\frac{n}{2}\\rfloor}{2},&\\text{if $n\\equiv 0\\ (\\!\\!\\!\\!\\!\\!\\mod 4)$;}\\\\\n\\frac{\\lfloor \\frac{n}{2}\\rfloor}{2}-\\frac{1}{2},&\\text{else.}\n\\end{cases}$$\n\\label{exam7}\n\\end{example}\n\\begin{proof} To prove Eq. \\eqref{USTC}, we need to estimate $\\delta\n(x,y,u,v)$ for any four vertices $x,y,u,v$ of the $n$-cycle graph\n$C_n.$  If there is a geodesic connecting two vertices and passing\nthrough all the four vertices $x,y,u,v$, we surely have $\\delta\n(x,y,u,v) =0$ just because we know that the hyperbolicity of a path\nis $0.$ So, we can assume that the cycle $C_n$ is $[c_0c_1\\ldots\nc_{n-1}]$ where $c_0=x,c_{xv}=v,c_{xv+vy}=y,c_{xv+vy+yu}=u$ and\n\\begin{equation}xv+vy+yu+ux=n;\n\\label{sum}\n\\end{equation} see Fig.     \\ref{n-cycle}.  With no loss of generality, we assume that\n\\begin{equation}xu-vy\\geq |xv-uy|.\n\\label{morning}\n\\end{equation} This implies  $ xu+uy \\geq   xv+vy\n$ and $ vx+xu\\geq  vy+yu$. According to the geometric distribution\nof the four points, we then come to\n$$xy=xv+vy\\ \\ \\text{and} \\ \\ vu=vy+yu.\n$$ It follows that\n\\begin{equation}xy+vu=(xv+yu)+2vy\\label{expo}\\end{equation} and\n$$xy+vu=xv+vy+vy+yu=(xv+vy+yu)+vy\\geq xu+vy.$$  At the moment, we\nsee that there are only two possibilities, either $xy+vu\\geq\nxu+vy>xv+yu$ or $xy+vu\\geq xv+yu \\geq xu+vy.$\n\n If the first case\nhappens, we have\n\\begin{equation}\n\\begin{array}{cll}\\delta(x,y,u,v)&=&\\frac{1}{2}(xy+vu- xu-vy)\n\\\\\n&= & \\frac{n}{2}-xu.\\ \\ \\text{(By Eqs. \\eqref{sum} and\n\\eqref{expo})}\n \\end{array}\n \\label{Kool}\n \\end{equation}\nBy\n Eqs.  \\eqref{sum}  and \\eqref{morning} and  $xu+vy>xv+yu$, we see\n that $xu\\geq\n \\begin{cases} \\lfloor \\frac{n}{4}\\rfloor +1,&\\text{if $n\\equiv 0,1,2\\ (\\!\\!\\!\\!\\mod  4)$,}\\\\\n \\lfloor \\frac{n}{4}\\rfloor +2,&\\text{if $n\\equiv 3\\ (\\!\\!\\!\\!\\mod  4)$,}\n\\end{cases}\n$\n and hence  Eq.  \\eqref{Kool}  forces\n \\begin{equation}\\label{cycle1}\\delta(x,y,u,v)= \\frac{n}{2} - xu  \\leq  \\begin{cases}\n \\lfloor \\frac{n}{4}\\rfloor -1,&\\text{if $n\\equiv 0\\ (\\!\\!\\!\\!\\!\\!\\mod 4)$;}\\\\\n \\lfloor \\frac{n}{4}\\rfloor ,&\\text{if $n\\equiv 2\\ (\\!\\!\\!\\!\\!\\!\\mod 4)$;}\\\\\n \\lfloor \\frac{n}{4}\\rfloor -\\frac{1}{2},&\\text{if $n\\equiv 1,3\\ (\\!\\!\\!\\!\\!\\!\\mod 4)$.}\n\\end{cases}\n\\end{equation}\n\n\n For the second case, we have\n$$\n\\begin{array}{cll}\\delta(x,y,u,v)&=&\\frac{1}{2}(xy+vu-\nxv-yu)\n\\\\\n&= & vy,\\ \\ \\text{(By Eq. \\eqref{expo})}\n\\end{array}$$\nand hence by\n Eqs.  \\eqref{sum}  and \\eqref{morning} and  $xv+yu\\geq vy +xu$, we further  obtain\n \\begin{equation}\\label{cycle2}  \\delta(x,y,u,v)=vy\\leq \\begin{cases}\n \\lfloor \\frac{n}{4}\\rfloor ,&\\text{if $n\\equiv 0,2,3\\ (\\!\\!\\!\\!\\!\\mod 4)$;}\\\\\n \\lfloor \\frac{n}{4}\\rfloor -1,&\\text{if $n\\equiv 1\\ (\\!\\!\\!\\!\\!\\mod 4)$.}\n\\end{cases}\n\\end{equation}\n\n\nCombining Eqs. \\eqref{cycle1} and \\eqref{cycle2}  yields\n\\begin{equation}\\label{quartet}\n\\delta (x,y,u,v)\\leq \\begin{cases} \\lfloor\n\\frac{n}{4}\\rfloor,&\\text{if $n\\equiv 0,2,3\\ (\\!\\!\\!\\!\\!\\mod 4)$;}\\\\\n\\lfloor\n\\frac{n}{4}\\rfloor-\\frac{1}{2},&\\text{if $n\\equiv 1\\ (\\!\\!\\!\\!\\!\\!\\mod  4)$.}\\\\\n\\end{cases}\n\\end{equation}\nTaking\n\\begin{equation*}\n (x,v,y,u)= \\begin{cases} (c_0,c_k,c_{2k},c_{3k}),&\\text{if $n\\equiv 0\\ (\\!\\!\\!\\!\\!\\!\\mod  4)$,}\\\\\n(c_0,c_k,c_{2k},c_{3k}),&\\text{if $n\\equiv 1\\ (\\!\\!\\!\\!\\!\\!\\mod  4)$,}\\\\\n(c_0,c_k,c_{2k},c_{3k+1}),&\\text{if $n\\equiv 2\\ (\\!\\!\\!\\!\\!\\!\\mod  4)$,}\\\\\n(c_0,c_{k+1},c_{2k+1},c_{3k+2}),&\\text{if $n\\equiv 3\\\n(\\!\\!\\!\\!\\!\\!\\mod 4)$,}\n\\end{cases}\n\\end{equation*}\nwe see that Eq. \\eqref{quartet} is tight and hence Eq.  \\eqref{USTC}\nis established.\n\\end{proof}\n\n\nFor any two graphs  $G_1$ and  $G_2,$ we define its {\\em Cartesian\nproduct } $G_1\\Box G_2$  to be the graph satisfying $V(G_1\\Box\nG_2)=V(G_1)\\times V(G_2)$ and $d_{G_1\\Box\nG_2}((u_1,u_2),(v_1,v_2))=d_{G_1}(u_1,v_1)+d_{G_2}(u_2,v_2)$\n\\cite[\\S 1.4]{IK}.\n\n\\begin{example}  \\label{exam3} Let $G_1$ and  $G_2$ be two graphs satisfying  $\\delta ^*(G_1)=\\delta^*\n(G_2)=0.$ Then $\\delta^*(G_1\\Box G_2)=\\min (D_1,D_2)$ where $D_1$\nand $D_2$ are the diameters of  $G_1$ and  $G_2$, respectively.\n\\end{example}\n\n\\begin{proof}  For any $v\\in V(G_1\\Box G_2)$, we often use\nthe convention that $v=(v_1,v_2)$ for $v_1\\in V(G_1)$ and $v_2\\in\nV(G_2)$.  For any $u,v\\in V(G_1\\Box G_2)$, we write  $uv$ for\n$d_{G_1\\Box G_2}(u,v)$, $(uv)_1$  for $d_{G_1}(u_1,v_1)$, $(uv)_2$\nfor $d_{G_2}(u_2,v_2)$  and we  use $\\delta$ for $\\delta_{G_1\\Box\nG_2}$.\n\nTake  $a,b\\in V(G_1)$ such that $d_{G_1}(a,b)=D_1$ and take $c,d\\in\nV(G_2)$ such that $d_{G_2}(c,d)=D_2$. Set\n$x=(a,c),y=(a,d),u=(b,c),v=(b,d)$. It is straightforward that\n$\\delta (x,y,u,v)=\\min (D_1, D_2)$.\n\nTo complete the proof, we pick any four vertices  $x,y,u,v$ of\n$G_1\\Box G_2$ and aim to show that \\begin{equation}\\delta\n(x,y,u,v)\\leq \\min (D_1, D_2). \\label{sunny}\\end{equation}  Let\n$A=xy+uv,$ $A_1=(xy)_1+(uv)_1$, $A_2=(xy)_2+(uv)_2$, $B=xu+yv,$\n$B_1=(xu)_1+(yv)_1$, $B_2=(xu)_2+(yv)_2$,   $C=xv+yu,$\n  $C_1=(xv)_1+(yu)_1$, $C_2=(xv)_2+(yu)_2.$ Because\n  $\\delta^*(G_1)=\\delta^*(G_2)=0$, we can suppose $A_1=\\max (B_1, C_1)$ and\n  $A_2=\\max (B_2,C_2)$.\n\n  If it happens either  $(A_1,A_2)=(B_1,B_2)$ or\n  $(A_1,A_2)=(C_1,C_2)$, we can immediately conclude that $\\delta\n  (x,y,u,v)=0.$ By symmetry between    $B$  and  $C$ and between $G_1$  and  $G_2,$   it thus remains to\n  deduce Eq.  \\eqref{sunny}  under the condition that\n\\begin{equation*}B\\geq C, A_1=B_1>C_1, \\ \\text{and}\\ A_2=C_2>B_2.\n\\label{KAIST}\n\\end{equation*}\n\n\n\nSince $A_1=B_1,$  we have $\\delta\n(x,y,u,v)=\\frac{A-B}{2}=\\frac{A_2-B_2}{2}$. We proceed with a direct\ncomputation and find \\begin{equation}\\label{eq14}\\delta (x,y,u,v)\n=\\frac{((xy)_2-(xu)_2)+((uv)_2-(yv)_2 )}{2}\\leq (yu)_2\\leq D_2.\n\\end{equation}\nMaking use of $A_2=C_2$ and  $B\\geq C,$  we can obtain  instead\n\\begin{equation}\\label{eq15}\n\\begin{array}{cll}\\delta(x,y,u,v)&\\leq&\\frac{A_2-B_2+B-C}{2}=\\frac{A_2-C_2+B_1-C_1}{2}=\\frac{B_1-C_1}{2}\n\\\\\n&= & \\frac{((xu)_1-(xv)_1)+((yv)_1-(yu)_1)}{2}\\leq (uv)_1\\leq D_1\n\\end{array}\n \\end{equation} Combining  Eqs. \\eqref{eq14} and \\eqref{eq15} we now get Eq. \\eqref{sunny}, as desired.\n\\end{proof}\n\n\n\\begin{remark}\\label{rem8}\nFor any $t$ natural numbers $m_1,\\ldots , m_t$, the  {\\em\n$t$-dimensional\n grid graph} $G_{m_1,\\ldots , m_t}$  is the graph with vertex set $\\{\n1,2,\\ldots,m_1\\}\\times \\cdots \\times \\{ 1,2,\\ldots,m_t\\}$ and\n$(i_1,\\ldots ,i_t)$ and $(j_1,\\ldots, j_t)$ are adjacent in\n$G_{m_1\\ldots,m_t}$ if any only if $\\sum_{p=1}^t(i_p-j_p)^2=1.$\nExample \\ref{exam3}  implies that $\\delta^*(G_{m_1,m_2})=\\min\n(m_1,m_2)-1$ and hence $G_{m,m}$ provides another example that the\nbound reported in Example \\ref{diam} is tight. It might be\ninteresting to determine the hyperbolicity of  $t$-dimensional\n grid graphs for $t\\geq 3.$\n\\end{remark}\n\n\n\n\n\\begin{remark}\\label{grid}\n Dourisboure and Gavoille   show  that the tree-length of $G_{n,m}$   is $ \\min (n,m)$ if $n\\not= m$ or $n=m$ is even and is $n-1$ if\n$n=m$ is odd \\cite[Theorem 3]{DG}. Remark \\ref{rem8}  tells us that\n$\\delta^*(G_{n,m})=\\min(m,n)-1$.  This says that Theorem \\ref{thm10}\nis   tight.\n\\end{remark}\n\n\n\n\n\n\n\n\n\n\\section{Chordality vs. hyperbolicity}\\label{MR}\n\n\\subsection{Main results}\\label{theorem}\n\n\nFirstly, we point out that\n a graph with low hyperbolicity may have large\nchordality. Indeed, take any graph $G$ and form the new graph $G'$\nby adding an additional vertex and connecting this new vertex with\nevery vertex of $G$. It is obvious that  $\\delta^*(G')\\leq 1$ while\n$\\mathbbm{l}\\mathbbm{c}(G')= \\mathbbm{l}\\mathbbm{c}(G)$ if  $G$ is\nnot a tree. Moreover, it is equally easy to see that $G'$ is even\n$\\frac{1}{2}$-hyperbolic if $G$ does not have any induced $4$-cycle\n\\cite[p. 695]{KM02}. Surely, this example does not preclude the\npossibility that for many important graph classes we can bound their\nchordality in terms of their hyperbolicity.\n\nOne of our  main results says that   hyperbolicity can be bounded\nfrom above in terms of chordality.\n\n\n\n\n\n\\begin{te} \\label{main} For each $k\\geq 4,$ all  $k$-chordal graphs are  $\\frac{\\lfloor\n\\frac{k}{2}\\rfloor}{2}$-hyperbolic.\n\\end{te}\n\n\n\n\\begin{remark}\n  A graph is {\\em bridged} \\cite{AF,LS}   if it\ndoes not contain any finite isometric cycles of length at least\nfour, or equivalently, if  it   is cop-win and has no chordless\ncycle of length $4$ or $5$. In contrast to    Theorem \\ref{main}, it\nis interesting to note that the hyperbolicity of bridged graphs can\nbe arbitrarily high \\cite[p. 684]{KM02}.\n\\end{remark}\n\n\n\\begin{remark}  Bandelt and Chepoi \\cite[\\S 5.2]{BC08}  make the remark that\n ``a characterization of\n all   $1$-hyperbolic   graphs   by   forbidden   isometric   subgraphs\nis   not\n in   sight,  in  as  much  as  isometric  cycles  of  lengths   up  to  $7$   may   occur,   thus\ncomplicating   the   picture''.  Note that our Theorem \\ref{main}\nsays that all $5$-chordal graphs are $1$-hyperbolic and hence the\nappearance of those chordless $6$-cycles and chordless $7$-cycles\nmay be a real headache to deal with in pursuing a characterization\nof all $1$-hyperbolic graphs.\n\\end{remark}\n\n\n\n\n\\begin{figure}\n\\hspace{-33mm}\n\\unitlength 1mm\n\\linethickness{0.4pt}\n\\ifx\\plotpoint\\undefined\\newsavebox{\\plotpoint}\\fi \\hspace{30mm}\n\\begin{center}\n\\begin{picture}(150,30)\n\n\n\\put(60,25){\\circle*{1}}\\put(60,28){\\makebox(0,0)[t]{$v_8$}}\n\n\\put(55,15){\\circle*{1}}\\put(51,15){\\makebox(0,0)[l]{$v_7$}}\n\n\\put(50,5){\\circle*{1}}\\put(50,1){\\makebox(0,0)[b]{$v_6$}}\n \\put(60,5){\\circle*{1}}\\put(60,1){\\makebox(0,0)[b]{$v_5$}}\n\\put(70,25){\\circle*{1}}\\put(70,29){\\makebox(0,0)[t]{$v_1$}}\n\\put(80,25){\\circle*{1}}\\put(80,28){\\makebox(0,0)[t]{$v_2$}}\n\\put(75,15){\\circle*{1}}\\put(79,15){\\makebox(0,0)[r]{$v_3$}}\n\n\\put(70,5){\\circle*{1}}\\put(70,1){\\makebox(0,0)[b]{$v_4$}}\n\n\\qbezier(60,25)(55,15)(50,5)\\qbezier(80,25)(75,15)(70,5)\n\\qbezier(55,15)(58,9)(60,5)\\qbezier(70,25)(73,19)(75,15)\\qbezier(50,5)(60,5)(70,5)\n\\qbezier(60,25)(70,25)(80,25)\n\\end{picture}\n\\end{center}\n\n\\caption{The outerplanar graph $F_2$ has  chordality $6$,\nhyperbolicity $\\frac{3}{2}$, and  tree-length $2$.}\\label{counter}\n\\end{figure}\n\n\n\n\n\\begin{example}\\label{exam8} For  any $t\\geq 2$ we set\n  $F_t$ to  be the  graph obtained from  the $4t$-cycle $[v_1v_2\\cdots v_{4t}]$ by adding the two\n edges $\\{v_1,v_3\\}$ and $ \\{v_{2t+1},v_{2t+3}\\}$;  see  Fig. \\ref{counter} for an illustartion of  $F_2$. Clearly,\n $\\delta(v_2, v_{t+2},v_{2t+2}, v_{3t+2})=t-\\frac{1}{2}$. Furthermore, we\n can check that\n  $\\mathbbm{l}\\mathbbm{c}(F_t)=4t-2$ and $\\delta^*(F_t)=t-\\frac{1}{2}=\\delta(v_2, v_{t+2},v_{2t+2},\n  v_{3t+2})=\\frac{\\mathbbm{l}\\mathbbm{c}(F_t)}{4}.$  $F_t$ is clearly an outerplanar graph. Thus, applying the result that   $\\mathbbm{t}\\mathbbm{l}(G)=\\lceil \\frac{\\mathbbm{l}\\mathbbm{c}(G)}{3}\\rceil$   for\nevery outerplanar graph $G$   \\cite[Theorem 1]{DG},  we even know\nthat  $\\mathbbm{t}\\mathbbm{l}(F_t)=\\lceil \\frac{4t-2}{3}\\rceil$.\n\\end{example}\n\n\n\n\n\nIt is clear that if the bound claimed by  Theorem \\ref{main} is\ntight for $k=4t$ ($k=4t-2$) then it is tight for $k=4t+1$\n($k=4t-1$). Consequently, Examples  \\ref{exam7} and \\ref{exam8}\nindeed mean that the bound reported in Theorem \\ref{main} is tight\nfor every $k\\geq 4.$\n      Surely,   the logical next   step would be to characterize     all those extremal\n      graphs $G$\nsatisfying\n\\begin{equation}\n\\delta^*(G)=\\frac{\\lfloor\n\\frac{\\mathbbm{l}\\mathbbm{c}(G)}{2}\\rfloor}{2}. \\label{extremal}\n\\end{equation}\n       However,   there seems to be still   a\nlong haul ahead in this direction.\n\n\\begin{remark} \\label{rem13}\nFor any graph $G$ and any positive number $t$, we put $S^t(G)$ to be\na {\\em subdivision graph} of $G$, which is obtained from $G$ by\nreplacing each edge $\\{u,v\\}$ of $G$ by a path $u,n_{u,v}^1,\\ldots,\n  n_{u,v}^{t-1}, v$ of length $t$ connecting $u$ and $v$ through a sequence of new vertices $n_{u,v}^1,\\ldots,\n   n_{u,v}^{t-1}$  (we surly require that  $n_{v,u}^q=n_{u,v}^{t-q}$).  For any four vertices $x,y,u,v\\in V(G)$,\nwe obviously have $\\delta_{S^t(G)}(x,y,u,v)=t\\delta_G(x,y,u,v)$ and\nso $\\delta^*(S^t(G))\\geq   t\\delta^*(G)$.\n Instead of the trivial fact    $\\mathbbm{l}\\mathbbm{c} (S^t(G))\\geq\n   t \\mathbbm{l}\\mathbbm{c}(G)$, if the good shape of  $G$ permits us to deduce  a good upper bound\n   of  $\\mathbbm{l}\\mathbbm{c} (S^t(G))$ in terms of\n   $\\mathbbm{l}\\mathbbm{c}(G)$, we will see that $\\delta^*(S^t(G))$\n   is   high relative to  $\\mathbbm{l}\\mathbbm{c} (S^t(G))$\n   provided so is $G.$  Recall that  the cycles whose lengths are divisible by $4$ as discussed  in Example\n   \\ref{exam7} are used to demonstrate  the tightness of the bound given in Theorem\n   \\ref{main}; also observe  that the graphs suggested by Example \\ref{exam8} is nothing but a slight\n   ``perturbation''\nof cycles of length divisible by $4.$ Since $C_{4t}=S^t(C_{4})$,\nthese examples can be said to be generated by the ``seed''  $C_4.$\nIt might\n   deserve to look for some other good ``seeds\" from which we can use\n   the above subdivision operation or its variant to produce graphs\n   satisfying Eq. \\eqref{extremal}.\n\\end{remark}\n\n\n\n\n\n\n\n\n\n\nLet  $C_4$,  $H_1$,  $H_2$, $H_3$, $H_4$  and  $H_5$ be the  graphs\ndisplayed in Fig. \\ref{fig0}.  It is simple to check that each of\nthem has hyperbolicity $1$ and is $5$-chordal.   Besides Theorem\n\\ref{main}, another main contribution of this paper is the\nfollowing, which says that  $5$-chordal graphs will be\n$\\frac{1}{2}$-hyperbolic as soon as these six obvious obstructions\ndo not occur.\n\n\n\n\n\n\n\n\n\\begin{figure}\n\\hspace{-33mm}\n\\unitlength 1mm\n\\linethickness{0.4pt}\n\\ifx\\plotpoint\\undefined\\newsavebox{\\plotpoint}\\fi \\hspace{30mm}\n\\begin{center}\n\\begin{picture}(150,100)\n\n\\put(30,60){\\circle*{1}}\\put(33,57){\\makebox(0,0)[br]{$y$}}\n\\put(0,60){\\circle*{1}}\\put(-3,57){\\makebox(0,0)[bl]{$u$}}\n\\put(30,90){\\circle*{1}}\\put(33,93){\\makebox(0,0)[tr]{$v$}}\n\\put(0,90){\\circle*{1}}\\put(-3,93){\\makebox(0,0)[tl]{$x$}}\n\\put(15,50){\\makebox(0,0)[l]{$C_{4}$}}\n\n\n\n\\qbezier(30,90)(20,90)(0,90)\\qbezier(0,90)(0,80)(0,60)\\qbezier(0,60)(20,60)(30,60)\\qbezier(30,60)(30,80)(30,90)\n\n\\put(70,60){\\circle*{1}}\\put(70,56){\\makebox(0,0)[b]{$y$}}\n\n\\put(50,80){\\circle*{1}}\\put(47,80){\\makebox(0,0)[l]{$u$}}\n\\put(90,80){\\circle*{1}}\\put(93,80){\\makebox(0,0)[r]{$v$}}\n\n\\put(70,100){\\circle*{1}}\\put(70,103){\\makebox(0,0)[t]{$x$}}\n\n\\put(60,90){\\circle*{1}}\\put(57,90){\\makebox(0,0)[l]{$a$}}\n\\put(80,90){\\circle*{1}}\\put(83,90){\\makebox(0,0)[r]{$b$}}\n\\put(60,70){\\circle*{1}}\\put(57,70){\\makebox(0,0)[l]{$c$}}\n\\put(80,70){\\circle*{1}}\\put(83,70){\\makebox(0,0)[r]{$d$}}\n\\put(70,50){\\makebox(0,0)[l]{$H_{1}$}}\n\n\n\\qbezier(50,80)(60,70)(70,60)\\qbezier(70,60)(80,70)(90,80)\\qbezier(90,80)(80,90)(70,100)\\qbezier(70,100)(60,90)(50,80)\n\\qbezier(60,90)(70,80)(80,70)\\qbezier(60,90)(60,80)(60,70)\\qbezier(60,70)(70,70)(80,70)\\qbezier(80,70)(80,80)(80,90)\n\\qbezier(60,90)(70,90)(80,90)\n\n\n\\put(120,60){\\circle*{1}}\\put(120,56){\\makebox(0,0)[b]{$y$}}\n\n\\put(100,80){\\circle*{1}}\\put(97,80){\\makebox(0,0)[l]{$u$}}\n\\put(140,80){\\circle*{1}}\\put(143,80){\\makebox(0,0)[r]{$v$}}\n\n\\put(120,100){\\circle*{1}}\\put(120,103){\\makebox(0,0)[t]{$x$}}\n\n\\put(110,90){\\circle*{1}}\\put(107,90){\\makebox(0,0)[l]{$a$}}\n\\put(130,90){\\circle*{1}}\\put(133,90){\\makebox(0,0)[r]{$b$}}\n\\put(110,70){\\circle*{1}}\\put(107,70){\\makebox(0,0)[l]{$c$}}\n\\put(130,70){\\circle*{1}}\\put(133,70){\\makebox(0,0)[r]{$d$}}\n\\put(120,50){\\makebox(0,0)[l]{$H_{2}$}}\n\n\n\\qbezier(100,80)(110,70)(120,60)\\qbezier(120,60)(130,70)(140,80)\\qbezier(140,80)(130,90)(120,100)\\qbezier(120,100)(110,90)(100,80)\n\\qbezier(110,90)(120,80)(130,70)\\qbezier(110,70)(120,80)(130,90)\\qbezier(130,90)(130,80)(130,70)\\qbezier(110,90)(120,90)(130,90)\n\\qbezier(110,90)(110,80)(110,70)\\qbezier(110,70)(120,70)(130,70)\n\n\n\n\\put(20,0){\\circle*{1}}\\put(20,-4){\\makebox(0,0)[b]{$y$}}\n\n\\put(0,20){\\circle*{1}}\\put(-3,20){\\makebox(0,0)[l]{$u$}}\n\\put(40,20){\\circle*{1}}\\put(43,20){\\makebox(0,0)[r]{$v$}}\n\n\\put(20,30){\\circle*{1}}\\put(20,33){\\makebox(0,0)[t]{$x$}}\n\n\n\\put(10,10){\\circle*{1}}\\put(7,10){\\makebox(0,0)[l]{$c$}}\n\\put(30,10){\\circle*{1}}\\put(33,10){\\makebox(0,0)[r]{$d$}}\n\\put(22,-6){\\makebox(0,0)[l]{$H_{3}$}}\n\n\n\\qbezier(0,20)(10,10)(20,0)\\qbezier(20,0)(30,10)(40,20)\\qbezier(40,20)(30,25)(20,30)\\qbezier(20,30)(10,25)(0,20)\n \\qbezier(10,10)(20,10)(30,10)\n\n\\put(70,0){\\circle*{1}}\\put(70,-4){\\makebox(0,0)[b]{$y$}}\n\n\\put(50,20){\\circle*{1}}\\put(47,20){\\makebox(0,0)[l]{$u$}}\n\\put(90,20){\\circle*{1}}\\put(93,20){\\makebox(0,0)[r]{$v$}}\n\n\\put(70,40){\\circle*{1}}\\put(70,43){\\makebox(0,0)[t]{$x$}}\n\n\\put(60,30){\\circle*{1}}\\put(57,30){\\makebox(0,0)[l]{$a$}}\n\\put(80,30){\\circle*{1}}\\put(83,30){\\makebox(0,0)[r]{$b$}}\n\\put(60,10){\\circle*{1}}\\put(57,10){\\makebox(0,0)[l]{$c$}}\n\\put(80,10){\\circle*{1}}\\put(83,10){\\makebox(0,0)[r]{$d$}}\n\\put(72,-6){\\makebox(0,0)[l]{$H_{4}$}}\n\n\\qbezier(50,20)(60,10)(70,0)\\qbezier(70,0)(80,10)(90,20)\\qbezier(90,20)(80,30)(70,40)\\qbezier(70,40)(60,30)(50,20)\n\\qbezier(60,30)(70,20)(80,10) \\qbezier(80,30)(70,20)(60,10)\n\n\n\\put(120,0){\\circle*{1}}\\put(120,-4){\\makebox(0,0)[b]{$y$}}\n\\put(110,10){\\circle*{1}}\\put(107,10){\\makebox(0,0)[l]{$c$}}\n\\put(110,30){\\circle*{1}}\\put(107,30){\\makebox(0,0)[l]{$a$}}\n\\put(130,30){\\circle*{1}}\\put(133,30){\\makebox(0,0)[r]{$b$}}\n\\put(130,10){\\circle*{1}}\\put(133,10){\\makebox(0,0)[r]{$d$}}\n\\put(100,20){\\circle*{1}}\\put(96,20){\\makebox(0,0)[l]{$u$}}\n\\put(140,20){\\circle*{1}}\\put(143,20){\\makebox(0,0)[r]{$v$}}\n\\put(120,40){\\circle*{1}}\\put(120,43){\\makebox(0,0)[t]{$x$}}\n\\qbezier(100,20)(110,10)(120,0)\\qbezier(140,20)(130,10)(120,0)\\qbezier(110,30)(120,20)(130,10)\n\\qbezier(120,40)(110,30)(100,20)\\qbezier(120,40)(130,30)(140,20)\n\\put(122,-6){$H_5$}\n \\end{picture}\n\n\\end{center}\n\\caption{Six  $5$-chordal graphs with hyperbolicity\n$1$.}\\label{fig0}\n\\end{figure}\n\n\n\n\n\n\n\n\n\\begin{te}\\label{main1}\n A $5$-chordal graph has hyperbolicity one  if and only if  one\nof $C_4,H_1,H_2,H_3,H_4, H_5$ appears  as an isometric subgraph of\nit.\n\\end{te}\n\n\n\n\n\nReturning to Remark \\ref{rem13}, it is natural to investigate if\nsome graphs mentioned in Theorem \\ref{main1} besides  $C_4$ can be\nused as ``good seeds''. The next example comes from Gavoille\n\\cite{CGa}.\n\n\n\n\n\\begin{figure}\n\\hspace{-33mm}\n\\unitlength 1mm\n\\linethickness{0.6pt}\n\\ifx\\plotpoint\\undefined\\newsavebox{\\plotpoint}\\fi \\hspace{30mm}\n\\begin{center}\n\\begin{picture}(50,40)\n\n\\put(20,0){\\circle*{1}}\\put(20,-4){\\makebox(0,0)[b]{$y$}}\n\\put(0,20){\\circle*{1}}\\put(-1,20){\\makebox(0,0)[r]{$u$}}\n\\put(40,20){\\circle*{1}}\\put(41,20){\\makebox(0,0)[l]{$v$}}\n\\put(20,40){\\circle*{1}}\\put(20,44){\\makebox(0,0)[t]{$x$}}\n\\put(10,30){\\circle*{1}}\\put(7,30){\\makebox(0,0)[l]{$a$}}\n\\put(10,10){\\circle*{1}}\\put(9,10){\\makebox(0,0)[r]{$c$}}\n\\put(30,10){\\circle*{1}}\\put(31,10){\\makebox(0,0)[l]{$d$}}\n\\put(30,30){\\circle*{1}}\\put(31,30){\\makebox(0,0)[l]{$b$}}\n\\put(5,25){\\circle*{1}}\\put(4,25){\\makebox(0,0)[r]{$u_a$}}\n\\put(5,15){\\circle*{1}}\\put(4,15){\\makebox(0,0)[r]{$u_c$}}\n\\put(15,5){\\circle*{1}}\\put(14,5){\\makebox(0,0)[r]{$y_c$}}\n\\put(25,5){\\circle*{1}}\\put(26,5){\\makebox(0,0)[l]{$y_d$}}\n\\put(35,15){\\circle*{1}}\\put(36,15){\\makebox(0,0)[l]{$v_d$}}\n\\put(35,25){\\circle*{1}}\\put(36,25){\\makebox(0,0)[l]{$v_b$}}\n\\put(15,35){\\circle*{1}}\\put(14,35){\\makebox(0,0)[r]{$x_a$}}\n\\put(25,35){\\circle*{1}}\\put(26,35){\\makebox(0,0)[l]{$x_b$}}\n\\qbezier[30](0,20)(10,30)(20,40)\\qbezier[30](20,40)(30,30)(40,20)\\qbezier[30](20,0)(30,10)(40,20)\\qbezier[30](20,0)(10,10)(0,20)\n\\qbezier[20](10,30)(10,30)(10,10)\n\\qbezier[20](10,30)(20,30)(30,30)\\qbezier[20](30,10)(30,10)(30,30)\n\\qbezier[20](10,10)(30,10)(30,10)\n\\qbezier(5,15)(5,20)(5,25)\\qbezier(15,5)(20,5)(25,5)\\qbezier(35,15)(35,20)(35,25)\\qbezier(15,35)(20,35)(25,35)\n\\qbezier[20](10,30)(20,20)(30,10)\\qbezier[20](10,10)(20,20)(30,30)\n \\end{picture}\n\\end{center}\n\\caption{$\\mathbb{G}_{4t}^q$.}\\label{example13}\n\\end{figure}\n\n\n\n\\begin{example}\\label{ep} \\cite{CGa} Let $t,q$ be two positive integer with\n$q<t$ and let $H_2$ be the graph shown in the  upper-right corner of\nFig. \\ref{fig0}. We construct a planar graph $\\mathbb{G}_{4t}^q$\nfrom $S^t(H_2)$ as follows: let $u_a=n_{u,a}^{q}$,\n$u_c=n_{c,u}^{q-1}$, $y_c=n_{y,c}^{q}$, $y_d=n_{d,y}^{q-1}$,\n$v_d=n_{v,d}^{q}$, $v_b=n_{b,v}^{q-1}$, $x_b=n_{x,b}^{q}$,\n$x_a=n_{a,x}^{q-1}$, and then add the new edges\n  $\\{u_a, u_c\\}$, $\\{x_a, x_b\\}$, $\\{y_c,\n y_d\\}$, $\\{v_b, v_d\\}$ to  $S^t(H_2)$; see Fig. \\ref{example13}. It\ncan be checked  that  $C=[u_a \\cdots a \\cdots x_ax_b\\cdots b \\cdots\nv_bv_d\\cdots d\\cdots y_dy_c\\cdots c \\cdots u_c]$ is an isometric\n  $4t$-cycle of $\\mathbb{G}_{4t}^q$ and that $\\mathbbm{l}\\mathbbm{c}(\\mathbb{G}_{4t}^q) =4t.$\nIt is also easy to see that $\\delta_{\\mathbb{G}_{4t}^q}(u,y,v,x)=t $\nand thus  Theorem \\ref{main} tells us that\n$\\delta^*(\\mathbb{G}_{4t}^q)=t$.\n\\end{example}\n\nMotivated by the above construction of Gavoille, we construct the\nnext graph family whose chordality parameters are $1$ modulo $4$.\n\n\n\n\\begin{example}\nBy deleting the edge $\\{ y_c, n_{d,y}^{q-1}\\}$ and adding a new edge\n$\\{ y_c,  n_{d,y}^{q}\\}$, we obtain   from $\\mathbb{G}_{4t}^q$ a\ngraph  $\\mathbb{G}_{4t+1}^q$. Using similar analysis like Example\n\\ref{ep}, we find that   $\\mathbbm{l}\\mathbbm{c}(\\mathbb{G}_{4t}^q)\n=4t+1$ and $\\delta^*(\\mathbb{G}_{4t+1}^q)=t=\\frac{\\lfloor\n\\frac{4t+1}{2}\\rfloor }{2}$.\n\\end{example}\n\n\n\nSimilar constructions using $F_2$ (see Fig. \\ref{counter})    as the\n``seed\" will lead to corresponding extremal graphs whose chordality\nparameters are $2$ or $3$ modulo $4$.\n\n\n\\begin{figure}\n\\hspace{-33mm}\n\\unitlength 1mm\n\\linethickness{0.4pt}\n\\ifx\\plotpoint\\undefined\\newsavebox{\\plotpoint}\\fi \\hspace{30mm}\n\\begin{center}\n\\begin{picture}(150,30)\n\n\n\\put(60,25){\\circle*{1}}\\put(60,28){\\makebox(0,0)[t]{$v_8$}}\n\n\\put(55,15){\\circle*{1}}\\put(51,15){\\makebox(0,0)[l]{$v_7$}}\n\n\\put(53,11){\\circle*{1}}\\put(52,11){\\makebox(0,0)[r]{$v_{67}$}}\n\n\\put(55,5){\\circle*{1}}\\put(55,4){\\makebox(0,0)[t]{$v_{65}$}}\n\n\\put(50,5){\\circle*{1}}\\put(50,1){\\makebox(0,0)[b]{$v_6$}}\n \\put(60,5){\\circle*{1}}\\put(61,1){\\makebox(0,0)[b]{$v_5$}}\n\\put(70,25){\\circle*{1}}\\put(69,26){\\makebox(0,0)[b]{$v_1$}}\n\\put(78,21){\\circle*{1}}\\put(79,21){\\makebox(0,0)[l]{$v_{23}$}}\n\\put(75,25){\\circle*{1}}\\put(75,26){\\makebox(0,0)[b]{$v_{21}$}}\n\\put(80,25){\\circle*{1}}\\put(81,28){\\makebox(0,0)[t]{$v_2$}}\n\\put(75,15){\\circle*{1}}\\put(79,15){\\makebox(0,0)[r]{$v_3$}}\n\n\\put(70,5){\\circle*{1}}\\put(70,1){\\makebox(0,0)[b]{$v_4$}}\n\n\\qbezier[20](60,25)(55,15)(50,5)\\qbezier[20](80,25)(75,15)(70,5)\n\\qbezier[20](55,15)(58,9)(60,5)\\qbezier[20](70,25)(73,19)(75,15)\\qbezier[20](50,5)(60,5)(70,5)\n\\qbezier[20](60,25)(70,25)(80,25)\\qbezier(53,11)(53,11)(55,5)\\qbezier(78,21)(78,21)(75,25)\n\\end{picture}\n\\end{center}\n\n\\caption{$\\mathbb{G}_{6(2t+1)}^q$.}\\label{expo}\n\\end{figure}\n\n\n\\begin{example} \\label{nuli} Let $t>q$ be two positive integers. We\nconstruct an outerplanar graph $\\mathbb{G}_{6(2t+1)}^q$ by adding\ntwo new edges   $\\{v_{21}, v_{23}\\}$  and  $\\{v_{65},\n v_{67}\\}$\n to the graph $S^{2t+1}(F_2)$\n where\n  $v_{21}=n_{v_2,v_1}^{q}$, $v_{23}=n_{v_3,v_2}^{q-1}$,\n$v_{65}=n_{v_6,v_5}^{q}$, $v_{67}=n_{v_7,v_6}^{q-1}$; see Fig.\n \\ref{expo} for an illustration. It is not hard to check that $\\mathbbm{l}\\mathbbm{c}(\\mathbb{G}_{6(2t+1)}^q)\n=6(2t+1)$ and  $\\delta^*(\\mathbb{G}_{6(2t+1)}^q)=3t+\\frac{3}{2}$.\nMoreover, if we replace the edge $\\{v_{21}, v_{23}\\}$\n by the edge $\\{v_{21}, n_{v_3,v_2}^{q}\\}      $, then we obtain from $\\mathbb{G}_{6(2t+1)}^q$\n another outerplanar graph $\\mathbb{G}_{6(2t+1)+1}^q$ for which we\n have\n$\\mathbbm{l}\\mathbbm{c}(\\mathbb{G}_{6(2t+1)+1}^q) =6(2t+1)+1$ and\n$\\delta^*(\\mathbb{G}_{6(2t+1)+1}^q)=3t+\\frac{3}{2}$.\n\\end{example}\n\n\n\n\n\n\n\n\nLet  $C_6,G_1,G_2,G_3$ be the graphs depicted in Fig.\n \\ref{figconjecture}. It is clear that\n $G_1,G_2,G_3,C_4,C_6,H_i,i=1,\\ldots,5,$ are $6$-chordal graphs with\n hyperbolicity $1$.\n\n\n\n\n\n\n\\begin{figure}\n\\hspace{-33mm}\n\\unitlength 1mm\n\\linethickness{0.4pt}\n\\ifx\\plotpoint\\undefined\\newsavebox{\\plotpoint}\\fi \\hspace{30mm}\n\\begin{center}\n\\begin{picture}(120,60)\n\n\\put(0,25){\\circle*{1}}\n\n\\put(10,15){\\circle*{1}} \\put(20,25){\\circle*{1}}\n\n\\put(20,5){\\circle*{1}}\n\n\\put(30,5){\\circle*{1}} \\put(30,25){\\circle*{1}}\n\\put(50,25){\\circle*{1}}\\put(40,15){\\circle*{1}}\n\n\\put(25,15){\\circle*{1}}\n\n\n\\qbezier(0,25)(10,15)(20,5)\\qbezier(20,5)(25,5)(30,5)\\qbezier(30,5)(40,15)(50,25)\\qbezier(50,25)(20,25)(0,25)\n\\qbezier(20,5)(25,15)(30,25)\n\\qbezier(20,25)(25,15)(30,5)\\qbezier(10,15)(25,15)(40,15)\\qbezier(20,25)(15,20)(10,15)\\qbezier(40,15)(35,20)(30,25)\n\n\\put(25,-5){\\makebox(0,0){$G_1$}}\n\n\n\n\\put(60,25){\\circle*{1}}\n\n\\put(55,15){\\circle*{1}} \\put(65,15){\\circle*{1}}\n\n\\put(50,5){\\circle*{1}}\n\n\\put(60,5){\\circle*{1}} \\put(70,25){\\circle*{1}}\n\\put(80,25){\\circle*{1}}\\put(75,15){\\circle*{1}}\n\n\\put(70,5){\\circle*{1}}\n\n\n\\qbezier(60,25)(55,15)(50,5)\\qbezier(60,5)(65,15)(70,25)\\qbezier(80,25)(75,15)(70,5)\\qbezier(55,15)(65,15)(75,15)\n\\qbezier(60,25)(65,15)(70,5)\n\\qbezier(70,25)(65,15)(60,5)\\qbezier(55,15)(58,9)(60,5)\\qbezier(70,25)(73,19)(75,15)\\qbezier(50,5)(60,5)(70,5)\n\\qbezier(60,25)(70,25)(80,25)\n\n\\put(60,-5){\\makebox(0,0){$G_2$}}\n\n\\put(55,45){\\circle*{1}}\\put(75,55){\\circle*{1}}\\put(75,35){\\circle*{1}}\\put(85,45){\\circle*{1}}\n\n\\put(65,55){\\circle*{1}}\\put(70,45){\\circle*{1}}\n\n\\put(65,35){\\circle*{1}}\n\\qbezier(55,45)(60,50)(65,55)\\qbezier(55,45)(60,40)(65,35)\\qbezier(65,35)(70,35)(75,35)\n\\qbezier(75,35)(80,40)(85,45)\\qbezier(75,55)(80,50)(85,45)\\qbezier(65,55)(70,55)(75,55)\n\\qbezier(65,55)(70,45)(75,35)\n\n\\put(70,30){\\makebox(0,0){$G_3$}}\n\n\\put(5,45){\\circle*{1}}\\put(15,35){\\circle*{1}}\\put(25,35){\\circle*{1}}\n\\put(35,45){\\circle*{1}}\\put(15,55){\\circle*{1}}\\put(25,55){\\circle*{1}}\n\\qbezier(15,35)(10,40)(5,45)\\qbezier(15,35)(20,35)(25,35)\\qbezier(25,35)(30,40)(35,45)\n\\qbezier(25,55)(30,50)(35,45)\\qbezier(15,55)(20,55)(25,55)\\qbezier(5,45)(10,50)(15,55)\n\\put(20,30){\\makebox(0,0){$C_6$}}\n \\end{picture}\n\\end{center}\n\n\\caption{Four  graphs with hyperbolicity $1$ and chordality\n$6$.}\\label{figconjecture}\n\\end{figure}\n\n\n\\begin{conjecture} \\label{conj14}  A  $6$-chordal graph   is\n$\\frac{1}{2}$-hyperbolic if and only if it does not contain any of a\nlist of ten special graphs $G_1,G_2,G_3,C_4,C_6,H_i,i=1,\\ldots,5,$\nas an isometric subgraph.\n\\end{conjecture}\n\n\n\n\n\n\n\n\n\n\nLet  $E_1$ and $E_2$ be the graphs  depicted  in Fig. \\ref{fig\nbridged}.     In comparison with Conjecture  \\ref{conj14}, when we\nremove the $6$-chordal restriction, we can present the following\ncharacterization   of all $\\frac{1}{2}$-hyperbolic graphs obtained\nby Bandelt and Chepoi \\cite{BC}.  We refer to \\cite[Fact 1]{BC} for\ntwo other characterizations; also see \\cite{FJ,SC}.\n\n\n \\begin{figure}\n\\hspace{-33mm}\n\\unitlength 1mm\n\\linethickness{0.4pt}\n\\ifx\\plotpoint\\undefined\\newsavebox{\\plotpoint}\\fi \\hspace{30mm}\n\\begin{center}\n\\begin{picture}(80,40)\n\\put(10,25){\\circle*{1}}\n\n\\put(5,15){\\circle*{1}} \\put(20,30){\\circle*{1}}\n\n\\put(0,5){\\circle*{1}}\n\n\\put(10,5){\\circle*{1}} \\put(20,15){\\circle*{1}}\n\\put(30,25){\\circle*{1}}\n\\put(20,5){\\circle*{1}}\\put(30,5){\\circle*{1}}\n\\put(30,35){\\circle*{1}}\\put(30,15){\\circle*{1}}\n\n\\qbezier(10,25)(5,15)(0,5)\\qbezier(10,5)(20,15)(30,25)\\qbezier(30,35)(30,15)(30,5)\\qbezier(5,15)(20,15)(30,15)\n\\qbezier(10,25)(15,20)(20,15)\n\\qbezier(20,30)(20,15)(20,5)\\qbezier(5,15)(8,9)(10,5)\\qbezier(10,25)(20,30)(30,35)\\qbezier(0,5)(10,5)(30,5)\n\\qbezier(20,5)(25,10)(30,15)\\qbezier(20,30)(26,27)(30,25)\n\n\\put(15,-5){\\makebox(0,0){$E_1$}}\n\n\\put(40,5){\\circle*{1}}\\put(50,5){\\circle*{1}}\\put(60,5){\\circle*{1}}\\put(70,5){\\circle*{1}}\n\n\\put(40,15){\\circle*{1}}\\put(70,15){\\circle*{1}}\\put(40,25){\\circle*{1}}\n\n\\put(70,25){\\circle*{1}}\\put(40,35){\\circle*{1}}\\put(50,35){\\circle*{1}}\\put(60,35){\\circle*{1}}\n\\put(70,35){\\circle*{1}}\\put(55,20){\\circle*{1}}\n\n\\qbezier(40,5)(60,5)(70,5)\\qbezier(40,5)(40,10)(40,35)\\qbezier(40,35)(50,35)(70,35)\n\\qbezier(70,35)(70,15)(70,5)\\qbezier(40,15)(55,20)(70,25)\\qbezier(40,25)(55,20)(70,15)\n\\qbezier(60,35)(55,20)(50,5)\\qbezier(50,35)(55,20)(60,5)\\qbezier(50,35)(45,30)(40,25)\n\\qbezier(40,15)(45,10)(50,5)\\qbezier(60,35)(65,30)(70,25)\n\n\\qbezier(60,5)(65,10)(70,15)\n\n\\put(55,-5){\\makebox(0,0){$E_2$}}\n \\end{picture}\n\\end{center}\n\n\\caption{Two bridged graphs with hyperbolicity $1$.}\\label{fig\nbridged}\n\\end{figure}\n\n\n\n\\begin{te}  \\cite[p. 325]{BC}  A graph  $G$ is $\\frac{1}{2}$-hyperbolic if and only if\n$G$ does not contain isometric $n$-cycles for any $n>5$,  for any\ntwo vertices $x$ and $y$ of  $G$ one cannot find two non-adjacent\nneighbors of  $x$  which are both closer to $y$ in  $G$ than $x$,\nand none of the six graphs $H_1,H_2,G_1,G_2,E_1,E_2$ occurs as an\nisometric subgraph of $G.$ \\label{thmBC}\n\\end{te}\n\n\n\n\\begin{remark} Instead of Theorem \\ref{thmBC}, it would be interesting to determine, if possible, a finite\n list of graphs such  that that a graph is $\\frac{1}{2}$-hyperbolic if and only if\nit does not include any graph from that list as an isometric\nsubgraph.  Koolen and Moulton point out a possible approach to\ndeduce such kind of a characterization in \\cite[p. 696]{KM02}.\n\\end{remark}\n\n\n\n\n\nNote that a $5$-chordal graph cannot contain any  isometric\n$n$-cycle for $n>5$.  It is also easy to see that\n$\\mathbbm{l}\\mathbbm{c}(G_1)=\\mathbbm{l}\\mathbbm{c}(G_2)=6,\\mathbbm{l}\\mathbbm{c}(E_1)=7,\n\\mathbbm{l}\\mathbbm{c}(E_2)=8$. Therefore, we obtain the following\neasy consequence of Theorem \\ref{thmBC}. It is interesting to\ncompare it with Theorems \\ref{main} and \\ref{main1}.\n\n\n\\begin{corollary}   A  $5$-chordal graph $G$ is $\\frac{1}{2}$-hyperbolic if and only if it does not\ncontain the graph   $H_1$ and  $H_2$ as isometric subgraphs and for\nany two vertices $x$ and $y$ of  $G$ the neighbors of  $x$ which are\ncloser to $y$ than  $x$ are pairwise adjacent.\n\\end{corollary}\n\n\n\n\n\n\n\\subsection{Some consequences}\\label{oct}\n\nNote that  $\\mathbbm{l}\\mathbbm{c}(C_4)=4,\n\\mathbbm{l}\\mathbbm{c}(H_1)=\\mathbbm{l}\\mathbbm{c}(H_2)=3,\n\\mathbbm{l}\\mathbbm{c}(H_3)=\\mathbbm{l}\\mathbbm{c}(H_4)=\\mathbbm{l}\\mathbbm{c}(H_5)=5.$\nThe next two results follow immediately from Theorem \\ref{main1}.\n\n\n\\begin{corollary}   Every  $4$-chordal graph must be $1$-hyperbolic and it has\nhyperbolicity one  if and only if it contains one of $C_4$, $H_1$\nand $H_2$ as an isometric subgraph.  \\label{cor7}\n\\end{corollary}\n\n\n\n\\begin{corollary}   \\cite[Theorem 1.1]{BKM01}  Every  chordal\ngraph is $1$-hyperbolic and it has hyperbolicity one if and only if\nit contains either $H_1$ or $H_2$ as an isometric\nsubgraph.\\label{BKM}\n\\end{corollary}\n\nWe remark that as long as every $4$-chordal graph is 1-hyperbolic is\nknown, Corollary \\ref{cor7} also immediately follows from Corollary\n\\ref{BKM}.  In addition, it is noteworthy  that the first part of\nCorollary \\ref{BKM}, namely every chordal graph is $1$-hyperbolic is\nimmediate from Theorem \\ref{thm10} as chordal graphs have\ntree-length $1$.\n\n\n\\begin{corollary}  Each  weakly chordal graph is $1$-hyperbolic and has\nhyperbolicty one if and only if it contains one of $C_4,H_1,H_2$ as\nan isometric subgraph.\n\\end{corollary}\n\n\\begin{proof} By definition, each weakly chordal graph is $4$-chordal. It is also easy to check that   that $C_4,H_1$ and $H_2$\nare all weakly chordal.  Hence, the result follows from  Corollary\n\\ref{cor7}.\n\\end{proof}\n\n\n\n\\begin{corollary} All strongly chordal graphs are\n$\\frac{1}{2}$-hyperbolic.\n\\end{corollary}\n\\begin{proof} Note that the cycle $C=[x,a,u,c,y,d,v,b]$ in $H_1$\nand $H_2$  does not have any odd chord and hence neither  $H_1$ nor\n$H_2$ can appear as an isometric subgraph of a strongly chordal\ngraph.  Since strongly chordal graphs must be chordal graphs, this\nresult holds by Corollary \\ref{BKM}.\n\\end{proof}\n\n\n\n\n\n\\begin{corollary} All threshold graphs are  $\\frac{1}{2}$-hyperbolic.\n\\end{corollary}\n\\begin{proof} It is obvious that    threshold graphs are chordal as they   contain neither   $4$-cycle\nnor path of length $3$ as induced subgraph.  Since the subgraph\ninduced by  $x,u,b,c$ in either $H_1$  or  $H_2$ is just the\ncomplement of $C_4$,   the result follows from Corollary \\ref{BKM}\nand the definition of a threshold graph.\n\\end{proof}\n\n\\begin{corollary}  Every  $AT$-free graph  is  $1$-hyperbolic and it has hyperbolicity one if and only if it contains $C_4$\nas an isometric subgraph. \\label{AT-free}\n\\end{corollary}\n\\begin{proof}   First observe that an\n $AT$-free graph  must be $5$-chordal. Further notice that\nthe triple $u,y,v$ is an $AT$ in any of the graphs $H_1,\\ldots,H_5.$\nNow, an application of Theorem  \\ref{main1}  concludes the proof.\n\\end{proof}\n\n\\begin{corollary} \\label{com}   A cocomparability graph  is  $1$-hyperbolic and has hyperbolicity one if and only if it contains $C_4$\nas an isometric subgraph.\n\\end{corollary}\n\\begin{proof} We know that cocomparability graphs are $AT$-free and\n $C_4$ is a cocomparability graph. Thus the result\ncomes directly  from Corollary  \\ref{AT-free}.   The deduction of\nthis result can also be made via Corollary \\ref{cor7} and the fact\nthat cocomparability graphs are $4$-chordal \\cite{BT1, Gallai}.\n\\end{proof}\n\n\n\n\n\n\\begin{corollary}  A permutation graph is $1$-hyperbolic and has hyperbolicity one if and only\nif it contains $C_4$ as an isometric subgraph.\n\\end{corollary}\n\n\\begin{proof} Every permutation graph is a cocomparability graph and   $C_4$ is a permutation\ngraph. So,  the result follows from Corollary \\ref{com}.\n\\end{proof}\n\n\n\n\n\n\\begin{corollary} \\cite[p. 16]{BC08}\nA distance-hereditary graph\n is always   $1$-hyperbolic and is    $\\frac{1}{2}$-hyperbolic exactly  when  it  is\n chordal, or equivalently, when it contains no induced $4$-cycle.\n \\label{distance}\n \\end{corollary}\n\n \\begin{proof} It is easy to see that distance-hereditary graphs\n must be $4$-chordal and can contain  neither  $H_1$ nor $H_2$ as an\nisometric subgraph. The result now follows from Corollary\n\\ref{cor7}.\n\\end{proof}\n\n\n\n\n\n\n\\begin{corollary} A cograph is $1$-hyperbolic and has hyperbolicity one if  and only if it contains\n$C_4$ as an isometric subgraph.\n\\end{corollary}\n\n\\begin{proof} We know that $C_4$ is a cograph and every  cograph is   ditance-hereditary. Applying  Corollary\n\\ref{distance}  yields  the  required result.\n\\end{proof}\n\n\n\n\n\n\n\n\n\\section{Relevant tree-likeness parameters}\\label{parameter}\n\n\\subsection{Tree-length}\n\n\n\n\n\n\nIt turns out that tree-length is a very useful concept for\nconnecting chordality and hyperbolicity.\n Indeed, the following theorem, which can be read from  Theorem \\ref{main} (Corollary \\ref{BKM}), comes directly from\n Theorems\n\\ref{thm9} and   \\ref{thm10}. This result is firstly notified to us\nby Dragan \\cite{Dragan} and is presumably in the folklore.\n\n\\begin{te} For any $k\\geq 3,$ every $k$-chordal graph is  $\\lfloor \\frac{k}{2}\\rfloor $-hyperbolic.\\label{June}\n\\end{te}\n\n\n\nIn view of Remark \\ref{grid}, to get better estimate than  Theorem\n\\ref{June} along the same approach\n  one may try to beef\nup   Theorem \\ref{thm9}. We point out that Dourisboure  and Gavoille\n\\cite[Question 1]{DG} posed as an open problem that whether or not\n\\begin{equation}  \\label{eq22} \\mathbbm{t}\\mathbbm{l}(G)\\leq \\lceil \\frac{\\mathbbm{l}\\mathbbm{c}(G)}{3}\\rceil\n\\end{equation} is true.\nThe {\\em $k$th-power} of a graph $G$, denoted  $G^k,$ is the graph\nwith $V(G)$ as vertex set and there is an edge connecting two\nvertices $u$ and $v$ if and only if $d_G(u,v)\\leq k.$  Let us\ninterpret the problem of Dourisboure  and Gavoille as a Chordal\nGraph Sandwich Problem:\n\n\\begin{question}\nFor any  graph  $G$,   is there always a chordal  graph $H$  such\nthat  $V(H)=V(G)=V(G^{\\lceil\n\\frac{\\mathbbm{l}\\mathbbm{c}(G)}{3}\\rceil})$ and $E(G)\\subseteq E(H)\n\\subseteq G^{\\lceil \\frac{\\mathbbm{l}\\mathbbm{c}(G)}{3}\\rceil}$?\n\\end{question}\n\n\nIf  \\eqref{eq22} can be established, it will be the best we can\nexpect in the sense that $\\mathbbm{t}\\mathbbm{l}(G)=\\lceil\n\\frac{\\mathbbm{l}\\mathbbm{c}(G)}{3}\\rceil$   for every outerplanar\ngraph $G$   \\cite[Theorem 1]{DG}.\n\n\n\n\n\n\n\n\n\n\\subsection{Approximating trees, slimness and thinness}\n\n\n\n\nWe introduce in this subsection two general approaches to connect\nchordality with hyperbolicity. A result is given together with a\nproof only when that proof is short and  when we do not find it\nappear very explicitly elsewhere. This section also aims to provide\nthe reader a warm-up before entering the longer proof in the main\npart of this paper.\n\n\nA\n result   weaker than Theorem \\ref{main} (Theorem \\ref{June}) and reported in\n\\cite[p. 64]{CE} as well as \\cite[p. 3]{CDEHVX} is that  each\n$k$-chordal graph is $k$-hyperbolic. The two approaches to be\nreported below by far basically only lead to\n this weaker result.\n Despite of this, it might be interesting  to see  different ways of bounding\n  hyperbolicity  in terms of chordality via the use of some other intermediate tree-likeness parameters.\n\nThe first approach is to look at distance approximating trees. A\ntree $T$ is a {\\em distance $t$-approximating tree} of a graph $G$\nprovided $V(T)=V(G)$ and $|d_G(u,v)-d_T(u,v)|\\leq t$ for any $u,v\\in\nV(G)$ \\cite{Balint,BCD,CD,DY}. It is well-known that a graph with a\ngood distance approximating tree    will have low hyperbolicity,\nwhich is briefly mentioned in  \\cite[p. 3]{CDEHVX}   and  \\cite[p.\n64]{CE}   and is in the same spirit of a general result on\nhyperbolic geodesic metric spaces \\cite[p. 402, Theorem 1.9]{BH}. We\nmake this point clear in the following simple lemma.\n\n\n\n\n\n\\begin{lemma}\\label{lem1}  Let $G$   be a graph and $t$ be a nonnegative integer.    If $G$  has a distance $t$-approximating tree\n$T$, then $G$ is $2t$-hyperbolic.\n\\end{lemma}\n\\begin{proof}\nFor any $x,y,u,v\\in V(G)$, our aim is to show that\n$\\delta_G(x,y,u,v)\\leq 2t.$    Assume, as we may, that\n$d_G(x,y)+d_G(u,v)\\geq d_G(x,u)+d_G(y,v)\\geq d_G(x,v)+d_G(y,u).$\nSince the tree metric  $d_T$ is a four-point  inequality   metric\n(or additive metric)   \\cite{DD}, we know that $\\delta^*(T)=0$ and\nso the following three cases are exhaustive.\n\n\n\n\\paragraph {\\sc Case 1:}\n$d_T(x,y)+d_T(u,v)= d_T(x,u)+d_T(y,v)\\geq d_T(x,v)+d_T(y,u).$\n\n\n\n\n\n\n\n$\\delta_G(x,y,u,v)=\\frac{1}{2}(d_G(x,y)+d_G(u,v))\n-\\frac{1}{2}(d_G(x,u)+d_G(y,v))\\leq \\frac{1}{2}(d_T(x,y)+d_T(u,v)+\n2t ) -\\frac{1}{2}(d_T(x,u)+d_T(y,v)-   2t)=2t. $\n\n\n\\paragraph {\\sc Case 2:}\n$d_T(x,y)+d_T(u,v)= d_T(x,v)+d_T(y,u)\\geq d_T(x,u)+d_T(y,v) $\n\n\n\n\n\n$\\delta_G(x,y,u,v)=\\frac{1}{2}(d_G(x,y)+d_G(u,v))\n-\\frac{1}{2}(d_G(x,u)+d_G(y,v))\\leq \\frac{1}{2}(d_G(x,y)+d_G(u,v))\n-\\frac{1}{2}(d_G(x,v)+d_G(y,u))\\leq  \\frac{1}{2} (d_T(x,y)+d_T(u,v)+\n2t ) -\\frac{1}{2}(d_T(x,v)+d_T(y,u)-   2t) = 2t. $\n\n\n\n\n\n\n\\paragraph {\\sc Case 3:}  $ d_T(x,v)+d_T(y,u)= d_T(x,u)+d_T(y,v) \\geq   d_T(x,y)+d_T(u,v).$\n\n\n\n\n$\\delta_G(x,y,u,v)=\\frac{1}{2}(d_G(x,y)+d_G(u,v))\n-\\frac{1}{2}(d_G(x,u)+d_G(y,v))\\leq \\frac{1}{2}(d_T(x,y)+d_T(u,v)+\n2t) -\\frac{1}{2}(d_T(x,u)+d_T(y,v)-  2t)\\leq\n\\frac{1}{2}(d_T(x,u)+d_T(y,v)+ 2t ) -\\frac{1}{2}(d_T(x,u)+d_T(y,v)-\n2t)= 2t. $\n\\end{proof}\n\n\nAfter showing that the existence of good distance approximating tree\n  guarantees low hyperbolicity, in order  to   connect chordality\nwith hyperbolicity,  we need to make sure  that low chordality\ngraphs have good distance approximating trees \\cite{BCD,CD}  . Here\nis an exact result.\n\n\n\n\n\\begin{te}  \\cite{CD}   Let  $G$ be a $k$-chordal graph. Then, there is a tree\n$T$ with $V(T)=V(G)$  such that for any $u,v\\in V(G)$ it holds\n\\begin{equation*} \\label{eq:1} |d_G(u,v)-d_T(u,v)|\\leq \\left\\{\n\\begin{aligned}\n          \\lfloor\n\\frac{k}{2 }\\rfloor +2, & \\ \\ \\ \\ \\ \\text{if}\\ \\   k=4,5, \\\\\n               \\lfloor  \\frac{k}{2 }\\rfloor +1, &  \\ \\ \\ \\ \\ \\text{else.}\n                          \\end{aligned} \\right.\n                          \\end{equation*}\n                           \\label{Chepoi}\n\\end{te}\n\n\n\nThe other  possible approach to connect hyperbolicity and chordality\nis via  the concept of the thinness\/slimness of geodesic triangles.\nThis approach also consists of two parts, one is to show that a\ngraph with low thinness\/slimness has low hyperbolicity, as\nsummarized in \\cite[Proposition 1]{CDEHV}, and the other part is to\nshow that low chordality implies low thinness\/slimness.\n\n\n\n\\begin{figure}\n\\hspace{-33mm}\n\\unitlength 1mm\n\\linethickness{0.4pt}\n\\ifx\\plotpoint\\undefined\\newsavebox{\\plotpoint}\\fi \\hspace{30mm}\n\\begin{center}\n\\begin{picture}(30,20)\n\\put(10,5.5){\\circle{11}}\n\\put(0,0){\\circle*{1}}\\put(0,-5){\\makebox(0,0)[b]{$x$}}\n\\put(20,0){\\circle*{1}}\\put(20,-5){\\makebox(0,0)[b]{$y$}}\n\\put(0,0){\\line(1,0){20}}\n\\put(10,17.3){\\circle*{1}}\\put(10,22){\\makebox(0,0)[t]{$z$}}\n\\put(5,8.5){\\circle*{1}}\\put(-5,8.5){\\makebox(0,0)[l]{$m_y^{z,x}$}}\n\\put(15,8.5){\\circle*{1}}\\put(25,8.5){\\makebox(0,0)[r]{$m_x^{y,z}$}}\n\\put(10,0){\\circle*{1}}\\put(10,-5){\\makebox(0,0)[b]{$m_z^{x,y}$}}\n\\qbezier(0,0)(5,8.5)(10,17.3)\\qbezier(20,0)(15,8.5)(10,17.3)\n \\end{picture}\n\\end{center}\n\\caption{An illustration of Gromov product.}\\label{fig-1}\n\\end{figure}\n\n\n\nGiven a   graph $G,$ we can put an orientation on it by choosing two\nmaps $\\partial_0$ and $\\partial_1$ from  $E(G)$ to $V(G)$ such that\neach edge $e$ just have $\\partial_0(e)$ and $\\partial_1(e)$ as its\ntwo endpoints.  The discrete metric space $(V(G),d_G)$ can then be\nnaturally embedded into the {\\em metric graph} \\cite[p. 7]{BH}\n$(X_G, \\widetilde{d_G})$, where $X_G$ is the quotient space of\n$E(G)\\times [0,1]$ under the identification of $(e,i)$ and $(e',i')$\nwhenever $\\partial _i(e)=\\partial_{i'}(e')$ for any $e,e'\\in E(G)$\nand $i,i'\\in \\{0,1\\},$ and $\\widetilde{d_G}$ is the   metric on\n$X_G$ satisfying $\\widetilde{d_G}((e,t),(e,t'))=\n    |t-t'|$ if $ e=e'$ and  $\\widetilde{d_G}((e,t),(e,t'))=\n\\min (d_G(\\partial_0(e),\\partial_0(e'))+t+t',\nd_G(\\partial_0(e),\\partial_1(e'))+t+1-t',\nd_G(\\partial_1(e),\\partial_0(e'))+1-t+t',d_G(\\partial_1(e),\\partial_1(e'))+2-t-t'\n)$ else. It is easy to see that the definition of  $(X_G,\n    \\widetilde{d_G})$ is indeed independent of the orientation of\n    $G.$  Also, any cycle of $G$ naturally corresponds to a\n    circle, namely one-dimensional sphere, embedded in $X_G.$\nFor any two points   $x,y\\in X_G$, there is a not necessarily unique\ngeodesic connecting them in $(X_G, \\widetilde{d_G})$, which we will\nuse the notation $[x,y]$ if no ambiguity arises.\n   We say that $(x,y,z)$ is {\\em $(\\delta_1,\\delta_2)$-thin} provided for any choice\n   of the geodesics $[x,y], [y,z], [z,x]$ and  $m_x^{y,z}\\in [y,z],m_y^{z,x}\\in [z,x],\n   m_z^{x,y}\\in [x,y]$ satisfying \\begin{equation}\\left\\{ \\begin{array}{l }\n    \\widetilde{d_G}(m_z^{x,y},x)=\\widetilde{d_G}(m_y^{z,x},x)=(y\\cdot z)_x,\\\\\n\\widetilde{d_G}(m_x^{y,z},y)=\\widetilde{d_G}(m_z^{x,y},y)=(z\\cdot x)_y,\\\\\n   \\widetilde{d_G}(m_y^{z,x},z)=\\widetilde{d_G}(m_x^{y,z},z)=(x\\cdot\n   y)_z,\n\\end{array}\\right.\\label{E3}\n\\end{equation}\n(Figure \\ref{fig-1} is an illustration of \\eqref{E3} as well as a\nwidely-used geometric interpretation of the Gromov product.)  the\nfollowing two conditions hold:\n   \\begin{itemize} \\item[(A)]\n$\\delta_1 \\geq \\min (\\widetilde{d_G}(m_x^{y,z},m_y^{z,x}),\n\\widetilde{d_G}(m_z^{x,y},   m_y^{z,x}))$;\n\\item[(B)]   $\\{p\\in X_G:\\ \\widetilde{d_G}(p,[y,z]\\cup [y,x])\\leq \\delta_2 \\}\\supseteq [x,z].$\n\\end{itemize}\n  Modifying  the original definition of Gromov slightly  \\cite[p. 8, Definition 1.5]{Alonso} \\cite[p.\n408]{BH} \\cite{Gromov}, we say     that a graph  $G$ is {\\em\n$(\\delta_1,\\delta_2)$-thin} provided every triple $(x,y,z)$ of its\nvertices is $(\\delta_1,\\delta_2)$-thin.\n\n\n\n\n\n\n\\begin{lemma}   Let $G$ be a   graph. If $G$ is\n$(\\delta_1,\\delta_2)$-thin, then it is\n$(\\delta_1+\\delta_2)$-hyperbolic. \\label{lemma1}\n\\end{lemma}\n\\begin{proof}  The proof is taken from\n  \\cite[p. 15, (2) implies (5)]{Alonso}.\n     It suffices to establish \\eqref{EQ} for any  $x,y,u,v\\in V(G)$.\nBy Eq. \\eqref{E3} and    Condition (A) for the\n$(\\delta_1,\\delta_2)$-thinness of $(x,u,y)$, we have\n\\begin{equation}\\label{4}(x\\cdot y)_u+\\delta_1 \\geq \\min (   \\widetilde{d_G}(u,\nm_x^{u,y})+ \\widetilde{d_G}(m_x^{u,y}, m_u^{y,x}),\n  \\widetilde{d_G}(u,\nm_y^{x,u})+ \\widetilde{d_G}(m_y^{x,u}, m_u^{y,x}) ) \\geq\n\\widetilde{d_G}(u, m_u^{y,x}).\\end{equation} By Condition (B) for\nthe  $(\\delta_1,\\delta_2)$-thinness of $(x,v,y)$, we can suppose,\nwithout loss of generality,  that there is $q\\in [y,v]$ such that\n\\begin{equation}\\label{5} \\delta_2 \\geq\n\\widetilde{d_G}(m_u^{y,x},q).\\end{equation} It follows from\n$\\widetilde{d_G}(q,v)+  \\widetilde{d_G}(q,y)=\\widetilde{d_G}(y,v)$,\n$\\widetilde{d_G}(u,q)+\\widetilde{d_G}(q,v)\\geq\n\\widetilde{d_G}(u,v)$, and\n$\\widetilde{d_G}(u,q)+\\widetilde{d_G}(q,y)\\geq \\widetilde{d_G}(u,y)$\nthat\n\\begin{equation}\\label{6}\\widetilde{d_G}(u,q)\\geq (y\\cdot v)_u.\\end{equation}\nWe surely  have\n\\begin{equation}\\label{7} \\widetilde{d_G}(u,\nm_u^{y,x})+ \\widetilde{d_G}( m_u^{y,x},q)\\geq\n\\widetilde{d_G}(u,q).\n\\end{equation}\nTo complete the proof, we just need to add together \\eqref{4},\n\\eqref{5}, \\eqref{6}, and \\eqref{7}.\n\\end{proof}\n\nAccording to Gromov    \\cite{Gromov},  Rips invents the concept of\nslimness: For any real number $\\delta,$  we say that a graph $G$ is\n{\\em $\\delta$-slim} if for every triple $(x,y,z)$ of vertices of $G,\n$ we have\n\\begin{equation}\\label{eq7} \\{p\\in X_G:\\ \\widetilde{d_G}(p,[y,z]\\cup\n[y,x])\\leq \\delta \\}\\supseteq [x,z].\n\\end{equation}\nAn easy observation is  that a $(\\delta_1,\\delta_2)$-thin graph is\n$\\delta_2$-slim.\n It is mentioned in  \\cite[Proposition 1]{CDEHV}  that every $\\delta$-slim graph is\n$8\\delta$-hyperbolic.  The next lemma gives a better bound.\n\n\\begin{lemma} If a graph is $\\delta$-slim, it must be $(2\\delta,\n\\delta)$-thin and hence $3\\delta$-hyperbolic.\n\\end{lemma}\n\\begin{proof}  By Lemma \\ref{lemma1}, our task is to prove that any $\\delta$-slim graph $G$ is $(2\\delta,\n\\delta)$-thin. For this purpose, it suffices to deduce $2\\delta \\geq\n\\min (\\widetilde{d_G}(m_x^{y,z},m_y^{z,x}),\n\\widetilde{d_G}(m_z^{x,y}, m_y^{z,x}))$ for any triple $(x,y,z)$ of\nvertices of $G.$ The following argument is almost word-for-word the\nsame as that given in \\cite[p. 13, (1) implies (3)]{Alonso}. By\n\\eqref{eq7}, we suppose, as we may, that there is $w\\in [y,x]$ such\nthat $\\widetilde{d_G}(m_y^{z,x},w)\\leq \\delta.$ Observe that\n$$\\widetilde{d_G}(x,w)\\geq\n \\widetilde{d_G}(m_y^{z,x},x)-\\widetilde{d_G}(m_y^{z,x},w)\\geq\n \\widetilde{d_G}(m_y^{z,x},x)-\\delta=\n \\widetilde{d_G}(m_z^{x,y},x)-\\delta$$\n and that\n$$\\widetilde{d_G}(x,w)\\leq\n \\widetilde{d_G}(m_y^{z,x},x)+\\widetilde{d_G}(m_y^{z,x},w)\\leq\n \\widetilde{d_G}(m_y^{z,x},x)+\\delta=\n \\widetilde{d_G}(m_z^{x,y},x)+\\delta.$$\nIt then follows $\\widetilde{d_G}(m_z^{x,y},w)\\leq \\delta$ and\nhenceforth $\\widetilde{d_G}(m_z^{x,y},m_y^{z,x})\\leq\n  \\widetilde{d_G}(m_z^{x,y},w)+ \\widetilde{d_G}(w,m_y^{z,x})\\leq\n  2\\delta,$ as desired.\n\\end{proof}\n\n\n\\begin{lemma}\n Every   $k$-chordal graph is\n$ (\\frac{k}{2},  \\frac{k}{2})$-thin.\\label{lemma2}\n\\end{lemma}\n\\begin{proof}\nConsider any triple $(x,y,z)$ of vertices of  $G.$ By an abuse of\nnotation as usual, denote by $[x,y],[y,z]$ and $[z,x]$ three\ngeodesic segments joining the corresponding endpoints and put\n$m_x^{y,z}\\in [y,z],$ $ m_y^{z,x}\\in [z,x]$ and $m_z^{x,y}\\in [x,y]$\nbe three points of $X_G$ satisfying Eq. \\eqref{E3}. For any\nnonnegative number $t\\leq (y\\cdot z)_x$, there is a unique point $u$\nlying in $ [x,y]$ such that $\\widetilde{d_G}(u,x)=t$; we use the\nnotation $(z;x)_t$ for this point $u.$ Similarly, we define\n$(y;x)_t$ for any $0\\leq t\\leq (y\\cdot z)_x$ and so on.\n By symmetry, it\nsuffices to show that $\\widetilde{d_G}((z;x)_t,(y;x)_t)\\leq\n\\frac{k}{2} $  for any $0\\leq t\\leq (y\\cdot z)_x$. Take the maximum\n$t'\\leq t$ such that $(z;x)_{t'}=(y;x)_{t'}$. The case of $t'=t$ is\ntrivial and so we assume  that $t'<t.$\n\n\\paragraph{\\sc Case 1:}\nThere exists  $t''$ such that $(z;x)_{t''}=(y;x)_{t''}$ and\n$(y\\cdot z)_x\\geq t''>t$. We can assume that\n$(z;x)_{t'''}\\not=(y;x)_{t'''}$  for any $t'<t'''<t''$.\n\nClearly, walking along $[x,y]$ from  $(z;x)_{t'}$ to $(z;x)_{t''}$\nand then go back to $(y;x)_{t'}$ along $[x,z]$ gives rise to a cycle\n$C$ in $G.$  This cycle might contain chords. But, surely $C$  has\nno chord which connects one point whose distance to $x$ is less than\n$t$ to another point whose distance to $x$ is larger than $t$. This\nmeans that $(z;x)_{t}$ and $(y;x)_{t}$\n must appear in a circle in  $X_G$ corresponding to a  chordless cycle of  $G$. Since $G$ is $k$-chordal, we arrive\n at  $\\widetilde{d_G}((z;x)_t,(y;x)_t)\\leq\n\\frac{k}{2}$.\n\n\n\n\\paragraph{\\sc Case 2:}  There exists no $t''$ such that $(z;x)_{t''}=(y;x)_{t''}$ and  $(y\\cdot z)_x\\geq t''>t$.\n\nLet $\\Lambda =  \\{(z;x)_s, (y;x)_s:\\   0\\leq s\\leq (y\\cdot x)_x\\}$\n  and   $\\Upsilon=\\{(z;y)_s:\\ 0\\leq s<   (z\\cdot x)_y\\}\\cup \\{(y;z)_s:\\ 0\\leq s<\n(x\\cdot y)_z\\}.$ For  any  $y\\in \\Upsilon,$\n$\\widetilde{d_G}(x,y)>(y\\cdot z)_x$ holds and for any $y\\in\n\\Lambda$,    $\\widetilde{d_G}(x,y)\\leq (y\\cdot z)_x$ holds. This\nsays that\n\\begin{equation}\\label{eq88}\\Lambda \\cap \\Upsilon=\\emptyset . \\end{equation}\nAnalogously, by considering both the\ndistance to $y$ and the distance to  $z$, we\n   have  \\begin{equation}\\label{eq99} \\Lambda\\cap [y,z]=\\emptyset\n.\\end{equation} Combining Eqs. \\eqref{eq88} and \\eqref{eq99}, we get\nthat there is a geodesic    $P$ connecting  $(z;x)_t$ and $(y;x)_t$\nwhose internal points fall inside  $ [y,z]\\cup \\Upsilon$.   We\nproduce a circle in  $X_G$ as follows: Walk along   $P$ from\n$(z;x)_t$ to $(y;x)_t$ and then go along $[x,z]$ from $(y;x)_t$ to\n$(y;x)_{t'}$ and finally return to $(z;x)_t$ by following $[x,y]$.\nThis circle naturally corresponds to a cycle of  $G.$ This cycle\nmight have chords. But for each chord which splits the circle into\ntwo smaller circles, our assumption guarantees that the two vertices\n$(z;x)_t$ and $(y;x)_t$ will still appear in one of them\nsimultaneously. This means that there is a circle of  $X_G$\ncorresponding to a chordless cycle of  $G$  and  passing from both\n$(z;x)_t$ and $(y;x)_t$. As $G$ is $k$-chordal,\n$\\widetilde{d_G}((z;x)_t,(y;x)_t)\\leq \\frac{k}{2}$ follows,\n as  expected.\n\\end{proof}\n\n\n\n\n\\section{Proofs}\\label{proofs}\n\n\\subsection{Lemmas}\\label{lemmas}\n\nThe proof of our main results, namely Theorems \\ref{main} and\n\\ref{main1},  is divided into a sequence of lemmas\/corollaries.\n\nIn the course of our proof, we will frequently make  use of the\ntriangle inequality for the shortest-path metric, namely $ab+bc\\geq\nac$, without any claim.  Besides this, we will also freely    apply\nthe ensuing simple observation, which is so simple that we need not\nbother to give any proof here.\n\n\\begin{lemma}  \\label{EASY} Let  $H$ be a vertex induced subgraph of a graph  $G.$\nThen $H$ is an isometric subgraph of  $G$ if and only if\n$d_{H}(u,v)=d_G(u,v)$ for each pair of vertices $(u,v)\\in V(G)\\times\nV(G)$ satisfying  $d_H(u,v)\\geq 3.$  In particular,  $H$ must be\nisometric   if its diameter   is at most $2$.\n\\end{lemma}\n\n\n\n\n\nOne small matter of convention here and in what follows. When we\nrefer to a graph, say a graph depicted in Fig. \\ref{fig0}, we\nsometimes indeed mean that graph together with the special labeling\nof its vertices as indicated when it is introduced and sometimes we\nmean a graph which is isomorphic to it.    We just leave it to\nreaders to decide from the context which usage it is. Two immediate\ncorollaries of Lemma \\ref{EASY}  are given subsequently. We state\nthem with the above convention and omit their routine proofs.\n\n\n\\begin{corollary}\\label{lem29}\nLet $G$ be a  graph. Let $H\\in \\{H_1,H_2,H_4\\}$ be an induced\nsubgraph of  $G$ such that $d_G(x,y)=d_G(u,v)=3$. Then $H$ is an\nisometric subgraph of  $G$.\n\\end{corollary}\n\n\n\n\n\n\\begin{corollary}\\label{lem30}\nLet $G$ be a  graph and $H_3$ be an induced subgraph of  $G$. If\n$d_G(x,y)=3$, then $H_3$ is an  isometric  subgraph of  $G$.\n\\end{corollary}\n\nIt is time to  deliver some formal proofs.\n\n\\begin{corollary}\\label{lem31}\nLet $G$ be a  graph and $H_5$ be an induced subgraph of  $G$. If\n$d_G(x,y)=d_G(u,v)=3$ and $d_G(b,c)=4$. Then $H_5$ is an isometric\nsubgraph of  $G$.\n\\end{corollary}\n\n\\begin{proof}\nBased on the fact that $d_G(b,c)=4$, we can derive from the triangle\ninequality that  $d_G(u,b)=d_G(y,b)=d_G(c,x)=d_G(c,v)=3$. The result\nthen follows from Lemma \\ref{EASY} as $$\\{x,y\\}, \\{ u,v\\},  \\{\nu,b\\}, \\{ y,b\\},  \\{ c,x\\},   \\{ c,v\\},  \\{ b,c\\}$$  are all pairs\ninside $V(H_5)\\choose 2$ which are of distance at least $3$ apart in\n$H_5$.\n\\end{proof}\n\n\n\n\n\n\\begin{lemma}\\label{first}   Let  $G$     be a graph and let $x,y,u,v\\in V(G)$. Then $\\delta_G\n(x,y,u,v)\\leq \\min (uv,xy,ux,yv,uy,xv)$. \\end{lemma}\n\n\n\n\n\n\\begin{proof}   Suppose that\n $d_G(x,S)=d_1, d_G(y,S)=d_2$, where $S=\\{ u,v\\}$. We can check the\n following:\n$$\\left\\{ \\begin{array}{l }\n    xy+uv\\leq (d_1+d_2+uv)+uv=d_1+d_2+2uv,\\\\\n  d_1+d_2\\leq  xu+yv\\leq (d_1+uv)+(d_2+uv)= d_1+d_2+2uv,\\\\\n    d_1+d_2\\leq xv+yu\\leq (d_1+uv)+(d_2+uv)=d_1+d_2+2uv,\n\\end{array}\\right.$$\n  from which we get $\\delta_G\n(x,y,u,v)\\leq uv$ and hence  our claim follows by symmetry.\n\\end{proof}\n\n\n\\begin{figure}\n\\hspace{-33mm}\n\\unitlength 1mm\n\\linethickness{0.4pt}\n\\ifx\\plotpoint\\undefined\\newsavebox{\\plotpoint}\\fi \\hspace{30mm}\n\\begin{center}\n\\begin{picture}(50,40)\n\n\\put(30,0){\\circle*{1}}\\put(30,-4){\\makebox(0,0)[b]{$y$}}\n\\put(10,20){\\circle*{1}}\\put(7,20){\\makebox(0,0)[l]{$u$}}\n\\put(50,20){\\circle*{1}}\\put(53,20){\\makebox(0,0)[r]{$v$}}\n\\put(30,40){\\circle*{1}}\\put(30,43){\\makebox(0,0)[t]{$x$}}\n\\put(20,30){\\circle*{1}}\\put(17,33){\\makebox(0,0)[l]{$a$}}\n\\put(40,30){\\circle*{1}}\\put(43,33){\\makebox(0,0)[r]{$b$}}\n\\put(20,10){\\circle*{1}}\\put(15,10){\\makebox(0,0)[l]{$c$}}\n\\put(40,10){\\circle*{1}}\\put(45,10){\\makebox(0,0)[r]{$d$}}\n\\put(35,-5){\\makebox(0,0)[l]{$H_{6}$}}\n\\qbezier(10,20)(20,10)(30,0)\\qbezier(30,0)(40,10)(50,20)\\qbezier(10,20)(20,30)(30,40)\\qbezier(30,40)(40,30)(50,20)\n\\qbezier(20,30)(20,20)(20,10)\n\\qbezier(20,30)(30,20)(40,10)\\qbezier(20,10)(30,10)(40,10)\n \\end{picture}\n\\end{center}\n\\caption{A  $5$-chordal graph with hyperbolicity $1$.}\\label{fig8}\n\\end{figure}\n\n\nThe next two simple lemmas concern the graph $H_6$ as given in Fig.\n\\ref{fig8}, which  is obviously a $5$-chordal graph with\nhyperbolicity $1.$\n\n\\begin{lemma}\\label{lead}  Let $H$ be a graph  satisfying\n  $V(H)=V(H_2)=V(H_5)=\\{x,y,u,v,a,b,c,d\\}$ and  $E(H_5)\\subseteq E(H)\\subseteq E(H_2)$.\nLet $t$ be the size  of   $E(H)\\cap \\{\\{a,b\\},\\{b,d\\}, \\{d,c\\},\n\\{c,a\\}\\}.$ If  $t\\in \\{1,2,3\\}$, then  either $H$ contains an\ninduced $C_4$ or there is an isomorphism from $H$ to $H_6$.\n\\end{lemma}\n\n\n\n\\begin{proof}  For any  $v_1,v_2\\in V(H)$,  $v_1v_2$  always refers to  $d_H(v_1,v_2)$ in the following.\n\n\n\\paragraph {\\sc Case 1:}  $bc=1.$\n\nLet us show  that  $H$  contains an induced $C_4$ in this case.\nSince $t\\in \\{1,2,3\\}$, by symmetry, we can assume that either\n$ac=1, cd\\not=1$ or  $cd=1,bd\\not= 1.$\n In the former case,\n$[acyd]$ is an induced $4$-cycle of  $H$ and in the latter case\n$[cdvb]$\n   is an induced $4$-cycle of  $H$.\n\n\n\\paragraph {\\sc Case 2:}  $bc\\not= 1.$\n\n\n\nFirst observe that replacing the two edges $\\{ a,c\\}$ and  $\\{d,c\\}$\nby the two new edges $\\{ a,b\\}$ and  $\\{d,b\\}$  will transform $H_6$\ninto another graph which is isomorphic to $H_6.$ Thus, by symmetry,\nthe condition that $t\\in \\{1,2,3\\}$ means  it is sufficient to\nconsider the case that $ac=1,cd>1$ and the case that $ac=cd=1,\nab=bd=2.$ For the first case, $[acyd]$ is an induced $4$-cycle of\n$H$; for the second case, $H$  itself is exactly $H_6$ after\nidentifying vertices of the same labels.\n\\end{proof}\n\n\n\\begin{lemma} \\label{lem10}\nSuppose that    $G$ is a     $5$-chordal graph  which has $H_6$ as\nan induced subgraph.  If  $d_G(x,y)=d_G(u,v)=3$, then  $G$ contains\neither $C_4$ or $H_2$ or $H_3$ as an isometric subgraph.\n\\end{lemma}\n\n\\begin{proof}   We can check that  the subgraph of   $H_6$, and hence of  $G,$ induced by $x,a,c,d,v,b$ is\nisomorphic to $H_3.$  If $d_G(b,c)=3$,   Corollary  \\ref{lem30}\nshows that  $G$  contains $H_3$ as an isometric subgraph.  Thus, in\nthe remaining discussions we will assume that\n\\begin{equation}  \\label{eq2+} d_G(b,c)=2.\n\\end{equation}\n\n\n\n\n\\paragraph {\\sc Case 1:} $\\min(d_G(b,u),d_G(b,y))=2$.\n\n\nAssume, as we may, that  $d_G(b,u)=2.$  Take, accordingly, $w\\in\nV(G)$ satisfying  $d_G(b,w)=d_G(w,u)=1.$ As $d_{H_6}(b,u)=3,$ we see\nthat  $w\\notin V(H_6)$. Observe that\n$$2=3-1=d_G(u,v)-d_G(u,w)\\leq d_G(v,w)\\leq d_G(v,b)+d_G(b,w)=2,$$\nwhich gives \\begin{equation}d_G(v,w)=2. \\label{China}\n\\end{equation}\n\n\n\n\\paragraph {\\sc Case 1.1:}  $d_G(w,d)=1$.\n\n\nIn this case, it follows from  Eq.  \\eqref{China} that $[wbvd]$  is\nan isometric $C_4$ of $G.$\n\n\n\\paragraph {\\sc Case 1.2:}   $d_G(w,d)\\geq  2.$\n\n\n\nSince  $G$ is $5$-chordal, we know that the $6$-cycle  $[wbvdau]$\ncannot be chordless in $G.$   By  Eq. \\eqref{China}  and the current\nassumption that   $d_G(w,d)\\geq  2,$  we can  draw the conclusion\nthat $d_G(w,a)=1 $ and hence find that the  subgraph of  $G$ induced\nby $w,u,a,d,v,b$ is isomorphic to $H_3. $  As we already\n   assumed that $d_G(u,v)=3$, this induced  $H_3$ is   even an isometric subgraph of\n$G,$  taking into account Corollary \\ref{lem30}.\n\n\n\\paragraph {\\sc Case 2:}  $d_G(b,u)=d_G(b,y)=3$.\n\nBy Eq.  \\eqref{eq2+},  we can choose  $w\\in V(G)$ such that\n$d_G(b,w)=d_G(w,c)=1.$   Since  $d_{H_6}(b,c)=3,$  we know that\n$w\\notin V(H).$ In addition, we have \\begin{equation}d_G(u,w)\\geq\nd_G(u,b)-d_G(b,w)=3-1=2, \\ \\text{and}\\ \\ d_G(y,w)\\geq\nd_G(y,b)-d_G(b,w)=3-1=2. \\label{full-moon}\n\\end{equation}\nClearly, the map which swaps  $u$ and  $y$, $a$ and $d$ and   $x$\nand  $v$ is an automorphism of $H_6$ and the requirement to specify\nour Case 2 will not be affected after applying this automorphism of\n$H_6.$ Therefore, noting  that  $d_G(w,v), d_G(w,d), d_G(w,x),\nd_G(w,a) \\in \\{1,2\\},$    we may  take advantage of this  symmetry\nof $H_6$ and merely consider\n the following   situations.\n\n\n\n\n\n\n\\begin{figure}\n\\hspace{-33mm}\n\\unitlength 1mm\n\\linethickness{0.5pt}\n\\ifx\\plotpoint\\undefined\\newsavebox{\\plotpoint}\\fi \\hspace{30mm}\n\\begin{center}\n\\begin{picture}(220,50)\n\n\n\n\n\n\\put(100,0){\\circle*{1}}\\put(100,-4){\\makebox(0,0)[b]{$y$}}\n\n\\put(80,20){\\circle*{1}}\\put(77,20){\\makebox(0,0)[l]{$u$}}\n\\put(120,20){\\circle*{1}}\\put(123,20){\\makebox(0,0)[r]{$v$}}\n\n\\put(100,40){\\circle*{1}}\\put(100,43){\\makebox(0,0)[t]{$x$}}\n\n\\put(90,30){\\circle*{1}}\\put(87,33){\\makebox(0,0)[l]{$a$}}\n\\put(110,30){\\circle*{1}}\\put(113,33){\\makebox(0,0)[r]{$w$}}\n\\put(90,10){\\circle*{1}}\\put(85,10){\\makebox(0,0)[l]{$c$}}\n\\put(110,10){\\circle*{1}}\\put(115,10){\\makebox(0,0)[r]{$d$}}\n\n\\qbezier(80,20)(90,10)(100,0)\\qbezier(100,0)(110,10)(120,20)\\qbezier(80,20)(90,30)(100,40)\\qbezier(100,40)(110,30)(120,20)\n\\qbezier(90,30)(90,20)(90,10)\n\\qbezier(90,30)(100,20)(110,10)\\qbezier(90,10)(100,10)(110,10)\\qbezier(90,30)(100,30)(110,30)\\qbezier(110,10)(110,20)(110,30)\\qbezier(90,10)(100,20)(110,30)\n \\end{picture}\n\\end{center}\n\n\\caption{Case 2.1 in the proof of Lemma \\ref{lem10}.}\\label{fig3}\n\\end{figure}\n\n\n\n\\paragraph {\\sc Case 2.1:}  $d_G(w,v)=d_G(w,d)=d_G(w,x)=d_G(w,a)=1.$\n\nFrom Eq. \\eqref{full-moon} and our assumption it follows  that\n the subgraph of  $G$ induced by\n$x,a,u,c,y,d,v,w$ is isomorphic to $H_2$;  see Fig.  \\ref{fig3}.\nBecause $d_G(x,y)=d_G(u,v)=3$, Corollary  \\ref{lem29} now tells us\nthat $G$ contains   $H_2$ as an isometric subgraph.\n\n\n\n\n\\paragraph {\\sc Case 2.2:}  $d_G(w,v)=d_G(w,d)=2$.\n\nIt is not difficult to check that the subgraph  of   $G$   induced\nby $w,b,v,d,c,y$ is isomorphic to $H_3$.  The condition  that\n$d_G(b,y)=3$ then enables us to appeal to  Corollary \\ref{lem30} and\nconclude that $G$ contains the graph $H_3$ as an isometric subgraph.\n\n\n\\paragraph {\\sc Case 2.3:}\n $d_G(w,v)=2$ and $ d_G(w,d)=1$.\n\n\n$G$ contains the induced $4$-cycle  $[wbvd]$.\n\n\\paragraph {\\sc Case 2.4:}\n $d_G(w,v)=1$ and $ d_G(w,d)=2$.\n\n $[wvdc]$ is a required induced $C_4$ of  $G.$\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\\begin{lemma}\\label{lem27}\nLet $G$ be a $5$-chordal graph which has   $H_5$ as an induced\nsubgraph.  If $d_G(x,y)=d_G(u,v)=3$, then $G$ contains at least one\nof the subgraphs $C_4,H_2,H_3$ and $H_5$ as an isometric subgraph.\n\\end{lemma}\n\n\n\n\n\n\\begin{proof}  Because  $H_5$ is an induced subgraph\n of  $G,$  it is clear that   $d_G(b,u),d_G(b,y),d_G(c,x),d_G(c,v)\\in\n \\{2,3\\}$.\nThere are thus two cases to consider.\n\n\n\n\\paragraph {\\sc Case 1:} $\\min  (d_G(b,y),d_G(b,u),d_G(c,x),d_G(c,v))=2$.\n\n\nWithout loss of generality, let us assume that  $d_G(b,y)=2.$  There\nis then a vertex  $w$ of   $G$ such that  $d_G(b,w)=d_G(w,y)=1.$\nObserve that\n\\begin{equation}d_G(w,x)\\geq d_G(x,y)-d_G(w,y)=3-1=2.\n\\label{John}\n\\end{equation}\n\n\\paragraph {\\sc Case 1.1:}  $d_G(w,a)=1.$\n\nBy Eq. \\eqref{John}, $[wbxa]$ is an induced $4$-cycle of  $G.$\n\n\n\n\n\\begin{figure}\n\\hspace{-33mm}\n\\unitlength 1mm\n\\linethickness{0.5pt}\n\\ifx\\plotpoint\\undefined\\newsavebox{\\plotpoint}\\fi \\hspace{30mm}\n\\begin{center}\n\\begin{picture}(70,50)\n\n\\put(10,10){\\circle*{1}}\\put(7,7){\\makebox(0,0)[l]{$y$}}\n\\put(10,20){\\circle*{1}}\\put(7,20){\\makebox(0,0)[l]{$c$}}\n\\put(20,30){\\circle*{1}}\\put(20,33){\\makebox(0,0)[t]{$a$}}\n\\put(30,20){\\circle*{1}}\\put(33,20){\\makebox(0,0)[r]{$b$}}\n\\put(20,10){\\circle*{1}}\\put(20,7){\\makebox(0,0)[r]{$d$}}\n\\put(10,30){\\circle*{1}}\\put(7,33){\\makebox(0,0)[l]{$u$}}\n\\put(30,10){\\circle*{1}}\\put(33,7){\\makebox(0,0)[r]{$v$}}\n\\put(30,30){\\circle*{1}}\\put(33,33){\\makebox(0,0)[r]{$x$}}\n\\put(20,0){\\circle*{1}}\\put(20,-3){\\makebox(0,0)[t]{$w$}}\n\n\\qbezier(10,30)(10,20)(10,10)\\qbezier(10,30)(20,30)(30,30)\\qbezier(30,30)(30,20)(30,10)\n\\qbezier(10,10)(20,10)(30,10)\\qbezier(20,10)(20,20)(20,30)\\qbezier(20,0)(15,5)(10,10)\\qbezier(20,0)(20,5)(20,10)\\qbezier(20,0)(50,10)(30,20)\n \\end{picture}\n\\end{center}\n\\caption{Case 1.2 in the proof of Lemma \\ref{lem27}.}\\label{fig2.2}\n\\end{figure}\n\n\n\\paragraph {\\sc Case 1.2:}  $d_G(w,a)>1.$\n\nConsider the $6$-cycle  $[bwydax].$  As $G$ is $5$-chordal, this\ncycle has a chord in  $G$. According to Eq. \\eqref{John} and our\nassumption that   $d_G(w,a)>1$, the only possibility is that such a\nchord connects $w$ and $d.$ We now examine the subgraph of $G$\ninduced by $w,b,x,a,d,y$ and realize that it is isomorphic with\n$H_3$; see Fig. \\ref{fig2.2}. Armed with Corollary \\ref{lem30}, our\nassumption that $d_G(x,y)=3$ shows that this $H_3$ is even an\nisometric subgraph of $G,$ as wanted.\n\n\n\n\n\n\n\n\n\n\\paragraph {\\sc Case 2:} $d_G(b,u)=d_G(b,y)=d_G(c,x)=d_G(c,v)=3$.\n\n\n\n\n\nBy Corollary   \\ref{lem31}, $H_5$ is an isometric subgraph of   $G$\nprovided  $d_G(b,c)=4$. Thus, we shall\n    restrict our attention to the cases that $ d_G(b,c)\\in \\{2, 3\\}.$\n\n\n\\paragraph{\\sc Case 2.1:} $d_G(b,c)=2$.\n\nPick a $w\\in V(G)$ such that $d_G(b,w)=d_G(w,c)=1.$  It is  clear\nthat  $[bwcydv]$ is a $6$-cycle in the $5$-chordal graph $G$ and\nhence must have a chord. We contend that  this chord can be nothing\nbut\n $\\{w, d\\}.$  To see this, one simply needs to notice the following:\n\\begin{equation*}\\left\\{ \\begin{array}{l }\n  d_G(w,y)\\geq d_G(b,y)-d_G(b,w)=3-1=2;\\\\\nd_G(w,v)\\geq d_G(c,v)-  d_G(c,w)=3-1=2.\n\\end{array}\\right.\n\\end{equation*}\n From the structure of the subgraph of  $G$ induced by  $b,w,c,y,d,v$  we deduce that    both\n $[cwdy]$ and $[bwdv]$ are  isometric $4$-cycles in  $G$,\n establishing our claim in this case.\n\n\n\\paragraph{\\sc Case 2.2:} $d_G(b,c)=3$.\n\n\n\nWe choose $p,q\\in V(G)$ such that $d_G(b,p)=d_G(p,q)=d_G(q,c)=1$. We\nfirst note that\n$$d_G(q,v)\\geq d_G(c,v)-d_G(c,q)=3-1=2.$$\nDue to the symmetry of  $H_5$, it is manifest then that\n\\begin{equation}  \\label{Texas}  \\min (d_G(q,v),d_G(p,y),d_G(q,x),d_G(p, u))\\geq 2.\n\\frac{}{}\\end{equation}\n               We also  have  $ d_G(q,y)\\in \\{1,2\\}$ as it holds\n$$2=1+1=d_G(q,c)+d_G(c,y) \\geq  d_G(q,y)\\geq\nd_G(b,y)-d_G(b,q)=3-2=1.$$ Arguing by analogy, we indeed have\n\\begin{equation} \\label{gang}\n d_G(q,y),d_G(q,u),d_G(p,v),d_G(p,x)\\in \\{1,2\\}.\n\\end{equation}\nEq.  \\eqref{Texas} along with Eq. \\eqref{gang}  shows that\n\\begin{equation}\\{p,q\\}\\cap V(H_5)=\\emptyset .\n\\label{xuhui}\n\\end{equation}\n\n\n\nAssume, as we may, that\n\\begin{equation}d_G(q,y)+d_G(p,v)\\geq d_G(q,u)+d_G(p,x).\n\\label{eqn3}\n\\end{equation}\n\n\n\n\\paragraph{\\sc Case 2.2.1:} $d_G(q,y)=d_G(p,v)=2$.\n\nWe start with two observations:  Thanks to Eq. \\eqref{Texas}, we\nhave $d_G(p,y)\\geq 2, d_G(q,v)\\geq 2$  while as $b,p,q,c$ is a\ngeodesic, we obtain $d_G(p,c)=d_G(q,b)=2.$ Now, let us take a look\nat the $7$-cycle $[bpqcydv]$  of the $5$-chordal graph\n $G$.  The cycle must have a chord, which, according to our previous observations  and our assumption that  $d_G(q,y)=d_G(p,v)=2$,\n  can only be the one  connecting $d$ to $p$ or to  $q.$ Without loss of\ngenerality, let $d_G(q,d)=1$. Then, we can find a $4$-cycle\n$[cqdy]$, as desired.\n\n\n\n\n\\begin{figure}\n\\hspace{-33mm}\n\\unitlength 1mm\n\\linethickness{0.5pt}\n\\ifx\\plotpoint\\undefined\\newsavebox{\\plotpoint}\\fi \\hspace{30mm}\n\\begin{center}\n\\begin{picture}(40,30)\n\n\\put(10,10){\\circle*{1}}\\put(7,10){\\makebox(0,0)[l]{$y$}}\n\\put(10,20){\\circle*{1}}\\put(7,20){\\makebox(0,0)[l]{$c$}}\n\\put(20,30){\\circle*{1}}\\put(20,33){\\makebox(0,0)[t]{$a$}}\n\\put(30,20){\\circle*{1}}\\put(33,20){\\makebox(0,0)[r]{$b$}}\n\\put(20,10){\\circle*{1}}\\put(20,7){\\makebox(0,0)[r]{$d$}}\n\\put(10,30){\\circle*{1}}\\put(7,33){\\makebox(0,0)[l]{$u$}}\n\\put(30,10){\\circle*{1}}\\put(33,10){\\makebox(0,0)[r]{$v$}}\n\\put(30,30){\\circle*{1}}\\put(33,33){\\makebox(0,0)[r]{$x$}}\n\\put(10,0){\\circle*{1}}\\put(10,-3){\\makebox(0,0)[t]{$q$}}\n\\put(30,0){\\circle*{1}}\\put(30,-3){\\makebox(0,0)[t]{$p$}}\n\\qbezier(10,30)(10,20)(10,10)\\qbezier(10,30)(20,30)(30,30)\\qbezier(30,30)(30,20)(30,0)\n\\qbezier(10,10)(20,10)(30,10)\\qbezier(20,10)(20,20)(20,30)\\qbezier(10,0)(20,0)(30,0)\\qbezier(10,0)(0,10)(10,20)\n\\qbezier(30,0)(25,5)(20,10)\\qbezier(30,0)(40,10)(30,20)\n \\end{picture}\n\\end{center}\n\\caption{Case 2.2.2 in the proof of Lemma\n\\ref{lem27}.}\\label{fig1.2.2}\n\\end{figure}\n\n\\paragraph{\\sc Case 2.2.2:} $d_G(q,y)=2, d_G(p,v)=1$.\n\n\nIn this case, the $5$-chordal graph   $G$ possesses  the  $6$-cycle\n $[pqcydv]$, which must have a chord.  We already assume that  $d_G(q,y)=2;$ as $b,p,q,c$ is a geodesic, we get $d_G(p,c)=2;$ finally, we have\n\\begin{equation*}\\left\\{ \\begin{array}{l }\n  d_G(p,y)\\geq d_G(b,y)-d_G(b,p)=3-1=2;\\\\\nd_G(q,v)\\geq d_G(c,v)-  d_G(c,q)=3-1=2.\n\\end{array}\\right.\n\\end{equation*}\nConsequently,  it happens either $d_G(q,d)=1$  or  $d_G(p,d)=1$.\n If $d_G(q,d)=1$, we will come to an isometric $4$-cycle  $[cqdy]$.  When $d_G(p,d)=1$ and $d_G(q,d)\\geq 2$,   the subgraph of $G$ induced by\n $p,q,c,y,d,v$ is isomorphic to $H_3$  as shown by Fig.  \\ref{fig1.2.2}, which is even an isomeric\n subgraph in view of Corollary \\ref{lem30} as well as the assumption  that\n $d_G(c,v)=3.$\n\n\n\n\n\n\n\n\n\\paragraph{\\sc Case 2.2.3} $d_G(q,y)=1, d_G(p,v)=2$.\n\n\n\n This case can be disposed of as Case 2.2.2.\n\n\\paragraph{\\sc Case 2.2.4:}   $d_G(q,y)=d_G(p,v)=1.$\n\n\n\n\n\nBy Eqs. \\eqref{gang}   and \\eqref{eqn3}, we obtain\n$d_G(q,u)=d_G(p,x)=1.$ Noting Eq.  \\eqref{xuhui}, we further get\n\\begin{equation}\\left\\{ \\begin{array}{l }\n 1\\leq d_G(p,d)\\leq d_G(p,v)+d_G(v,d)=1+1=2;\\\\\n1\\leq d_G(q,d)\\leq d_G(q,y)+d_G(y,d)=1+1=2;\\\\\n1\\leq d_G(p,a)\\leq d_G(p,x)+d_G(x,a)=1+1=2;\\\\\n1\\leq d_G(q,a)\\leq d_G(q,u)+d_G(u,a)=1+1=2.\\\\\n\\end{array}\\right.\n\\label{picb}\n\\end{equation}\nBecause of Eq.  \\eqref{picb}, it is only necessary to consider the\nfollowing three cases, since all others would follow by symmetry.\n\n\n\\paragraph{\\sc Case 2.2.4.1:}  $d_G(p,d)=d_G(q,d)=2$.\n\n\n\n\n\n\n\nThe subgraph of  $G$ induced by   $c,y,q,p,v,d$ is isomorphic to\n$H_3$; see Fig.   \\ref{fig1.2.4.1}.  But $d_G(c,v)=3$ is among the\nstanding assumptions for Case 2  and hence Corollary\n     \\ref{lem30}  demonstrates that  $G$ has this $H_3$ as an\n     isometric subgraph.\n\n\n\n\n\n\\begin{figure}\n\\hspace{-33mm}\n\\unitlength 1mm\n\\linethickness{0.5pt}\n\\ifx\\plotpoint\\undefined\\newsavebox{\\plotpoint}\\fi \\hspace{30mm}\n\\begin{center}\n\\begin{picture}(40,30)\n\n\\put(10,10){\\circle*{1}}\\put(7,10){\\makebox(0,0)[l]{$y$}}\n\\put(10,20){\\circle*{1}}\\put(7,20){\\makebox(0,0)[l]{$c$}}\n\\put(20,30){\\circle*{1}}\\put(20,33){\\makebox(0,0)[t]{$a$}}\n\\put(30,20){\\circle*{1}}\\put(33,20){\\makebox(0,0)[r]{$b$}}\n\\put(20,10){\\circle*{1}}\\put(20,7){\\makebox(0,0)[r]{$d$}}\n\\put(10,30){\\circle*{1}}\\put(7,33){\\makebox(0,0)[l]{$u$}}\n\\put(30,10){\\circle*{1}}\\put(33,10){\\makebox(0,0)[r]{$v$}}\n\\put(30,30){\\circle*{1}}\\put(33,33){\\makebox(0,0)[r]{$x$}}\n\\put(10,0){\\circle*{1}}\\put(10,-3){\\makebox(0,0)[t]{$q$}}\n\\put(30,0){\\circle*{1}}\\put(30,-3){\\makebox(0,0)[t]{$p$}}\n\\qbezier(10,30)(10,20)(10,0)\\qbezier(10,30)(20,30)(30,30)\\qbezier(30,30)(30,20)(30,0)\n\\qbezier(10,10)(20,10)(30,10)\\qbezier(20,10)(20,20)(20,30)\\qbezier(10,0)(20,0)(30,0)\\qbezier(10,0)(-10,15)(10,30)\\qbezier(10,0)(0,10)(10,20)\n\\qbezier(30,0)(50,15)(30,30)\\qbezier(30,0)(40,10)(30,20)\n \\end{picture}\n\n\\end{center}\n\\caption{Case 2.2.4.1 in the proof of Lemma\n\\ref{lem27}.}\\label{fig1.2.4.1}\n\\end{figure}\n\n\n     \\paragraph{\\sc Case 2.2.4.2:}\n$\\{d_G(p,d),d_G(q,d)\\}=\\{ 1, 2\\}$.\n\nThere is no loss of generality in assuming that $d_G(p,d)=1$  and\n$d_G(q,d)=2.$  In such a situation,  by recalling from  Eq.\n\\eqref{Texas}\n     that $d_G(p,y)\\geq 2$, we find that   $[pdyq]$  is an induced\n$4$-cycle of  $G$, as wanted.\n\n\n\n\n\n\n\\begin{figure}\n\\hspace{-33mm}\n\\unitlength 1mm\n\\linethickness{0.5pt}\n\\ifx\\plotpoint\\undefined\\newsavebox{\\plotpoint}\\fi \\hspace{30mm}\n\\begin{center}\n\\begin{picture}(50,50)\n\n\n\n\n\n\n\\put(20,0){\\circle*{1}}\\put(20,-4){\\makebox(0,0)[b]{$y$}}\n\n\\put(0,20){\\circle*{1}}\\put(-3,20){\\makebox(0,0)[l]{$u$}}\n\\put(40,20){\\circle*{1}}\\put(43,20){\\makebox(0,0)[r]{$v$}}\n\n\\put(20,40){\\circle*{1}}\\put(20,43){\\makebox(0,0)[t]{$x$}}\n\n\\put(10,30){\\circle*{1}}\\put(7,33){\\makebox(0,0)[l]{$a$}}\n\\put(30,30){\\circle*{1}}\\put(33,33){\\makebox(0,0)[r]{$p$}}\n\\put(10,10){\\circle*{1}}\\put(7,7){\\makebox(0,0)[l]{$q$}}\n\\put(30,10){\\circle*{1}}\\put(33,7){\\makebox(0,0)[r]{$d$}}\n\n\n\\qbezier(0,20)(10,10)(20,0)\\qbezier(20,0)(30,10)(40,20)\\qbezier(40,20)(30,30)(20,40)\\qbezier(20,40)(10,30)(0,20)\n\\qbezier(10,30)(20,20)(30,10)\\qbezier(10,10)(20,20)(30,30)\\qbezier(30,30)(30,20)(30,10)\\qbezier(30,10)(20,10)(10,10)\n\\qbezier(10,10)(10,20)(10,30)\\qbezier(10,30)(20,30)(30,30)\n\n\n\n \\end{picture}\n\n\\end{center}\n\\caption{Case 2.2.4.3 in the proof of Lemma\n\\ref{lem27}.}\\label{fig1.2.4.3}\n\\end{figure}\n\n   \\paragraph{\\sc Case 2.2.4.3:}    $d_G(p,d)=d_G(q,d)=d_G(p,a)=d_G(q,a)=1$.\n\n\n\n\n\nAfter checking all those  existing  assumptions on pairs of adjacent\nvertices  as well as the fact that $\\{q,v\\},\\{ p,y\\}\\notin E(G)$ as\nguaranteed by Eq. \\eqref{Texas}, we are led to the conclusion that\nthe subgraph of $G$ induced by $x,a,u,q,y,d,v,p$ is just $H_2$; see\nFig. \\ref{fig1.2.4.3}. Noting further our governing assumption in\nthe lemma that\n $d_G(x,y)=d_G(u,v)=3$,     Corollary  \\ref{lem29}  then enables us reach the conclusion that this $H_2$ is indeed an isometric subgraph\n of  $G,$  as was to be shown.\n\\end{proof}\n\n\n\n\n\n\n\n\nThe next simple result resembles \\cite[Lemma 2.2]{BKM01} closely.\n\n\n\n\n\\begin{lemma} \\label{lem2.1} Let $G$ be a $k$-chordal graph and let  $C=[x_1\\cdots\nx_kx_{k+1}\\cdots x_{k+t}]$ be  a cycle of $G$.\n If no chord of  $C$ has an endpoint in $\\{ x_2,\\cdots,x_{k-1}\\}$, then $x_1x_{k}=1.$\n  \\end{lemma}\n\n\n\\begin{proof}   We consider the induced subgraph\n$H=G[x_1,x_{k}, x_{k+1},\\ldots ,x_{k+t}]$. There must exist a\nshortest path in $H$ connecting $x_1$ and  $x_{k}$, say $P$. If the\nlength of $P$ is greater than $1$, then we walk along $P$ from\n$x_{k}$ to $x_1$ and then continue with $x_2,x_3,\\ldots,$ and\nfinally get back to $x_{k}$, creating a chordless cycle of length at\nleast $k+1,$ which is absurd as $G$ is $k$-chordal.  This proves\nthat $x_1x_k=1,$ as desired.\n\\end{proof}\n\n\n\n\n\n\nLet $G$ be a graph.   When studying  $\\delta _G(x,y,u,v)$ for some\nvertices $x,y,u,v$ of $G,$ it is natural to look at a  {\\em geodesic\nquadrangle}   $\\mathcal {Q}(x,u,y,v)$  with {\\em corners} $x,u,y$\nand $v$, which is just the subgraph of  $G$   induced by the union\nof all those vertices on four geodesics connecting $x$ and $u,$ $u$\nand  $y$,  $y$ and  $v$, and  $v$ and  $x$, respectively. Let us fix\nsome notation to be used throughout the paper.\n\n\n\n\\begin{figure}\n\\hspace{-33mm}\n\\unitlength 1mm\n\\linethickness{0.4pt}\n\\ifx\\plotpoint\\undefined\\newsavebox{\\plotpoint}\\fi \\hspace{30mm}\n\\begin{center}\n\\begin{picture}(60,60)\n\n\\put(20,0){\\circle*{1}}\\put(20,-4){\\makebox(0,0)[b]{$y$}}\n\\put(0,20){\\circle*{1}}\\put(-3,20){\\makebox(0,0)[l]{$u$}}\n\\put(40,20){\\circle*{1}}\\put(42,20){\\makebox(0,0)[l]{$v$}}\n\\put(20,40){\\circle*{1}}\\put(20,43){\\makebox(0,0)[t]{$x$}}\n\n\\put(5,25){\\circle*{1}}\\put(-5,25){\\makebox(0,0)[l]{$a_{xu-1}$}}\n\n\\put(15,35){\\circle*{1}}\\put(10,35){\\makebox(0,0)[l]{$a_{1}$}}\n\\put(5,15){\\circle*{1}}\\put(-5,15){\\makebox(0,0)[l]{$c_{yu-1}$}}\n\n\\put(15,5){\\circle*{1}}\\put(10,5){\\makebox(0,0)[l]{$c_{1}$}}\n\\put(25,5){\\circle*{1}}\\put(30,5){\\makebox(0,0)[r]{$d_{1}$}}\n\n\\put(35,15){\\circle*{1}}\\put(45,15){\\makebox(0,0)[r]{$d_{yv-1}$}}\n\n\\put(25,35){\\circle*{1}}\\put(30,35){\\makebox(0,0)[r]{$b_{1}$}}\n\n\\put(35,25){\\circle*{1}}\\put(45,25){\\makebox(0,0)[r]{$b_{xv-1}$}}\n\n\n\\qbezier[7](5,15)(10,10)(15,5)\\qbezier(0,20)(2,22)(5,25)\\qbezier(0,20)(2,18)(5,15)\n\\qbezier(15,5)(18,2)(20,0)\\qbezier(20,0)(22,2)(25,5)\\qbezier(35,15)(38,18)(40,20)\n\\qbezier[7](25,5)(30,10)(35,15)\\qbezier(15,35)(18,38)(20,40)\\qbezier(20,40)(22,38)(25,35)\n\\qbezier(35,25)(38,22)(40,20)\\qbezier[7](25,35)(30,30)(35,25)\\qbezier[7](5,25)(10,30)(15,35)\n\n \\end{picture}\\end{center}\n\\caption{The geodesic quadrangle   $\\mathcal\n{Q}(x,u,y,v)$.}\\label{fig1}\n\\end{figure}\n\n\n\n\\paragraph{\\sc \\bf  Assumption I:}\n Let  us assume that  $x,u,y,v$ are four different vertices of a graph  $G$ and\n  the    four geodesics corresponding to  the  geodesic quadrangle  $\\mathcal\n {Q}(x,u,y,v)$  are\n\\begin{equation*}\\left\\{ \\begin{array}{l }\n   P_{a}: x=a_0,a_1,\\ldots,a_{xu}=u;\\\\\n P_b: x=b_0,b_1,\\ldots,b_{xv}=v;\\\\\n   P_c: y=c_0,c_1,\\ldots,c_{yu}=u;\\\\\nP_d: y=d_0,d_1,\\ldots,d_{yv}=v.\n\\end{array}\\right.\n\\end{equation*}\nWe call $P_a$, $P_b, P_c$  and $P_d$ the four {\\em sides} of\n$\\mathcal {Q}(x,u,y,v)$  and often just refer to them as vertex\nsubsets of  $V(G)$  rather than vertex sequences. We write $\\mathcal\n{P}(x,u,y,v)$ for the pseudo-cycle $$[x, a_1,\\ldots , a_{xu-1}, u,\nc_{yu-1},\\ldots,c_1,y,d_1,\\ldots,d_{yv-1},v,b_{xv-1},\\ldots, b_1].$$\nNote that $\\mathcal {P}(x,u,y,v)$ is not necessarily a cycle as the\nvertices appearing in the sequence may not be all different. Let us\nsay that $x$ is {\\em opposite} to $P_c$ and $P_d$, say that $x$ and\n$y$ are {\\em opposite corners}, say that $x$ and $v$ are {\\em\nadjacent corners}, say that $x$ is the {\\em common peak } of $P_a$\nand $P_b$, say that $P_a$ and $P_b$ are {\\em adjacent} to each\nother, say that $P_a$ and $P_d$ are {\\em opposite} to each other,\n  and say that\n    those vertices inside  $P_a\\setminus \\{x,u\\}$  are {\\em ordinary vertices} of\n    $P_a,$  etc..  An edge of $\\mathcal\n {Q}(x,u,y,v)$   which intersects with two adjacent sides      but do not lie in any single  side  is called an $\\mathbb{A}$-edge and an edge of\n    $\\mathcal\n {Q}(x,u,y,v)$ which intersect with two opposite sides but does not lie in any single side is called an\n $\\mathbb{H}$-edge.\n\n\n Suppose that $a_i=v_0,v_1,\\ldots, v_{a_id_j}=d_j $ is a geodesic\n connecting  $a_i$  and  $d_j$ in  $G.$ We call the two\n  walks $$x,a_1,\\ldots ,a_i,v_1,\\ldots, v_{a_id_j-1},  d_j,d_{j-1},\\ldots ,d_1,y$$\n  and\n  $$u,a_{xu-1},\\ldots,a_i,v_1,\\ldots, v_{a_id_j-1}, d_j,d_{j+1},\\ldots,d_{yv-1},v$$\n {\\em   $\\mathcal\n {Z}$-walks of   $\\mathcal\n {Q}(x,u,y,v)$   through $\\{a_i,d_j\\}$} or just  {\\em   $\\mathcal\n {Z}$-walks of   $\\mathcal\n {Q}(x,u,y,v)$  between $P_a$  and  $P_d.$} In an apparent way, we\n define   similar  concepts for {\\em   $\\mathcal\n {Z}$-walks of   $\\mathcal\n {Q}(x,u,y,v)$    between $P_b$  and  $ P_c$.}\n\n\n\n\n\n\n\n\\rz\n\n\n\n\\begin{lemma} \\label{lem11} Let  $G$ be a graph and let    $\\mathcal\n {Q}(x,u,y,v)$ be one of its geodesic quadrangles for which Assumption I holds.\n Suppose any two adjacent sides of   $\\mathcal\n {Q}(x,u,y,v)$ has only one common vertex and that vertex is  their common peak.\nThen $\\mathcal\n {Q}(x,u,y,v)$ contains  a   cycle on which $b_1,x, a_1$ appear in that\norder consecutively.   Moreover, if $\\min(d(P_a,P_d),d(P_b,\nP_c))\\geq t$ for some $t$, then we may even require that the length\nof the cycle is no shorter than $4t$.\n\\end{lemma}\n\n\n\\begin{proof}\nIf $\\min(d(P_a,P_d),d(P_b, P_c))\\geq t$ for $t>0$, then $\\mathcal\n {P}(x,u,y,v)$\nitself gives rise to  a required cycle.  Otherwise, without loss of\ngenerality, assume that $d(P_a, P_d)=0.$  Take the minimum $i$ such\nthat $d(a_i, P_d)=0.$  There is a unique $j>0$ such that $a_i=d_j$.\nIt is plain that  $i>0$  and  $j<yv.$\n This then shows that $[\nd_j\\cdots d_{yv-1}b_{xv}\\cdots b_1xa_1\\cdots a_{i-1}]$ is a required\ncycle.\n   \\end{proof}\n\n\n\n\n\n\\begin{lemma}  \\cite[p. 67, Claim 2]{BKM01}  We make Assumption I.    Further assume that  \\begin{equation}\\label{eq1}ub_i=1\\end{equation}  for some  $i\\geq 1$ and  that\n\\begin{equation}\\label{eq2} xy+uv\\geq xv+yu+2.\\end{equation} Then, $b_1u<xu.$ \\label{koolen2}\n\\end{lemma}\n\n\n\\begin{proof} We first check the following: \\begin{equation*}\\begin{array}{cll} xu+uv-2 & \\geq  &\n(xy-yu)+uv-2\\\\ & \\geq & xv \\ \\ \\ \\text{(By Eq. \\eqref{eq2})}\\\\ & = &\nxb_i+b_iv\\\\ & = & (xb_i+b_iu)+(ub_i+b_iv)-2    \\ \\ \\ \\text{(By Eq.\n\\eqref{eq1})}\\\\\n & \\geq & xu+uv-2.\n\\end{array}  \\end{equation*}\nClearly, equalities have to  hold throughout all the above\ninequalities. In particular, we have $xu=xb_i+b_iu$. This implies\nthat there is a geodesic between    $x$ and  $u$ passing through\n$b_1$ and hence it is straightforward to see   $b_1u<xu,$ as wanted.\n\\end{proof}\n\n\n\nThe next lemma is some variation of  Lemma \\ref{first} and will play\nan   important role in our short proof of Theorem \\ref{main} as to\nbe presented in Section \\ref{Proof}.\n\n\n\n\\begin{lemma}  Let  $G$ be a graph and let    $\\mathcal\n {Q}(x,u,y,v)$ be one of its geodesic quadrangles for which Assumptions I   holds.\nIf \\begin{equation}\\label{Ringel} 2\\delta_G\n(x,y,u,v)=(xy+uv)-\\max(xu+yv,xv+yu),\n\\end{equation}\n then $\\delta_G(x,y,u,v)\\leq\n\\min(d(P_a,P_d),d(P_b,P_c))$.\\label{lem14}\n\\end{lemma}\n\n\n\\begin{figure}\n\\hspace{-33mm}\n\\unitlength 1mm\n\\linethickness{0.4pt}\n\\ifx\\plotpoint\\undefined\\newsavebox{\\plotpoint}\\fi \\hspace{30mm}\n\\begin{center}\n\\begin{picture}(60,60)\n\n\\put(20,0){\\circle*{1}}\\put(20,-4){\\makebox(0,0)[b]{$y$}}\n\\put(0,20){\\circle*{1}}\\put(-3,20){\\makebox(0,0)[l]{$u$}}\n\\put(40,20){\\circle*{1}}\\put(42,20){\\makebox(0,0)[l]{$v$}}\n\\put(20,40){\\circle*{1}}\\put(20,43){\\makebox(0,0)[t]{$x$}}\n\\put(10,30){\\circle*{1}}\\put(5,30){\\makebox(0,0)[l]{$a_{i}$}}\n\\put(30,10){\\circle*{1}}\\put(35,10){\\makebox(0,0)[r]{$d_{j}$}}\n\\put(5,25){\\circle*{1}}\\put(-5,25){\\makebox(0,0)[l]{$a_{xu-1}$}}\n\n\\put(15,35){\\circle*{1}}\\put(10,35){\\makebox(0,0)[l]{$a_{1}$}}\n\\put(5,15){\\circle*{1}}\\put(-5,15){\\makebox(0,0)[l]{$c_{yu-1}$}}\n\n\\put(15,5){\\circle*{1}}\\put(10,5){\\makebox(0,0)[l]{$c_{1}$}}\n\\put(25,5){\\circle*{1}}\\put(30,5){\\makebox(0,0)[r]{$d_{1}$}}\n\n\\put(35,15){\\circle*{1}}\\put(45,15){\\makebox(0,0)[r]{$d_{yv-1}$}}\n\n\\put(25,35){\\circle*{1}}\\put(30,35){\\makebox(0,0)[r]{$b_{1}$}}\n\n\\put(35,25){\\circle*{1}}\\put(45,25){\\makebox(0,0)[r]{$b_{xv-1}$}}\n\\put(10,30){\\vector(1,-1){5}}\\put(20,40){\\vector(-1,-1){3}}\n\\put(25,5){\\vector(-1,-1){3}}\n\n\\qbezier[7](5,15)(10,10)(15,5)\\qbezier(0,20)(2,22)(5,25)\\qbezier(0,20)(2,18)(5,15)\n\\qbezier(15,5)(18,2)(20,0)\\qbezier(20,0)(22,2)(25,5)\\qbezier(35,15)(38,18)(40,20)\n\\qbezier[7](25,5)(30,10)(35,15)\\qbezier(15,35)(18,38)(20,40)\\qbezier(20,40)(22,38)(25,35)\n\\qbezier(35,25)(38,22)(40,20)\\qbezier[7](25,35)(30,30)(35,25)\\qbezier[7](5,25)(10,30)(15,35)\n\n\\qbezier[10](10,30)(20,20)(30,10)\n \\end{picture}\n\\end{center}\n\\caption{$xy\\leq i+a_id_j+j$.}\\label{fig2}\n\\end{figure}\n\\begin{proof} Without loss of generality, we assume that   there exist $i$ and  $j$ such that  \\begin{equation}\n\\label{Kanazawa}a_id_j=\\min(d(P_a,P_d),d(P_b,P_c)).\n\\end{equation}\nFocusing on a $\\mathcal\n {Z}$-walk  of   $\\mathcal\n {Q}(x,u,y,v)$   through $\\{a_i,d_j\\}$ connecting $x$  and  $y$, we\n find that \\begin{equation}xy\\leq xa_i+a_id_j+d_jy=i+a_id_j+j;\n\\label{eq917}\n\\end{equation} see Fig. \\ref{fig2}. Analogously, we have \\begin{equation}   uv\\leq\n(xu-i)+a_id_j+(yv-j). \\label{eq917+}\\end{equation} Henceforth, we\narrive at the following:\n\\begin{equation*} \\begin{array}{cll} 2\\delta\n(x,y,u,v)&=&(xy+uv)-\\max(xu+yv,xv+yu)\\ \\ \\text{(By Eq.\n\\eqref{Ringel})}\n\\\\\n&\\leq& (xy+uv)-(xu+yv)\n\\\\\n&\\leq& (i+a_id_j+j)+((xu-i)+a_id_j+(yv-j))-(xu+yv)\\ \\ \\text{(By\n Eqs.  \\eqref{eq917}  and \\eqref{eq917+})   }\\\\&=& 2a_id_j.\\label{eq918}\\end{array} \\end{equation*}\n Combining  this with Eq. \\eqref{Kanazawa},    we finish  the proof of the lemma.\n\\end{proof}\n\n\n\nAs regards the inequality asserted in Lemma \\ref{lem14}, we need to\nsay something more for the purpose of deriving Theorem \\ref{main1}.\n\n\n\\begin{lemma} \\label{lemma41} Let  $G$ be a graph and let    $\\mathcal\n {Q}(x,u,y,v)$ be one of its geodesic quadrangles for which both Assumption I and Eq.  \\eqref{Ringel} hold.\n\n  (i)  If $\\delta (x,y,u,v)=d(P_a, P_d)=a_id_j,$    then\n\\begin{itemize}\n\\item any $\\mathcal\n {Z}$-walk of   $\\mathcal {Q}(x,u,y,v)$   through $\\{a_i,d_j\\}$\n between $P_a$  and  $P_d$ must be a geodesic, and\n \\item  under the additional assumption that\n$a_id_j=1$ and $G$ is $5$-chordal,  either   $G$ has an isometric\n$4$-cycle or $\\{a_i,d_j\\}$ is the only edge intersecting both $P_a$\nand  $P_d$.\n\\end{itemize}\n\n(ii)\n   If   $\\delta (x,y,u,v)=d(P_b, P_c)=b_pc_q ,$   then\n\\begin{itemize}\n\\item any $\\mathcal\n {Z}$-walk of   $\\mathcal {Q}(x,u,y,v)$   through $\\{b_p,c_q\\}$\n between $P_b$  and  $ P_c$ must be a geodesic, and\n \\item\nunder the additional assumption that $b_pc_q=1$ and $G$ is\n$5$-chordal,  either   $G$ has an isometric $4$-cycle or\n$\\{b_p,c_q\\}$ is the only edge intersecting both $P_b$ and  $P_c$.\n\\end{itemize}\n\\end{lemma}\n\n\n\\begin{proof} (i) Let us  continue   our discussion  launched in the proof of Lemma\n\\ref{lem14}.\n\nIn the event that    $\\delta (x,y,u,v)= d(P_a, P_d)=a_id_j,$ the\nequalities in both Eq. \\eqref{eq917} and Eq. \\eqref{eq917+} must\noccur, which clearly shows that any $\\mathcal\n {Z}$-walk of   $\\mathcal\n {Q}(x,u,y,v)$   through $\\{a_i,d_j\\}$ between $P_a$  and  $\n P_d$ must be a geodesic.\n\nWe now  further assume that $a_id_j=d(P_a, P_d)=1$ and $G$ is\n$5$-chordal. Set\n$$\\mathcal {I}=\\{ (k,\\ell):\\ a_{k}d_{\\ell}= 1\\}.$$ For any  $(k,\\ell)\\in  \\mathcal {I} $, by considering each\n$\\mathcal{Z}$-walk through $\\{a_k,d_\\ell\\}$ connecting $x$ and  $y$,\nwhich is a geodesic as we already know, we come to\n \\begin{equation}  \\mathcal {I}= \\{  (k,\\ell):\\\na_{k}d_{\\ell}= 1\\} \\subseteq \\{ (k,\\ell):\\ k+\\ell =xy-1 \\} .\n\\label{eqn8}\n\\end{equation} Eq. \\eqref{eqn8}  means that there exists $(i',j')$\nand  $0=t_0<t_1<\\cdots <t_{m}$ such that\n$$\\mathcal {I}=\\{  (i'-t_\\alpha,j'+t_\\alpha):\\  \\alpha =0,1,\\ldots, m  \\}.$$\nSuppose  that $\\{a_i,d_j\\}$ is not the only edge intersecting both\n$P_a$ and  $P_d$. This means that\n $|\\mathcal {I}|-1=m \\geq 1$ and then we see that  $$[a_{i'}a_{i'-1}\\cdots\na_{i'-t_1}b_{j'+t_1}b_{j'+t_1-1}\\cdots b_{j'}]$$ is a chordless\ncycle of length $2t_1+2$.  Since  $G$ is $5$-chordal and $t_1$ is a\npositive integer, this cycle can only be an isometric $4$-cycle of\n$G,$ finishing the proof  of  (i).\n\n(ii)  The proof can be carried out in the same way as that of (i).\n\\end{proof}\n\n\n\n\n\\begin{lemma}  Let $G$ be a graph and we will adopt Assumption  I.\n\n We choose $j$ to be the\nmaximum number such that $a_jb_j\\leq 1$, $i$ the minimum number such\nthat $b_id_{yv-xv+i}\\leq 1,$   $\\ell$ the maximum number such that\n$c_\\ell d_\\ell \\leq 1$, and $m$ the minimum number such that\n$a_mc_{yu-xu+m}\\leq 1.$ Put \\begin{equation}\\left\\{ \\begin{array}{l\n}\n   \\pi (b)=i-j+\\frac{a_jb_j+b_id_{yv-xv+i}}{2},\\\\\n \\pi(d)=(yv-xv+i)-\\ell+\\frac{b_id_{yv-xv+i}+c_\\ell d_\\ell}{2},\\\\\n   \\pi (c)=(yu-xu+m)-\\ell+\\frac{a_mc_{yu-xu+m}+c_\\ell d_\\ell}{2},\\\\\n   \\pi (a)=m-j+\\frac{a_jb_j+a_m c_{yu-xu+m}}{2}.\n\\end{array}\\right.\n\\label{PI}\n\\end{equation}\nIf Eq. \\eqref{Ringel}   is valid, then $\\delta_G(x,y,u,v)\\leq \\min\n(\\pi (a), \\pi (b), \\pi (c), \\pi (d))\\leq 1+\\min ( i-j,\n(yv-xv+i)-\\ell,\n   (yu-xu+m)-\\ell, m-j).$\n     \\label{lem49}\n\\end{lemma}\n\n\\begin{figure}\n\\hspace{-33mm}\n\\unitlength 1mm\n\\linethickness{0.4pt}\n\\ifx\\plotpoint\\undefined\\newsavebox{\\plotpoint}\\fi \\hspace{30mm}\n\\begin{center}\n\\begin{picture}(60,60)\n\n\\put(20,0){\\circle*{1}}\\put(20,-4){\\makebox(0,0)[b]{$y$}}\n\\put(0,20){\\circle*{1}}\\put(-3,20){\\makebox(0,0)[l]{$u$}}\n\\put(40,20){\\circle*{1}}\\put(41,20){\\makebox(0,0)[l]{$v$}}\n\\put(20,40){\\circle*{1}}\\put(20,43){\\makebox(0,0)[t]{$x$}}\n\\put(12,32){\\circle*{1}}\\put(7,32){\\makebox(0,0)[l]{$a_{j}$}}\n\\put(32,28){\\circle*{1}}\\put(36,28){\\makebox(0,0)[r]{$b_{i}$}}\n\\put(5,25){\\circle*{1}}\\put(-5,25){\\makebox(0,0)[l]{$a_{xu-1}$}}\n\\put(28,32){\\circle*{1}}\\put(32,32){\\makebox(0,0)[r]{$b_{j}$}}\n\\put(32,12){\\circle*{1}}\\put(48,12){\\makebox(0,0)[r]{$d_{yv-xv+i}$}}\n\\put(15,35){\\circle*{1}}\\put(11,35){\\makebox(0,0)[l]{$a_{1}$}}\n\\put(5,15){\\circle*{1}}\\put(-5,15){\\makebox(0,0)[l]{$c_{yu-1}$}}\n\n\\put(15,5){\\circle*{1}}\\put(10,5){\\makebox(0,0)[l]{$c_{1}$}}\n\\put(25,5){\\circle*{1}}\\put(30,5){\\makebox(0,0)[r]{$d_{1}$}}\n\\put(12,32){\\vector(1,0){15}} \\put(32,28){\\vector(0,-1){15}}\n\\put(0,20){\\vector(1,1){3}}\\put(20,40){\\vector(1,-1){3}}\n\\put(25,5){\\vector(-1,-1){3}}\\put(35,25){\\vector(1,-1){3}}\n\\put(35,15){\\circle*{1}}\\put(45,15){\\makebox(0,0)[r]{$d_{yv-1}$}}\n\n\\put(25,35){\\circle*{1}}\\put(29,35){\\makebox(0,0)[br]{$b_{1}$}}\n\n\\put(35,25){\\circle*{1}}\\put(45,25){\\makebox(0,0)[r]{$b_{xv-1}$}}\n\n\n\\qbezier[7](5,15)(10,10)(15,5)\\qbezier(0,20)(2,22)(5,25)\\qbezier(0,20)(2,18)(5,15)\n\\qbezier(15,5)(18,2)(20,0)\\qbezier(20,0)(22,2)(25,5)\\qbezier(35,15)(38,18)(40,20)\n\\qbezier[7](25,5)(30,10)(35,15)\\qbezier(15,35)(18,38)(20,40)\\qbezier(20,40)(22,38)(25,35)\n\\qbezier(35,25)(38,22)(40,20)\\qbezier[7](25,35)(30,30)(35,25)\\qbezier[7](5,25)(10,30)(15,35)\n\n\\qbezier(32,12)(32,20)(32,28)\\qbezier(12,32)(20,32)(28,32)\n\n\n\n \\end{picture}\\end{center}\n\\caption{$xy\\leq i+b_id_{yv-xv+i}+(yv-(xv-i))$, $uv\\leq\n(xu-j)+a_jb_j+(xv-j)$.} \\label{fig5}\n\\end{figure}\n\\begin{proof} By symmetry, we only need to show that $\\delta_G(x,y,u,v)\\leq \\pi (b).$\n\n Taking into account the fact that we can walk from $x$ to  $y$\nin $G$ by following $x,b_1,\\ldots,b_i$ and then moving in at most\none step from $b_i$ to $d_{yv-xv+i}$ and finally traversing from\n$d_{yv-xv+i}$ to $y$ along $P_d$, we get that \\begin{equation}xy\\leq\ni+b_id_{yv-xv+i}+(yv-(xv-i)). \\label{jian}\\end{equation} Similarly,\nstarting from $u,$ we can first walk along $P_a$ and then jump from\n$a_j$ to $b_j$ in at most one step and then walk along $P_b$ to\narrive at $v$. This gives us \\begin{equation}uv\\leq\n(xu-j)+a_jb_j+(xv-j). \\label{guo}\\end{equation} See Fig. \\ref{fig5}.\nAccordingly, we have\n\\begin{equation}\\begin{array}{cll} 2\\delta _G(x,y,u,v) & = &\n(xy+uv)- \\max(xu+yv,xv+yu)   \\ \\ \\ \\ \\text{(By Eq.  \\eqref{Ringel})}\\\\ & \\leq & (xy+uv)-(xu+yv)\\\\\n& \\leq & (i+b_id_{yv-xv+i} +(yv-(xv-i)))+((xu-j)+a_jb_j+(xv-j))-(xu+yv)\\\\\n& = & 2\\pi (b),\n\\end{array} \\label{eq53}  \\end{equation}\nwhich is exactly what we want.\n\\end{proof}\n\n\n\n\n\n\nBrinkmann,  Koolen and Moulton \\cite{BKM01} introduced an\nextremality argument to deduce upper bounds of hyperbolicity of\ngraphs. We follow their approach and make the following  standing\nassumption in the main step of proving Theorems \\ref{main} and\n\\ref{main1} and thus in several subsequent lemmas.\n\n\\paragraph{\\sc \\bf Assumption II:}  We assume  $x,y,u,v$ are four different vertices of $G$ such that\n the sum $xy+uv$ is minimal\nsubject to the condition\n\\begin{equation}\\label{key}  xy+uv=\\max ( xu+yv,xv+yu )+2\\delta^*(G).\n\\end{equation}\n\n\n\n\n  \\rz\n\n\n\n\n\n\n\n\\begin{lemma}  \\cite[p. 67, Claim 1]{BKM01}   \\cite[p. 690, Claim 1]{KM02}  Let $G$ be any graph\nand $u,v,x,y\\in V(G)$.    Under the Assumptions I and II, we have\n            $a_{1}v\\geq xv$, $a_{xu-1}y\\geq uy$, $b_{1}u\\geq xu$, $b_{xv-1}y\\geq\nvy$, $c_{1}v\\geq yv$, $c_{yu-1}x\\geq ux$,  $d_{1}u\\geq yu$,\n$d_{yv-1}x\\geq vx$. \\label{koolen1}\n\\end{lemma}\n\n\n\\begin{proof} By symmetry, we only need to show that  $a_{1}v\\geq\nxv$.  If $a_{1}v< xv$, then,  as a result of  $a_1v\\geq\nxv-xa_1=xv-1$, we have\n\\begin{equation}\\label{eq4}a_{1}v=xv-1.\n\\end{equation} Notice the obvious fact that\n\\begin{equation}\\label{eq4+}a_{1}u=xu-1.\n\\end{equation} We then  come to  the following:\n\\begin{equation}\\begin{array}{cll} a_{1}y+uv & \\geq  &\n(xy-xa_1)+uv\\\\ & = & (xy-1)+uv\\\\& = & (xy+uv)-1\\\\ & = & \\max\n(xu+yv-1, xv+yu-1\n  )+2\\delta^{*}(G)\\ \\ \\ \\ \\ \\ \\   \\text{(By Eq. \\eqref{key})}\\\\ & = &\n\\max  ( a_1u+yv, a_1v+yu )+2\\delta^{*}(G).\\ \\ \\ \\ \\ \\ \\ \\ \\text{(By\nEqs. \\eqref{eq4}  and \\eqref{eq4+})}\n\\end{array}  \\label{eq3}\\end{equation}\n\n\n\n\nAccording to the definition of $\\delta^{*}(G)$,  we read from  Eq.\n\\eqref{eq3} that    $a_{1}y+uv= \\max  (a_1u+yv, a_1v+yu\n  )+2\\delta^{*}(G)$   and hence that      $a_{1}y+uv = xy+uv-1$. This\n contrasts with\n  the minimality of the sum $xy+uv$ (Assumption II), completing the\nproof.\n\\end{proof}\n\n\n\n\\begin{corollary} \\cite[p. 67, Claim 2]{BKM01}       Under Assumptions I and II and stipulating that\n $\\delta^* (G)\\geq 1,$   we have that each corner  of  $\\mathcal\n {Q}(x,u,y,v)$  is not adjacent to its opposite corner and any ordinary vertex\n of  its  opposite sides and hence has degree $2$ in  $\\mathcal\n {Q}(x,u,y,v)$.  \\label{cor2.1}  \\end{corollary}\n\n\n\n\\begin{proof}  By symmetry, we only need to prove the claim for the\ncorner $u$, which\n  directly  follows from Lemmas \\ref{koolen2}  and \\ref{koolen1}.\\end{proof}\n\n\n\n\n\n\n\n\n\\begin{lemma} Suppose that Assumptions I and II are met.  (i)  Any two adjacent sides of $\\mathcal\n {Q}(x,u,y,v)$ only intersect at their common peak.  In particular,\n  no corner of         $\\mathcal\n {Q}(x,u,y,v)$  can be an ordinary vertex of some side of  $\\mathcal\n {Q}(x,u,y,v)$.   (ii)  For any geodesic  connecting two adjacent corners of $\\mathcal\n {Q}(x,u,y,v)$, say  $P=w_0,w_1,\\ldots ,w_m$,  no corner of  $\\mathcal\n {Q}(x,u,y,v)$ can be found among  $\\{w_1,\\ldots ,w_{m-1}\\}.$    (iii)  Let\n$w$ be the common peak of two adjacent sides $P$ and $P'$ of\n$\\mathcal\n {Q}(x,u,y,v)$.\nLet $\\alpha \\in P\\setminus \\{w\\}$ and $\\alpha '\\in P'\\setminus\n\\{w\\}$ be two vertices of $\\mathcal\n {Q}(x,u,y,v)$  such that  $\\alpha \\alpha '=1$, then\n  $\\alpha w=\\alpha 'w$.\n \\label{lem2.4}\n\\end{lemma}\n\n\n\n\\begin{proof}\n(i)  By symmetry, it suffices to prove that  $a_p\\not= b_q$ for any\n$p\\geq q >0.$ Suppose otherwise, it   then follows   that\n$b_1,b_2,\\ldots, b_q=a_p, a_{p+1},\\ldots,a_{xu}=u$ is a path\nconnecting $b_1$ and $u$ and so $b_1u<xu ,$ violating  Lemma\n\\ref{koolen1}.\n\n(ii)  Assume the contrary, we can replace the side of $\\mathcal\n {Q}(x,u,y,v)$ connecting the two asserted adjacent corners by the\n geodesic $P$ and get to a new geodesic quadrangle for which Assumptions I and II still\n hold  but for which\na corner   appears as an ordinary vertex in the side $P,$ yielding a\ncontradiction to (i).\n\n\n(iii)\n It is no loss to merely   prove that if $i,j>0$ and $a_ib_j=1$\nthen\n $i=j$.    In the case of   $i>j,$  $b_1,b_2,\\ldots, b_j,\na_{i},a_{i+1},\\ldots,a_{xu}=u$ is a path connecting $b_1$ and $u$ of\nlength smaller than $xu ,$ contrary to Lemma \\ref{koolen1}.\nSimilarly, $i<j$\n is impossible as well.\n\\end{proof}\n\n\n\n\n\\begin{corollary} \\label{cor45} Let $G$ be a   graph with  $\\delta^*(G)>0$  and let $\\mathcal\n {Q}(x,u,y,v)$ be a geodesic quadrangle for which  Assumptions  I and II\n hold. Then $\\mathcal\n {P}(x,u,y,v)$ is a cycle.  Moreover, if $\\delta^*(G)>\\frac{1}{2}$,\n then all chords of $\\mathcal\n {P}(x,u,y,v)$ must be either $\\mathbb{A}$-edges   or $\\mathbb{H}$-edges.\n\\end{corollary}\n\\begin{proof} This follows from Lemma   \\ref{lem14}, Lemma \\ref{lem2.4} (i)\nand    Corollary \\ref{cor2.1}   in a straightforward fashion.\n\\end{proof}\n\n\nThe next result is very essential to our proof of Theorem\n\\ref{main1} and both its statement and its  proof have their origin\nin  \\cite[p. 65, Prop. 3.1]{BKM01} and \\cite[p. 691, Claim 2]{KM02}.\n\n\n\n\n\\begin{lemma}\nSuppose that $G$ is a graph for which Assumptions I and II are met\nand $\\mathcal\n {Q}(x,u,y,v)$   has  at least one\n$\\mathbb{A}$-edge. Then we have $xu+yv=xv+yu$.\\label{cor12}\n\\end{lemma}\n\n\\begin{proof}\n If the claim were false, without loss of generality, we suppose that\n\\begin{equation}xu+yv>xv+yu.\n\\label{eq5}\n\\end{equation}\nBy symmetry    and because of  Lemma   \\ref{lem2.4}  (iii), let us\nwork under the assumption  that $a_ib_i=1$. It  clearly holds\n\\begin{equation}a_i\\not= x. \\label{clear}\n\\end{equation}\nBefore moving on, let us prove that\n\\begin{equation} a_i\\not= u.\\label{SJTU}\n\\end{equation}\nSuppose for a contradiction that $a_i=u$, we find that $$\n\\begin{array}{cll} yv & \\geq  &\nyu+xv-xu+1\\ \\ \\ \\ \\ \\ \\   \\text{(By Eq. \\eqref{eq5})}\\\\ & = &\nyu+xv-xa_i+1\\\\\n& = &  yu+xv-i+1\\\\\n& = & yu+b_iv+1.\n\\end{array}\n$$\n But we surely have\n$yv\\leq yu+ub_i+b_iv=yu+b_iv+1$ and so we conclude that we can  get\na geodesic $P$ connecting  $y$ to  $v$ in  $G$  by first walking\nalong $P_c$ to go from $y$ to $u$, then moving from $u$ to $b_i$ in\none step and finally traversing  along $P_b$ from $b_i$ to $v$.\nSince this geodesic passes through $u$ in the middle, we obtain a\ncontradiction to Lemma \\ref{lem2.4} (ii) and hence establish Eq.\n\\eqref{SJTU}.\n\nTo go one step further, let us check the following:\n\\begin{equation}\\begin{array}{cll} xv+1  & =  &  xb_i+b_iv+1= xb_i+b_iv+a_ib_i\\\\ & \\geq  &\nxb_i+a_iv=xa_i+a_iv \\\\\n& =  &xa_1+a_1a_i+a_iv \\ \\ \\ \\ \\ \\ \\ \\text{(By Eq.  \\eqref{clear})}\\\\\n& =  & 1+a_1a_i+a_iv\\geq\n 1+a_1v\\\\ & \\geq  & 1+xv.  \\ \\ \\ \\ \\ \\ \\ \\text{(By Lemma\n\\ref{koolen1})}\n\\end{array}  \\label{autumn}\\end{equation}\nClearly,    equalities hold throughout Eq.  \\eqref{autumn}.\n In particular, we have\n\\begin{equation}b_iv+1=a_iv.\n\\label{eqn12}\n\\end{equation}\n From  Eq. \\eqref{eqn12}  and  $xv=b_iv+i$  we   deduce that\n\\begin{equation}\\label{eq8} a_iv=xv-(i-1).\\end{equation}\n  Here comes the punch line of the proof:\n\\begin{equation}\\begin{array}{cll} a_iy+uv & \\geq  &\n(xy-xa_i)+uv\\\\& =  & (xy-i)+uv\\\\ & = & \\max ( xu+yv,xv+yu )\n+2\\delta^*(G)-i \\ \\ \\ \\ \\ \\ \\text{(By Eq. \\eqref{key})}\n\\\\ & = &\n\\max ( (xu-i)+yv, xv+yu-(i-1) ) +2\\delta^*(G)   \\ \\ \\ \\ \\ \\\n\\text{(By Eq. \\eqref{eq5})}\n\\\\ & = &\n\\max\n ((xu-i)+yv,a_iv+yu) +2\\delta^*(G)   \\ \\ \\ \\ \\ \\\n\\text{(By Eq. \\eqref{eq8})}\n\\\\ & = & \\max\n (a_iu+yv,a_iv+yu) +2\\delta^*(G).\n\\end{array}  \\label{eq6}\\end{equation}\nAccording to   Eqs. \\eqref{clear} and \\eqref{SJTU}, we can apply\nLemma \\ref{lem2.4} (i) to find that $a_i,y,u,v$ are four different\nvertices. We further conclude from the definition of $\\delta^*(G)$\nthat Eq. \\eqref{eq6} should hold equalities throughout, hence that\n$xy+uv\\leq a_iy+uv $ as a result of the minimality  of $xy+uv$\n as indicated in  our Assumption II, and finally that the first inequality in Eq.\n\\eqref{eq6}   must be strict in light of Eq. \\eqref{clear}, getting\na contradiction with the assertion that  all equalities in Eq.\n\\eqref{eq6}  hold. This is the end of the proof.\n\\end{proof}\n\n\n\\begin{lemma}  \\label{lem15}  Let  $G$ be a graph     for which Assumptions I and II\nare required.   Suppose that  $\\mathcal\n {Q}(x,u,y,v)$ has  an     $\\mathbb{A}$-edge. If there is $1\\leq i\\leq xu-1$ and\n  $0\\leq j\\leq yv$  such that  $a_id_j\\leq 1$, then  $a_id_j= 1$,\n $a_iu+d_jy=yu$ and  $a_ix+d_jv=xv.$\n\\end{lemma}\n\n\n\\begin{proof} We start with an easy observation:\n\\begin{equation}\\label{EWN} \\left\\{ \\begin{array}{l }\n   a_iu+d_jy=a_{xu-1}a_i+1+d_jy\\geq a_{xu-1}a_i+a_id_j+d_jy\\geq a_{xu-1}y,\\\\\n  a_ix+d_jv=a_{1}a_i+1+d_jv\\geq a_{1}a_i+a_id_j+d_jv\\geq a_{1}v.\n\\end{array}\\right.\n\\end{equation}\n In view of    Lemma \\ref{koolen1}, this says that\n\\begin{equation}a_iu+d_jy\\geq uy,  \\ \\    a_ix+d_jv\\geq xv.\\label{eq13}\n\\end{equation}\n Adding together the two inequalities in Eq. \\eqref{eq13}, we obtain\n\\begin{equation}\\label{eqn14}xu+yv\\geq xv+yu.\n\\end{equation} But, it follows from\n Lemma   \\ref{cor12}  and  the\nexistence of an $\\mathbb{A}$-edge of   $\\mathcal\n {Q}(x,u,y,v)$ that the equality in Eq. \\eqref{eqn14} must occur.\n Consequently, none of the inequalities  in Eqs. \\eqref{EWN}  and \\eqref{eq13} can be strict, which\n is\n exactly what we want to prove.\n\\end{proof}\n\n\n\n\n\n    With a little bit of luck, the forthcoming lemma contributes the number  $ \\frac{ \\lfloor \\frac{k}{2}\\rfloor}{2}$, which\n    is just the mysterious one we find\nin Theorem\n \\ref{main}. Note that $ \\frac{ \\lfloor\n \\frac{k}{2}\\rfloor}{2}$ is  the smallest half integer that is\n greater than $\\frac{k-2}{4}$.\n\n\\begin{lemma} \\label{lemma54} Let $G$ be a $k$-chordal graph for some  $k\\geq 4$ and let $\\mathcal\n {Q}(x,u,y,v)$ be a geodesic quadrangle for which  Assumptions  I and II hold.\n Then we have  $\\delta^*(G)\\leq \\frac{ \\lfloor \\frac{k}{2}\\rfloor}{2}$ provided\n\\begin{equation} \\min(d(P_a,P_d),d(P_b,P_c))> 1.\\label{eq00}\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof} Take $i,j,m,\\ell$ as specified in Lemma \\ref{lem49} and follow all the convention made\nin the statement and the proof of  Lemma \\ref{lem49}.\n Surely, the result is a direct consequence of   Lemma \\ref{lem49}  when\n     \\begin{equation}\\min (\\pi (a), \\pi (b), \\pi (c), \\pi (d))\\leq     \\frac{ \\lfloor \\frac{k}{2}\\rfloor}{2}.\n   \\label{xiang}\\end{equation}\n\n\\begin{figure}\n\\hspace{-33mm}\n\\unitlength 1mm\n\\linethickness{0.4pt}\n\\ifx\\plotpoint\\undefined\\newsavebox{\\plotpoint}\\fi \\hspace{30mm}\n\\begin{center}\n\\begin{picture}(60,60)\n\n\\put(20,0){\\circle*{1}}\\put(20,-4){\\makebox(0,0)[b]{$y$}}\n\\put(0,20){\\circle*{1}}\\put(-3,20){\\makebox(0,0)[l]{$u$}}\n\\put(40,20){\\circle*{1}}\\put(42,20){\\makebox(0,0)[l]{$v$}}\n\\put(20,40){\\circle*{1}}\\put(20,43){\\makebox(0,0)[t]{$x$}}\n\n\\put(7,27){\\circle*{1}}\\put(2,27){\\makebox(0,0)[l]{$a_{m}$}}\n\n\n\\put(15,35){\\circle*{1}}\\put(10,35){\\makebox(0,0)[l]{$a_{j}$}}\n\n\\put(7,13){\\circle*{1}}\\put(-10,13){\\makebox(0,0)[l]{$c_{yu-xu+m}$}}\n\n\n\\put(15,5){\\circle*{1}}\\put(10,5){\\makebox(0,0)[l]{$c_{\\ell}$}}\n\\put(25,5){\\circle*{1}}\\put(30,5){\\makebox(0,0)[r]{$d_{\\ell}$}}\n\n\n\\put(33,13){\\circle*{1}}\\put(35,13){\\makebox(0,0)[l]{$d_{yv-xv+i}$}}\n\n\n\\put(25,35){\\circle*{1}}\\put(30,35){\\makebox(0,0)[r]{$b_{j}$}}\n\n\n\\put(33,27){\\circle*{1}}\\put(38,27){\\makebox(0,0)[r]{$b_{i}$}}\n\n\n\n\\qbezier(15,35)(15,35)(25,35)\\qbezier(7,27)(7,13)(7,13)\\qbezier(7,27)(7,27)(9,29)\n\\qbezier[9](9,29)(9,29)(13,33)\\qbezier(13,33)(13,33)(15,35)\\qbezier(7,13)(7,13)(9,11)\n\\qbezier(25,5)(25,5)(27,7)\\qbezier[9](27,7)(27,7)(31,11)\\qbezier[7](0,20)(5,15)(7,13)\\qbezier(33,13)(33,13)(31,11)\n\\qbezier[7](0,20)(2,22)(7,27)\\qbezier(15,5)(15,5)(13,7)\\qbezier[9](13,7)(9,11)(9,11)\\qbezier(15,5)(15,5)(25,5)\n\\qbezier[7](15,5)(18,2)(20,0)\\qbezier[7](20,0)(22,2)(25,5)\\qbezier[7](33,13)(38,18)(40,20)\n\\qbezier(25,5)(25,5)(27,7)\\qbezier[9](27,7)(27,7)(31,11)\\qbezier[7](15,35)(18,38)(20,40)\\qbezier[7](20,40)(22,38)(25,35)\n\\qbezier[7](33,27)(38,22)(40,20)\\qbezier[9](27,33)(27,33)(31,29)\\qbezier(25,35)(25,35)(27,33)\\qbezier(33,27)(33,27)(31,29)\n\n\\qbezier(33,27)(33,20)(33,13)\n\n \\end{picture}\\end{center}\n\\caption{A chordless cycle in   $\\mathcal\n{Q}(x,u,y,v)$.}\\label{figlem48}\n\\end{figure}\n\nSuppose, for a contradiction,   that the inequality \\eqref{xiang}\ndoes not hold. In this event, as $\\frac{ \\lfloor\n\\frac{k}{2}\\rfloor}{2}\\geq 1$, we know that $\\min ( i-j,\n(yv-xv+i)-\\ell,\n   (yu-xu+m)-\\ell, m-j) \\geq \\min (\\pi(a),\\pi(b),\\pi(c),\\pi(d))-1> 0$.    By virtue of  Lemma   \\ref{lem2.4} (i)  and Eq.  \\eqref{eq00},\n   this implies that\n\\begin{equation}C=[a_jb_jb_{j+1}\\cdots b_id_{yv-xv+i}\\cdots\nd_{\\ell-1}d_{\\ell}c_{\\ell}c_{\\ell+1}\\cdots\nc_{yu-xu+m}a_ma_{m-1}\\cdots a_{j+1}] \\label{cycle}\\end{equation} is\na cycle, where the redundant $a_j$ should be deleted from the above\nnotation when $a_j=b_j=x$,  the redundant $b_i$ should be deleted\nfrom the above notation when $b_i=d_{yv-xv+i}=v$, etc.;    see  Fig.\n  \\ref{figlem48}. Moreover, by Lemma \\ref{lem2.4}  (iii), Eq. \\eqref{eq00}\nand the choice of $i,j,\\ell, m,$ we know that $C$ is even a\nchordless cycle. But the length of $C$ is just $\\pi (a)+\\pi (b)+\\pi\n(c)+\\pi (d)$, which, as the assumption is that \\eqref{xiang} is\nviolated, is no smaller than $4(\\frac{1}{2}+ \\frac{ \\lfloor\n\\frac{k}{2}\\rfloor}{2})$ and hence is at least $k+1.$ This\ncontradicts the assumption that $G$ is $k$-chordal, finishing the\nproof.\n\\end{proof}\n\n\n\\begin{lemma}   Let    $G$ be a  $5$-chordal  graph and\nwe demand that\n Assumptions I and II hold.\nTake  $i,j,\\ell,m$  to be the  numbers as specified in Lemma\n\\ref{lem49}. Suppose that $\\mathcal\n {Q}(x,u,y,v)$ has no  $\\mathbb{H}$-edges and  $\\min(xu,xv,$ $yu,yv$, $2\\delta^*(G))$ $\\geq\n 2$.\nThen we have\n\\begin{equation}a_jb_j+b_id_{yv-xv+i}+c_{\\ell}d_{\\ell}+a_mc_{yu-xu+m}\\geq\n2. \\label{Nippon}\n\\end{equation} Furthermore, we have the following conclusions:    if $a_jb_j=1$, then   $(i,m)\\in  \\{  (j,j), (j, xu), (xv,j)\\};$\nif $b_id_{yv-xv+i}=1$, then  $(j,\\ell)\\in \\{  (i, yv-xv+i), (i,0),\n(0, yv-xv+i) \\}$; if  $c_{\\ell}d_{\\ell}=1,  $    then $(yu-xu+m,\nyv-xv+i)\\in \\{(\\ell , \\ell), (\\ell ,yv), (yu, \\ell )$; if  $a_m\nc_{yu-xu+m}=1,$ then  $(j, \\ell)\\in \\{  (m, yu-xu+m), (m, 0), (0,\nyu-xu+m)\\}$. \\label{lem92}\n\\end{lemma}\n\n\n\\begin{proof}\n Since  $\\delta^*(G)\\geq 1$, it\nfollows from Lemma \\ref{lem49} that\n\\begin{equation}  \\min (i-j,\n(yv-xv+i)-\\ell, (yu-xu+m)-\\ell, m-j)\\geq 0. \\label{mgs}\n\\end{equation}\nUsing Lemma  \\ref{lem14} instead, we obtain from $\\delta^*(G)\\geq 1$\n that $\\min (d(P_a,P_d),d(P_b,P_c))\\geq 1$.\nConsidering Lemma   \\ref{lem2.4} (i) additionally, this says  that\n$G$ has a cycle $C$  as displayed in Eq. \\eqref{cycle} whose length\nis $\\pi (a)+\\pi (b)+\\pi (c)+\\pi (d)$,   where  $\\pi (a), \\pi (b),\n\\pi (c), \\pi (d)$ stand for  the numbers   introduced in Eq.\n\\eqref{PI}. As $\\mathcal\n {Q}(x,u,y,v)$ has no  $\\mathbb{H}$-edges, by the choice of  $i,j,\\ell,m$ and by Lemma  \\ref{lem2.4} (iii), we see that     $C$ is chordless and are thus led\n  to $\\pi (a)+\\pi (b)+\\pi (c)+\\pi (d)\\leq 5.$\n\n\nWe first observe that\n$a_jb_j+b_id_{yv-xv+i}+c_{\\ell}d_{\\ell}+a_mc_{yu-xu+m}>0$; as\notherwise $C$ will be a chordless cycle of length $xv+xu+yv+yu\\geq\n8,$ contradicting $\\mathbbm{l}\\mathbbm{c}(G)\\leq 5.$ Let us proceed\nto consider the case that\n$(a_jb_j,b_id_{yv-xv+i},c_{\\ell}d_{\\ell},a_mc_{yu-xu+m})=(1,0,0,0)$.\nNote that Corollary \\ref{cor2.1} guarantees that $0< j\n<\\min(xu,xv)$. Evoking our assumption $\\min (yv,yu)\\geq 2$, it is\nthen obvious that the cycle $C$ contains at least $6$ different\nvertices $a_j,b_j,v,d_1,y,c_1,u$, which is absurd as $G$ is\n$5$-chordal.  By symmetry, Eq. \\eqref{Nippon} is thus established.\n\nAmong the four conclusions, let us now only deal with the one\naccompanied  with the  assumption   that $a_jb_j=1.$ If\n$a_{m}c_{yu-xu+m}=b_id_{yv-xv+i}=0,$ Eq. \\eqref{Nippon}\n implies $c_{\\ell}d_{\\ell}=1$ and so  $C$ will have at least $6$ different vertices\n$u,a_j,b_j,v,d_{\\ell},c_{\\ell}$, contrary to\n$\\mathbbm{l}\\mathbbm{c}(G)\\leq 5$. To this point, we can conclude\nthat $\\max (b_id_{yv-xv+i}, a_mc_{yu-xu+m})=1$ and so it suffices to\nprove that  $i\\in \\{ j, xv\\}$  and  $m\\in \\{ j, xu\\}.$ We only prove\nthe first claim and the second one will follow by symmetry. Since\nwe already have $i\\geq j$ as guaranteed by Eq. \\eqref{mgs}, our task\nis now to get  from $i>j$ to $i=xv.$ If this\n is not true,  the chordless cycle  $C$ will already have four different\n vertices $a_j,b_j,b_i, d_{yv-xv+i}$, which are all outside of $P_c$\n according to Corollary  \\ref{cor2.1}. Consequently, due to Corollary  \\ref{cor2.1} and $\\mathbbm{l}\\mathbbm{c}(G)\\leq\n 5,$ we find that $C$ must have the fifth vertex $c\\in P_c\\setminus \\{ u,y\\}$ such\n that   $ca_j=cd_{yv-xv+i}=1.$  In view of  Lemma \\ref{lem2.4}\n (iii), we then see that  $$cu=a_ju,cy=\n d_{yv-xv+i}y,a_jx=b_jx,b_iv=d_{yv-xv+1}v.$$ We sum them up and yield\n  $xv+yu=xu+yv+b_ib_j>xu+yv$, which is a contradiction with Lemma\n  \\ref{cor12}.\n This completes the\nproof of the lemma.\n\\end{proof}\n\n\\begin{lemma}\\label{cor10}  Let $G$ be a  graph for which we will make  Assumptions I and II.\nLet $P$ and $P'$    be two adjacent sides of  $\\mathcal\n {Q}(x,u,y,v)$  whose common peak is   $w$.\nLet $\\alpha,  \\beta\\in P\\setminus \\{w\\}$ and $\\alpha', \\beta'\\in\nP'\\setminus \\{w\\}$ be four vertices of $\\mathcal\n {Q}(x,u,y,v)$  such that  $\\alpha\\alpha'=1$ and\n  $\\beta w=\\beta 'w<\\alpha w=\\alpha 'w$. Then it holds   $\\beta \\beta\n  '=1$ in the case that  $G$ is $4$-chordal as well as in the case that   $G$ is $5$-chordal and  $\\beta w=\\beta\n  'w>1$.\n\\end{lemma}\n\n\n\\begin{proof}  By symmetry, we only need to show that  for any $i\\geq\n3$ ($i\\geq 2$) we can obtain from $a_ib_i=1$ that $a_{i-1}b_{i-1}=1$\nprovided $G$ is $5$-chordal ($4$-chordal). But   Lemma \\ref{lem2.4}\n(i) states that $C=[a_{i-1}a_ib_ib_{i-1}\\cdots b_1a_0a_1$ $\\cdots\na_{i-2}]$ is a\n  cycle of length at\n least $7$ ($5$). Thus,  Lemma\n \\ref{lem2.1}  in conjunction with Lemma   \\ref{lem2.4} (iii) applies to give\n $a_{i-1}b_{i-1}=1$, as wanted.\n\\end{proof}\n\n\n\\begin{corollary} \\label{cor15}   Let $G$ be a $5$-chordal graph without isometric $C_4$ for which we will make  Assumptions I and\nII. If there is an   $\\mathbb{A}$-edge connecting $\\alpha$ and\n$\\alpha '$ lying in two adjacent sides $P$ and $P'$ with common peak\n$w$, respectively, then   this  is the only $\\mathbb{A}$-edge\nbetween  $P$  and  $P'$  and  $\\alpha w=\\alpha 'w \\leq 2.$\n\\end{corollary}\n\n\\begin{proof}\nThis follows directly from Lemmas \\ref{lem2.4} (iii) and\n\\ref{cor10}.\n\\end{proof}\n\n\n\n\n\n\\begin{lemma} \\label{lem23}        Let  $G$ be a $5$-chordal  graph with $\\delta^*(G)\\geq 1$ and let  Assumptions I and II hold.\nAssume that  $\\mathcal {Q}(x,u,y,v)$\n has no $\\mathbb{A}$-edges. (i)\n  If  there  is an $\\mathbb{H}$-edge\n between $P_b$ and  $P_c$, then $\\max (xu, yv)\\leq 2.$  (ii)  If  there  is an $\\mathbb{H}$-edge\n between $P_a$ and  $P_d$, then $\\max (xv, yu)\\leq 2.$\n\\end{lemma}\n\n\n\n\\begin{proof}  We only prove  $xu\\leq 2$   under the assumption that   there  is an $\\mathbb{H}$-edge\n between $P_b$ and  $P_c$ and all the other claims follow similarly.\nTake the minimum $i$ such that $b_i$ is incident with an\n$\\mathbb{H}$-edge and then pick the maximum  $j$ such that $b_ic_j$\nis an $\\mathbb{H}$-edge.  By Corollary  \\ref{cor2.1}, we have $\\min\n(i,  yu-j)\\geq 1$.\n      Since $\\mathcal {Q}(x,u,y,v)$\n has no $\\mathbb{A}$-edges, we find that  $$[b_1\\cdots b_ic_jc_{j+1}\\cdots c_{yu-1}ua_{xu-1}\\cdots\n a_1x]$$\n is a chordless cycle of length $xu+1+i+(yu-j)\\geq xu+3$. Finally, because  $G$\n is\n $5$-chordal, we conclude that $xu\\leq 2,$ as desired.\n\\end{proof}\n\n\n\n\n\\begin{lemma}\\label{lem19}\nLet  $G$ be a  $5$-chordal  graph with   $\\delta^*(G)\\geq 1$.  We\nkeep Assumptions I and II. In addition, assume that  $\\mathcal\n {Q}(x,u,y,v)$ has a side of length one. Then,  $G$ contains at least one\n  graph among $C_4,H_3$ and  $H_5$  as an isometric subgraph.\n\\end{lemma}\n\n\\begin{proof}\nIt involves  no restriction  of generality in assuming that $xv=1.$\nOwning to Lemma  \\ref{lem2.4} (i), the walk along $P_a, P_c$ and\n$P_d$ will connect $x$ and  $v$ without passing through $x$ or $v$\nin the middle and hence there is a shortest path connecting $x$ and\n$v$ in the graph obtained from $\\mathcal\n {Q}(x,u,y,v)$ by deleting the edge $\\{x,v\\}$. This\n says that $\\mathcal\n {Q}(x,u,y,v)$  has an induced cycle passing through $x$ and  $v$\n contiguously, say  $C=[w_1w_2\\cdots w_n]$, where  $w_1=x$ and $w_2=v.$\nFrom Corollary \\ref{cor2.1} we know that $w_3=d_{yv-1}\\not= a_1=w_n$\nand hence  $n>3$. Since $G$ is $5$-chordal, our task is to derive\nthat if  $n=5$ then  $G$ contains an isometric $C_4,$ $H_3$ or\n$H_5$.\n\n\n\n\n\n\n\\begin{figure}\n\\hspace{-33mm}\n\\unitlength 1mm\n\\linethickness{0.4pt}\n\\ifx\\plotpoint\\undefined\\newsavebox{\\plotpoint}\\fi \\hspace{30mm}\n\\begin{center}\n\\begin{picture}(80,25)\n\n\\put(20,5){\\circle*{1}}\\put(24,1){\\makebox(0,0)[b]{$y=w_3$}}\n\\put(10,5){\\circle*{1}}\\put(10,1){\\makebox(0,0)[b]{$w_4$}}\n\\put(10,15){\\circle*{1}}\\put(5,17){\\makebox(0,0)[l]{$w_5$}}\n\\put(0,5){\\circle*{1}}\\put(-3,3){\\makebox(0,0)[l]{$u$}}\n\\put(30,15){\\circle*{1}}\\put(32,15){\\makebox(0,0)[l]{$v=w_2$}}\n\\put(15,20){\\circle*{1}}\\put(20,23){\\makebox(0,0)[t]{$x=w_1$}}\n\\put(10,-5){\\makebox(0,0)[b]{$H_3$}}\n\\qbezier(0,5)(10,15)(15,20)\\qbezier(0,5)(0,5)(20,5)\\qbezier(10,15)(10,10)(10,5)\n\\qbezier(20,5)(25,10)(30,15)\\qbezier(30,15)(20,18)(15,20)\n\n\n\n\n\\put(70,5){\\circle*{1}}\\put(75,1){\\makebox(0,0)[b]{$y=w_3$}}\n\\put(60,5){\\circle*{1}}\\put(60,1){\\makebox(0,0)[b]{$w_4$}}\n\\put(60,17){\\circle*{1}}\\put(55,19){\\makebox(0,0)[l]{$w_5$}}\n\\put(40,5){\\circle*{1}}\\put(37,3){\\makebox(0,0)[l]{$u$}}\n\\put(80,15){\\circle*{1}}\\put(82,15){\\makebox(0,0)[l]{$v=w_2$}}\n\\put(65,20){\\circle*{1}}\\put(70,23){\\makebox(0,0)[t]{$x=w_1$}}\n\n\\put(50,5){\\circle*{1}}\\put(50,1){\\makebox(0,0)[b]{$c_2$}}\n\\put(50,11){\\circle*{1}}\\put(45,12){\\makebox(0,0)[l]{$a_2$}}\n\n\\qbezier(40,5)(60,17)(65,20)\\qbezier(40,5)(60,5)(70,5)\\qbezier(70,5)(75,10)(80,15)\n\\qbezier(80,15)(70,18)(65,20)\\qbezier(60,5)(60,10)(60,17)\n\\put(55,-5){\\makebox(0,0)[b]{$H_5$}}\n \\end{picture}\n\\end{center}\n \\caption{Case 1 in the proof  of Lemma \\ref{lem19}.}\\label{case1}\n\\end{figure}\n\n\\paragraph{\\sc Case 1:}  $w_3$  is a\ncorner of  $\\mathcal\n {Q}(x,u,y,v)$, namely  $yv=1$.\n\nIn light of Corollary \\ref{cor2.1}, we have  $w_4=c_1$. If $c_1=u$\nor  $w_5=u$ occurs,  then   $\\mathcal\n {Q}(x,u,y,v) $ turns out to be a $5$-cycle and hence has hyperbolicity $\\frac{1}{2}$.  This is impossible as Assumption II means that this hyperbolicity\n can be no smaller than  $\\delta^*(G)\\geq 1$. Accordingly, by\n   Lemma \\ref{lem2.4} (iii)   we know that  $w_4u$  and   $w_5u$ have a common value, say  $m.$\n\n   If $m>3 $   or  there are two  $\\mathbb{A}$-edges between  $P_a$ and  $P_c$,     Corollary \\ref{cor15} says that\n $G$  contains an isometric $C_4.$\n\n\nWhen $m=2$ and there are no two $\\mathbb{A}$-edges between  $P_a$\nand  $P_c$,  the graph $H_5$ as depicted on the right of Fig.\n\\ref{case1} is an induced graph of $G$. Utilizing Eq.  \\eqref{key}\nand the assumption that  $\\delta^*(G)\\geq 1$, we find that\n$$4=3+1=ux+xv\\geq uv\\geq xv+uy+2\\delta^*(G)-xy=2+2\\delta^*(G)\\geq 4.$$ This\nillustrates that $uv=4$.  It follows from Lemma \\ref{koolen1} that\n$a_2y\\geq uy=3$. In addition, we have $a_2y\\leq\na_2a_1+a_1c_1+c_1y=3$ and so we see that $a_2y=3.$ Similarly, we\nhave $c_2x=3.$ Getting that $a_2y=c_2x=3$ and $uv=4,$ we apply\n  Corollary  \\ref{lem31} and conclude that the above-mentioned  $H_5$ must be an isometric\n subgraph of  $G.$\n\n\n\nWhen $m=1,$ the graph $H_3$ as depicted on the left of Fig.\n\\ref{case1} is an induced graph of $G$.  As in the case of  $m=2$,\nwe make use of  Eq.  \\eqref{key} and  $\\delta^*(G)\\geq 1$ to get an\nimportant  information: $$3=uw_5+w_5x+xv \\geq uv\\geq xv+uy\n+2\\delta^*(G)-xy=1+2\\delta^*(G)\\geq 3.$$ This implies $uv=3$ and\nhence we deduce from    Corollary \\ref{lem30} that this $H_3$ is\neven an isometric subgraph of $G$.\n\n\n\n\n \\paragraph{\\sc Case 2:}  $w_5$  is a\ncorner of  $\\mathcal\n {Q}(x,u,y,v)$, namely $xu=1$.\n\nThe analysis is symmetric to that of Case 1.\n\n \\paragraph{\\sc Case 3:} Neither $w_3$ nor $w_5$  is a\ncorner.  In this case, Corollary \\ref{cor2.1} ensures that $w_4$ is\nnot a corner as well.  We proceed to show that this case indeed\ncannot happen.\n\n \\paragraph{\\sc Case 3.1:}  $w_4\n \\in P_a$.\n\n Since  $P_a$ is a geodesic, we get that $w_4=a_2.$\n    It is easy to see that\n $xy\\leq vy+vx=vy+1$ and that\n$uv=uw_2\\leq w_2w_3+w_3w_4+w_4u=2+a_2u=xu.$\n Adding together, we obtain      by Assumption II that\n\\begin{equation*}   2\\delta^*(G)=(xy+uv)-\\max(xv+yu,vy+xu)\\leq (xy+uv)-(vy+xu)\\leq\n 1,\n\\end{equation*}   violating the assumption that $\\delta^*(G)\\geq 1.$\n\n \\paragraph{\\sc Case 3.2:}  $w_4\n \\in P_d$.\n\n\n\n\nReasoning as in Case 3.1   rules out the possibility that this case\nmay happen.\n\n\n \\paragraph{\\sc Case 3.3:}  $w_4\n \\in P_c$.\n\n\n\nIn this case,  $\\mathcal\n {Q}(x,u,y,v)$   contains  $\\mathbb{A}$-edges  and hence  Lemma\n \\ref{cor12} tells us\n \\begin{equation}\\label{e33}\n  xu+yv=xv+yu.\n \\end{equation} But, by\nLemma  \\ref{lem2.4} (iii) we have  $xu-1=uw_5=uw_4$ and  $yv-1=w_3y=\nw_4y$. We therefore get\n  that $xu+yv=2+uw_4+w_4y=2+yu=1+xv+yu,$ which contradicts    Eq. \\eqref{e33} and finishes the proof.\n\\end{proof}\n\n\n\n\\begin{lemma}     Let  $G$ be a $5$-chordal  graph  and let  Assumptions I and II hold.\nIf\n $\\min (ux,xv,uy,yv,2\\delta^*(G))\\geq 2$, and $\\mathcal {Q}(x,u,y,v)$\n has no $\\mathbb{A}$-edges, then $\\delta^*(G)= 1$  and   either   $ux=xv=uy=yv=2$  or  $G$ has an isometric $4$-cycle.\n \\label{lem2.6}\\end{lemma}\n\n\\begin{proof} By\nCorollary \\ref{cor45}, $\\mathcal {P}(x,u,y,v)$ is a cycle of length\nat least $8$. As $G$ is  $5$-chordal, this cycle must have chords,\nwhich, by Corollary \\ref{cor45} again and by the fact that\n$\\mathcal {Q}(x,u,y,v)$\n has no $\\mathbb{A}$-edges, must\nbe\n  $\\mathbb{H}$-edges. So, without loss of generality, suppose that  $\\mathcal {Q}(x,u,y,v)$\n has an  $\\mathbb{H}$-edge between $P_a$ and  $ P_d$.\nOn the one hand, we can thus go to  Lemma   \\ref{lem23} and get\n\\begin{equation}\\label{eq24}   xv=yu=2.\n\\end{equation}\nOn the other hand, this allows us to apply\n Lemmas \\ref{lem14}  and  \\ref{lemma41}  to deduce that  $\\delta^*(G)= 1$  and that\n either $G$ has an isometric  $C_4$ or  has exactly  one  $\\mathbb{H}$-edge\n between  $P_a$ and  $P_d.$  If the latter case happens, say  we\n have  an  $\\mathbb{H}$-edge connecting  $a_i$ and  $d_j,$ we will\n get two chordless cycles of  $G$, $[a_ia_{i-1}\\cdots xb_1\\cdots b_{xv-1}vd_{yv-1}\\cdots d_j]$\nand   $[a_ia_{i+1}\\cdots uc_{yu-1}\\cdots c_1yd_1\\cdots d_j]$. Since\nneither of these two chordless cycles can be longer than $5$, it\nfollows from   Eq. \\eqref{eq24} that $a_ix+d_jv\\leq 2$ and\n$ua_i+yd_j\\leq 2.$ Taking into account additionally     that $2\\leq\nux=ua_i+a_ix$ and $2\\leq yv=yd_j+d_jv$, we thus have  $ xu=yv=2.$\nThis is the end of the proof.\n\\end{proof}\n\n\n\n\n\n\\begin{lemma}  We take a $5$-chordal graph  $G$ satisfying $\\delta^*(G)=1$ and   require  Assumptions I and II. Suppose that  $\\mathcal\n {Q}(x,u,y,v)$ has no  $\\mathbb{H}$-edge and   $[ua_{xu-1}b_{xu-1}d_{yu-1}c_{yu-1}]$  is an induced $5$-cycle of  $G$; see Fig. \\ref{fig924}. Then $G$ has at\n least one graph among $C_4$, $H_3$   and  $H_5$ as an isometric subgraph.\n \\label{lem54}\n\\end{lemma}\n\n\n\n\n\\begin{figure}\n\\hspace{-33mm}\n\\unitlength 1mm\n\\linethickness{0.4pt}\n\\ifx\\plotpoint\\undefined\\newsavebox{\\plotpoint}\\fi \\hspace{30mm}\n\\begin{center}\n\\begin{picture}(60,60)\n\n\\put(20,0){\\circle*{1}}\\put(20,-4){\\makebox(0,0)[b]{$y$}}\n\\put(0,20){\\circle*{1}}\\put(-3,20){\\makebox(0,0)[l]{$u$}}\n\\put(40,20){\\circle*{1}}\\put(42,20){\\makebox(0,0)[l]{$v$}}\n\\put(20,40){\\circle*{1}}\\put(20,43){\\makebox(0,0)[t]{$x$}}\n\n\\put(5,25){\\circle*{1}}\\put(-6,25){\\makebox(0,0)[l]{$a_{xu-1}$}}\n\n\\put(15,35){\\circle*{1}}\\put(10,35){\\makebox(0,0)[l]{$a_{1}$}}\n\\put(5,15){\\circle*{1}}\\put(-6,15){\\makebox(0,0)[l]{$c_{yu-1}$}}\n\n\\put(15,5){\\circle*{1}}\\put(10,5){\\makebox(0,0)[l]{$c_{1}$}}\n\\put(25,5){\\circle*{1}}\\put(30,5){\\makebox(0,0)[r]{$d_{1}$}}\n\n\\put(30,10){\\circle*{1}}\\put(42,10){\\makebox(0,0)[r]{$d_{yu-1}$}}\n\n\\put(25,35){\\circle*{1}}\\put(30,35){\\makebox(0,0)[r]{$b_{1}$}}\n\n\\put(30,30){\\circle*{1}}\\put(40,30){\\makebox(0,0)[r]{$b_{xu-1}$}}\n\n\\qbezier[7](5,15)(10,10)(15,5)\n\\qbezier(0,20)(5,25)(5,25)\\qbezier(0,20)(5,15)(5,15)\n\\qbezier(15,5)(18,2)(20,0)\\qbezier(20,0)(22,2)(25,5)\\qbezier[7](30,30)(40,20)(40,20)\\qbezier(35,25)(40,20)(40,20)\n\\qbezier[7](25,5)(30,10)(35,15)\\qbezier(15,35)(18,38)(20,40)\\qbezier(20,40)(22,38)(25,35)\n\\qbezier[7](30,10)(30,10)(40,20)\\qbezier(35,15)(30,10)(40,20)\\qbezier[7](25,35)(30,30)(35,25)\\qbezier[7](5,25)(10,30)(15,35)\n\\qbezier(5,25)(17,27.5)(30,30)\\qbezier(5,15)(15,13)(30,10)\\qbezier(30,30)(30,20)(30,10)\n \\end{picture}\\end{center}\n\\caption{$[ua_{xu-1}b_{xu-1}d_{yu-1}c_{yu-1}]$ is an induced\n$5$-cycle in $\\mathcal\n {Q}(x,u,y,v)$.}\\label{fig924}\n\\end{figure}\n\n\\begin{proof}\nBy  Corollary  \\ref{cor2.1} and Lemma  \\ref{lem2.4} (iii),\n it will be enough to consider the following  cases,\n$b_{xu-1}v=d_{yu-1}v>3 $ or $b_{xu-1}v=d_{yu-1}v\\in \\{ 1,2\\}.$\n\n\n\n\\paragraph{\\sc Case 1:} $b_{xu-1}v=d_{yu-1}v>3.$\n\nCorollary \\ref{cor15} implies that $G$ contains an isometric $C_4$.\n\n\n\n\n\n\n\\paragraph{\\sc Case 2:} $b_{xu-1}v=d_{yu-1}v\\in \\{1,2\\}.$\n\n\n\n\nBefore jumping into the analysis of two separate subcases, we make\nsome general observations. Note that\n\\begin{equation}\\begin{array}{cll} xu+yv+2 & = &\n(xa_{xu-1}+ua_{xu-1})+(yd_{yu-1}+ d_{yu-1}v)+2\n\\\\ &=&xb_{xu-1}+ua_{xu-1}+yd_{yu-1}+b_{xu-1}v+(a_{xu-1}b_{xu-1}+b_{xu-1}d_{yu-1})\n\\\\ &=& (xb_{xu-1}+b_{xu-1}d_{yu-1}+yd_{yu-1})+(ua_{xu-1}+a_{xu-1}b_{xu-1}+b_{xu-1}v)\\\\\n &\\geq& xy+(ua_{xu-1}+a_{xu-1}v) \\\\   &\\geq& xy+uv \\\\ &=&\\max(xu+yv,xv+yu)+2\\delta^*(G)   \\ \\ \\ \\ \\text{(By Assumption II)}\\\\\n& \\geq & xu+yv+2.    \\ \\ \\ \\ \\text{(By $\\delta^*(G)=1$)}\n\\end{array} \\label{eq924}  \\end{equation}\nIt follows that all  inequalities in Eq. \\eqref{eq924}  are best\npossible     and hence  we have\n\\begin{equation}uv=ua_{xu-1}+a_{xu-1}b_{xu-1}+b_{xu-1}v=2+b_{xu-1}v\\label{925}\n\\end{equation}\nand\n\\begin{equation}a_{xu-1}v=a_{xu-1}b_{xu-1}+b_{xu-1}v=1+b_{xu-1}v.\\label{108}\n\\end{equation}\n\n\\paragraph{\\sc Case 2.1:} $b_{xu-1}v=d_{yu-1}v=1$.\n\nWe derive from Corollary  \\ref{cor2.1} that the subgraph  of  $G$\ninduced by $u,a_{xu-1},b_{xu-1},v,d_{yu-1},c_{yu-1}$ is isomorphic\nto $H_3$ in an obvious way. Thanks to Corollary \\ref{lem30}, in\norder to check that this  $H_3$ is isometric, our task is to show\nthat $uv=3.$ But $uv=3$  is an immediate result of\n Eq.  \\eqref{925},  proving the claim in this case.\n\n\\paragraph{\\sc Case 2.2:} $b_{xu-1}v=d_{yu-1}v=2$.\n\n\nTo start things off  we look at  the following:\n\\begin{equation}\\begin{array}{cll} b_{xu-1}v+1 & = &  d_{yu-1}v+1   =d_{yu-1}d_{yv-1}+2  \\\\ &=&\nd_{yu-1}d_{yv-1}+   a_{xu-1}b_{xu-1}+b_{xu-1}d_{yu-1}\n\\\\ &\\geq &a_{xu-1}d_{yv-1}   \\ \\ \\ \\ \\text{(By the triangle inequality)}\n\\\\ &\\geq & xd_{yv-1}-xa_{xu-1}  \\ \\ \\ \\ \\text{(By the triangle inequality)}  \\\\\n&\\geq& xv -xa_{xu-1}   \\ \\ \\ \\ \\text{(By Lemma   \\ref{koolen1})}          \\\\ &=&  (xb_{xu-1}+b_{xu-1}v)-xa_{xu-1} \\\\\n& = & b_{xu-1}v.\n\\end{array} \\label{KIM}  \\end{equation}\nA consequence of  Eq. \\eqref{KIM} is  that\n\\begin{equation}b_{xu-1}v+1\\geq a_{xu-1}d_{yv-1}\\geq\nb_{xu-1}v.\\label{44}\n\\end{equation}\nBy symmetry, we also have\n\\begin{equation}b_{xu-1}v+1= d_{yu-1}v+1  \\geq c_{yu-1}b_{xv-1}\\geq d_{yu-1}v=b_{xu-1}v.\n\\label{45}\n\\end{equation}\nAs a result of  Eqs.   \\eqref{44} and \\eqref{45} we\n  obtain\n \\begin{equation}3\\geq \\max (a_{xu-1}d_{yv-1},   c_{yu-1}b_{xv-1})\\geq \\min   (a_{xu-1}d_{yv-1},   c_{yu-1}b_{xv-1}) \\geq\n  2.\\label{eq47}\n  \\end{equation}\nFinally,  let us remark    that  $b_{xu-1}v=2$ implies $xu-1=xv-2$\nand hence\n  $b_{xu-1}b_{xv-1}=d_{xu-1}d_{xv-1}=1.$\n\nAccording to Eq. \\eqref{eq47}, the following two subcases are\nexhaustive.\n\n\n\n\n\\paragraph{\\sc Case 2.2.1:}\n$\\min(a_{xu-1}d_{yv-1},c_{yu-1}b_{xv-1})=2$.\n\n\n Without loss of\ngenerality, we suppose that there is a vertex $w\\in V(G)$ such that\n$a_{xu-1}w=wd_{yv-1}=1$. Note that Lemma  \\ref{lem2.4} (iii) says\nthat $w\\notin \\{a_{xu-1},b_{xu-1},b_{xv-1},v,d_{yv-1}\\} $  and hence\n    $C= [a_{xu-1}b_{xu-1}b_{xv-1}vd_{yv-1}w]$ is a $6$-cycle in  $G$.\nBecause $G$ is $5$-chordal, $C$  has at least one chord. Observe\nthat   Eq.   \\eqref{108} says that\n\\begin{equation} a_{xu-1}v=1+2=3\\label{eq57}\n\\end{equation}\nand so\n$$wv\\geq   a_{xu-1}v-a_{xu-1}w =3-1=2.$$\nIn consequence, by  virtue of   Lemma  \\ref{lem2.4} (iii), we have\n  $\\min ( wb_{xu-1}, wb_{xv-1},  b_{xv-1}d_{yv-1} )=1.$  There are\n  three cases to dwell on.\n\\paragraph{\\sc Case 2.2.1.1:} If  $b_{xv-1}d_{yv-1}=1,$ then\n$[b_{xu-1}b_{xv-1}d_{yv-1}d_{xu-1}]$ is a required isometric\n$4$-cycle.\n\n\\paragraph{\\sc Case 2.2.1.2:}  If $wb_{xv-1}=1$ and  $b_{xv-1}d_{yv-1}>1$, we find that\n$[b_{xv-1}vd_{yv-1}w]$ is an isometric $4$-cycle, as desired.\n\n\n\n\n\\paragraph{\\sc Case 2.2.1.3:}\nIf $\\min (wb_{xv-1}, b_{xv-1}d_{yv-1})    >wb_{xu-1}=1$, as a result\nof Eq.  \\eqref{eq57},\n we can   make use of Corollary \\ref{lem30}\nto yield that the subgraph induced by\n$a_{xu-1},b_{xu-1},b_{xv-1},v,d_{yv-1},w$ is an isometric $H_3$ in\n$G$; see Fig. \\ref{fig9.24}.\n\n\n\n\n\\begin{figure}\n\\hspace{-33mm}\n\\unitlength 1mm\n\\linethickness{0.4pt}\n\\ifx\\plotpoint\\undefined\\newsavebox{\\plotpoint}\\fi \\hspace{30mm}\n\\begin{center}\n\\begin{picture}(30,20)\n\\put(0,15){\\circle*{1}}\\put(-5,18){\\makebox(0,0)[l]{$a_{xu-1}$}}\n\\put(0,0){\\circle*{1}}\\put(0,-4){\\makebox(0,0)[b]{$w$}}\n\\put(15,5){\\circle*{1}}\\put(19,5){\\makebox(0,0)[r]{$v$}}\n\\put(10,0){\\circle*{1}}\\put(10,-4){\\makebox(0,0)[b]{$d_{yv-1}$}}\n\n\\put(10,10){\\circle*{1}}\\put(20,10){\\makebox(0,0)[r]{$b_{xv-1}$}}\n\\put(5,15){\\circle*{1}}\\put(15,15){\\makebox(0,0)[r]{$b_{xu-1}$}}\n\n\\qbezier(0,15)(0,15)(0,0)\\qbezier(0,0)(10,0)(10,0)\\qbezier(0,15)(0,15)(5,15)\n\\qbezier(5,15)(5,15)(15,5)\\qbezier(15,5)(15,5)(10,0)\\qbezier(0,0)(5,15)(5,15)\n \\end{picture}\\end{center}\n\\caption{Case 2.2.1.3 in the proof of Lemma\n\\ref{lem54}.}\\label{fig9.24}\n\\end{figure}\n\n\\paragraph{\\sc Case 2.2.2:}           $a_{xu-1}d_{yv-1}=c_{yu-1}b_{xv-1}=3$.\n\nFrom Eq.  \\eqref{925} we obtain $uv=4.$ This, together with the\nstanding assumption of Case 2.2, enables us to deduce from Corollary\n\\ref{lem31}   that the subgraph induced by\n$$u,a_{xu-1},b_{xu-1},b_{xv-1},v,d_{yv-1},d_{yu-1},c_{yu-1}$$\nis an isometric $H_5$.\n\\end{proof}\n\n\n\n\n\n\\subsection{Proofs of Theorems \\ref{main} and\n\\ref{main1}}\\label{Proof}\n\n\nWe now have all necessary tools to prove our main results.\n\n\\rz\n\n\n\n\n\n\n\\par \\noindent \\textbf{Proof of  Theorem \\ref{main}: }\nUsing typical compactness argument, it suffices to prove that every\n connected finite  induced  subgraph of a $k$-chordal graph $G$ is $\\frac{\\lfloor\n\\frac{k}{2}\\rfloor}{2}$-hyperbolic. If $G$ has less than $4$\nvertices, the result is trivial.  Thus, we can simply assume that\n$4\\leq |V(G)| <\\infty $   and henceforth there surely exists\n a\ngeodesic quadrangle  $\\mathcal\n {Q}(x,u,y,v)$ in  $G$  fulfilling\nAssumptions I and II. When\n $\\min(d(P_a,P_d),d(P_b,P_c))\\leq 1$, the result is direct from  Lemma\n\\ref{lem14} and the fact that  $1\\leq \\frac{\\lfloor\n\\frac{k}{2}\\rfloor}{2}$         while when\n $\\min(d(P_a,P_d),d(P_b,P_c))>\n1$ we are done by  Lemma \\ref{lemma54}. {\\QED\\par \\bigskip \\par}\n\n\n\n\n\n\n\\par \\noindent \\textbf{Proof of  Theorem \\ref{main1}: }\nConsider a $5$-chordal graph $G$ with  $\\delta^*(G)=1$.   We surely\ncan get a  geodesic quadrangle  $\\mathcal\n {Q}(x,u,y,v)$ in  $G$  for which   Assumption I and Assumption II\n hold. Passing to the proof that $G$ contains one graph from  Fig. \\ref{fig0} as an isometric\n subgraph, we have to distinguish four main  cases.\n\n\n\\paragraph {\\sc Case 1:}\n $\\min (xu,xv,yu,yv)=1. $\n\n  Lemma  \\ref{lem19} tells\nus that $G$ has either  an isometric $C_4$ or an isometric $H_3$ or\nan isometric  $H_5$.\n\n\n\n\\paragraph {\\sc Case 2:}\n $\\min (xu,xv,yu,yv)\\geq 2$ and there exist   no\n$\\mathbb{A}$-edges.\n\n\n\n\\paragraph {\\sc Case 2.1:}   $\\max (xu,xv,yu,yv)>2$.\n\nBy Lemma \\ref{lem2.6},  $G$ must have an isometric\n $C_4$.\n\n\n\n\n\n\n\\paragraph {\\sc Case 2.2:}  $xu=xv=yu=yv=2$.\n\n\n\nBy Corollary \\ref{cor45},  $\\mathcal\n {Q}(x,u,y,v)$  must have an $\\mathbb{H}$-edge. By Corollary\n \\ref{cor2.1}, we may assume, without loss of generality, that\n $a_1d_1=1.$  It then follows from Lemma \\ref{lemma41} (i) that $xy=uv=3.$\n\n\n\n\\paragraph {\\sc Case 2.2.1:}\n $\\mathcal\n {Q}(x,u,y,v)$  has only one  $\\mathbb{H}$-edge  and hence the subgraph of  $G$ induced by  its vertices is isomorphic to  $H_5$.\n\nBy     Lemma  \\ref{lem27}, $G$ has one of $C_4,H_2,H_3$   and  $H_5$\nas an isometric subgraph.\n\n\\paragraph {\\sc Case 2.2.2:}\n $\\mathcal\n {Q}(x,u,y,v)$  has  two  $\\mathbb{H}$-edges    and hence the subgraph of  $G$ induced by  its vertices is isomorphic to  $H_4$.\n\n\nBy Corollary \\ref{lem29},  $G$  contains $H_4$ as an isometric\nsubgraph.\n\n\n\n\n\n\n\n\\paragraph {\\sc Case 3:}   $\\min (xu,xv,yu,yv)\\geq 2$ and there exist no\n$\\mathbb{H}$-edges.\n\n\n\nTake  $i,j,\\ell,m$  to be the  numbers as specified in Lemma\n\\ref{lem49}. By  Lemma \\ref{lem92}, Eq. \\eqref{Nippon} holds. So,\nwithout loss of generality, we can assume that $i=j,$ $a_jb_j=1$ and\n$b_jd_{yv-xv+j}=1.$\n\n\\paragraph {\\sc Case 3.1:}  $d_{\\ell}c_{\\ell}=a_mc_{yu-xu+m}=1$.\n\n\n\nBy Lemma  \\ref{lem92}, the chordless cycle displayed  in Eq.\n\\eqref{cycle} is an isometric $C_4.$\n\n\n\n\\begin{figure}\n\\hspace{-33mm}\n\\unitlength 1mm\n\\linethickness{0.4pt}\n\\ifx\\plotpoint\\undefined\\newsavebox{\\plotpoint}\\fi \\hspace{30mm}\n\\begin{center}\n\\begin{picture}(60,60)\n\n\\put(20,0){\\circle*{1}}\\put(20,-4){\\makebox(0,0)[b]{$y$}}\n\\put(0,20){\\circle*{1}}\\put(-3,20){\\makebox(0,0)[l]{$u$}}\n\\put(40,20){\\circle*{1}}\\put(41,20){\\makebox(0,0)[l]{$v$}}\n\\put(20,40){\\circle*{1}}\\put(20,43){\\makebox(0,0)[t]{$x$}}\n\\put(12,32){\\circle*{1}}\\put(7,32){\\makebox(0,0)[l]{$a_{j}$}}\n\n\n\\put(28,32){\\circle*{1}}\\put(32,32){\\makebox(0,0)[r]{$b_{j}$}}\n\\put(28,8){\\circle*{1}}\\put(30,8){\\makebox(0,0)[l]{$d_{yv-xv+j}$}}\n\n\\put(12,8){\\circle*{1}}\\put(10,8){\\makebox(0,0)[r]{$c_{yu-xu+j}$}}\n\n\\qbezier(12,32)(12,20)(12,8)\n\\qbezier[7](5,15)(10,10)(15,5)\\qbezier(0,20)(2,22)(5,25)\\qbezier(0,20)(2,18)(5,15)\n\\qbezier(12,8)(18,2)(20,0)\\qbezier(20,0)(22,2)(28,8)\\qbezier(35,15)(38,18)(40,20)\n\\qbezier[7](25,5)(30,10)(35,15)\\qbezier(15,35)(18,38)(20,40)\\qbezier(20,40)(22,38)(25,35)\n\\qbezier(35,25)(38,22)(40,20)\\qbezier[7](25,35)(30,30)(35,25)\\qbezier[7](5,25)(10,30)(15,35)\n\n\\qbezier(28,32)(28,20)(28,8)\\qbezier(12,32)(20,32)(28,32)\n\n\n \\end{picture}\\end{center}\n\\caption{Case 3.2 in the proof of Theorem \\ref{main1}.}\n\\label{case3.2}\n\\end{figure}\n\n\n\n\\paragraph {\\sc Case 3.2:}  $(d_{\\ell}c_{\\ell}, a_mc_{yu-xu+m})=\n(0,1)$ or  $(1,0)$.\n\n\nWe only consider the case that  $(d_{\\ell}c_{\\ell}, a_mc_{yu-xu+m})=\n(0,1)$.  For now, the chordless cycle   shown  in Eq. \\eqref{cycle}\nis just  the  $5$-cycle $[a_jb_jd_{yv-xv+j}yc_{yu-xu+j}];$ see\n  Fig.  \\ref{case3.2}. Lemma \\ref{lem54}  demonstrates that  $G$   contains one\ngraph among $C_4,H_3$ and  $H_5$ as an isometric subgraph.\n\n\n\n\\paragraph {\\sc Case 3.3:}  $d_{\\ell}c_{\\ell}=a_mc_{yu-xu+m}=0$.\n\nThis case is impossible as the chordless cycle  demonstrated  in Eq.\n\\eqref{cycle} will contain $6$ different vertices  $a_j,b_j,\nd_{yv-xv+j}, y,c_1, u$.\n\n\n\\paragraph {\\sc Case 4:}   $\\min (xu,xv,yu,yv)\\geq 2$ and there exist both $\\mathbb{H}$-edges\nand $\\mathbb{A}$-edges.\n\n\n\n\nBefore delving into the case by case analysis, here are some general\nobservations.\n First note that\n Lemma  \\ref{cor12} can be applied to give\n \\begin{equation} \\label{eq30}  xu+yv=xv+yu.\n \\end{equation}\nSecondly, according to Corollary \\ref{cor2.1},\n we can suppose that\nthere are \\begin{equation}1\\leq i\\leq xu-1\\  \\ \\text{ and} \\ \\ 1\\leq\nj\\leq yv-1 \\label{eq31}\\end{equation} such that $a_id_j=1$ and,\n by Lemma \\ref{lem15},  hence  that\n\\begin{equation}a_iu+d_jy=yu\\ \\ \\text{and} \\ \\  a_ix+d_jv=xv.\n\\label{eq23}\n\\end{equation}\n Thirdly, as    $\\delta^*(G)=a_id_j=1$,  Lemma \\ref{lem14} gives\n \\begin{equation}d(P_a,P_d)=1.\n \\label{JAIST}\n \\end{equation}\nFinally,\n  Lemma \\ref{lemma41} (i) says that the  $\\mathcal\n {Z}$-walks of   $\\mathcal\n {Q}(x,u,y,v)$   through the  $\\mathbb{H}$-edge  $\\{a_i,d_j\\}$ must\n be\n geodesics.  Since any subpath of a geodesic is still a geodesic, we\n come to\n \\begin{equation}\nud_j=ua_i+a_id_j=ua_i+1 \\ \\text{and } \\ \\  a_iy=a_id_j+d_jy=1+d_jy.\n\\label{eqn33}\n\\end{equation}\n\n\n\n\n\\paragraph {\\sc Case 4.1:} $yu=xv=2$.\n\nIn this case,  Eq.  \\eqref{eq30} forces $xu=xv=yu=yv=2$ and so Eq.\n\\eqref{eq31} tells us that    $i=j=1.$      It follows that  $\\max\n(xy, uv)\\leq 3$ due to the existence of the path $x,a_1,d_1,y$ and\nthe path $u,a_1,d_1,v.$ For the moment, in view of Eq. \\eqref{key},\nwe can get  \\begin{equation}xy=uv=3.\\label{eq33}\n\\end{equation}\n\n\nIdentifying  $a_1,b_1,c_1,d_1$ with $a,b,c,d,$ respectively,\nCorollary \\ref{cor2.1} says that  $\\mathcal\n {Q}(x,u,y,v)$ is obtained from the graph  $H_5$ as depicted in Fig.   \\ref{fig0}  by adding\n $t$ additional edges among\n$\\{a,b\\},\\{b,d\\},  \\{d,c\\}, \\{c,a\\}$, where $t\\in \\{1,2,3,4\\}$, and\nadding possibly the edge $\\{b,c\\}$.\n\n\n\n\nIf $t=4$ and $bc=1$,   we easily infer from Eq. \\eqref{eq33} and\nCorollary \\ref{lem29}  that $\\mathcal\n {Q}(x,u,y,v)$  is an isometric\nsubgraph of $G$ which is isomorphic to $H_2$.\n\n\nIf   $t=4$  and $bc>1$, we can check that   $\\mathcal\n {Q}(x,u,y,v)$  is an induced\nsubgraph of $G$ isomorphic to   $H_1$  and then, again    by\n  Eq.  \\eqref{eq33}\nand   Corollary\n \\ref{lem29},  $G$  contains an  isometric   $H_1$.\n\n\n\nIf $t<4$, as a consequence of Lemma \\ref{lead},  either $C_4$\n is   an induced subgraph of  $G$ or   $\\mathcal\n {Q}(x,u,y,v)$ is isomorphic with $H_6$.          Accordingly,  Eq.  \\eqref{eq33} together with    Lemma  \\ref{lem10}   implies  that  $G$\n has an isometric  subgraph which is isomorphic to either  $C_4$  or\n  $H_2$  or  $H_3.$\n\n\n   \\paragraph {\\sc Case 4.2:} $\\max (yu, xv) >2$.\n\n\n\n\nWe will show that $G$ contains an isometric subgraph which is\nisomorphic to    $H_3$, under the assumption that $G$ has no\nisometric $C_4$.  Note that the nonexistence of an isometric $C_4$\nin $G$  together with Eq.  \\eqref{JAIST} yields that there exists\nexactly one $\\mathbb{H}$-edge between $P_a$ and $P_d$, namely\n$\\{a_i,d_j\\}$, as a result of Lemma \\ref{lemma41} (i).\n\nIt is no loss of generality in setting\n\\begin{equation}      yu>2.  \\label{eqn35}\\end{equation}\n By Lemma \\ref{lem2.4} (i)  and Eq.  \\eqref{JAIST}, the following is a set   of pairwise different\nvertices:\n $$y,c_1,\\ldots, c_{yu-1},u,\na_{xu-1},\\ldots,a_i,d_j,d_{j-1},\\ldots ,d_1.$$ In the subgraph $F$\ninduced by these  vertices  in  $G$, $a_i$ and $d_j$ are connected\nby a path disjoint from the edge $\\{a_i,d_j\\}.$ This means that\nthere is a chordless cycle $[w_1w_2\\cdots w_n]$ in $F$ where $n\\geq\n3$ and $w_1 =a_i,w_2=d_j$.   Recall that it is already stipulated\nthat the $5$-chordal graph $G$ has no isometric $4$-cycle  and hence\n$n$ can only take value either $3$ or  $5$.\n\n\n\n\\begin{figure}\n\\hspace{-33mm}\n\\unitlength 1mm\n\\linethickness{0.4pt}\n\\ifx\\plotpoint\\undefined\\newsavebox{\\plotpoint}\\fi \\hspace{30mm}\n\\begin{center}\n\\begin{picture}(50,50)\n\n\n\n\n\n\\put(30,0){\\circle*{1}}\\put(30,-4){\\makebox(0,0)[b]{$y$}}\n\n\\put(20,0){\\circle*{1}}\\put(20,-4){\\makebox(0,0)[b]{$c_1$}}\n\\put(10,0){\\circle*{1}}\\put(10,-4){\\makebox(0,0)[b]{$c_2$}}\n\n\\put(0,20){\\circle*{1}}\\put(-4,23){\\makebox(0,0)[lt]{$a_i$}}\n\n\\put(30,10){\\circle*{1}}\\put(35,10){\\makebox(0,0)[r]{$d_1$}}\n\\put(30,20){\\circle*{1}}\\put(35,23){\\makebox(0,0)[rt]{$d_2$}}\n\n\n\\qbezier(10,0)(5,10)(0,20)\\qbezier(0,20)(10,20)(30,20)\\qbezier(30,20)(30,10)(30,0)\\qbezier(30,0)(20,0)(10,0)\n\\qbezier(10,0)(20,10)(30,20)\n\n \\end{picture}\n\\end{center}\n\n\\caption{Case 4.2.1 in the proof of Theorem\n\\ref{main1}.}\\label{fig4.2.1}\n\\end{figure}\n\n\n \\paragraph {\\sc Case 4.2.1:}  $n=3$.\n\nSince there is exactly one $\\mathbb{H}$-edge between $P_a$ and\n$P_d$, $w_3$ is neither on\n  $P_a$ nor on   $P_d$. Hence,  there is   $0< q < yu$ such that   $w_3=c_q$. It follows from\n   Lemma \\ref{lem2.4} (iii) that $ua_i=uc_q$ and  $yc_q=yd_j$.\nFrom  Eq. \\eqref{eqn35} we obtain $\\max (yc_q, c_qu) \\geq 2.$\nWithout loss of generality, assume that $yc_q=\\max (yc_q, c_qu) \\geq\n2$. Since $G$ contains no isometric $C_4$, we infer from Corollary\n\\ref{cor15} that $q=j=2$ and $c_1d_1=2$. This then demonstrates that\nthe subgraph induced by the vertices $a_i,d_2,d_1,y,c_1,c_2$ is\n isomorphic    to $H_3$;  see Fig.  \\ref{fig4.2.1}.\nGranting that $a_iy=3$,\n Corollary  \\ref{lem30} will bring to us that $G$ contains $H_3$ as an\n isometric subgraph.\n But $a_iy=3$  follows from  Eq. \\eqref{eqn33} and  $d_jy=j=2.$\n\n\n\n\n\n\n\n \\paragraph {\\sc Case 4.2.2:}  $n=5$.\n\n\n\n  We aim to prove that this case will never happen by deducing contradictions in all the following subcases.\n\n\n\n \\paragraph {\\sc Case 4.2.2.1:}  Both $w_3$  and  $w_5$ belong to $P_c.$\n\n\nFirst consider the case that both $w_3$  and  $w_5$ are ordinary\n vertices of $P_c.$\n From Lemma   \\ref{lem2.4} (iii) we obtain $a_iu=uw_5$ and  $d_jy=w_3y.$\n It then follows  $uy=uw_5+w_3y$   by means of Eq. \\eqref{eq23}. Since $w_3$ and  $w_5$ are on the same geodesic connecting\n $u$ and  $y$, this is  possible only when\n $w_3=w_5$, yielding a contradiction.\n\n Next the case that at least one of  $w_3$  and  $w_5$ is a  corner.  We could assume that\n  $w_3$ is a corner, and then, in view of Corollary   \\ref{cor2.1},\nit holds  $w_3=y$. This implies that $w_5\\not= u$, as otherwise we\nobtain $yu=w_3w_5= 2,$ contradicting Eq.  \\eqref{eqn35}.\nAccordingly, it follows from  Lemma   \\ref{lem2.4} (iii) that\n$a_iu=uw_5$. But, we surely have\n $ 2 = w_5w_3=w_5y$  and  $  yd_j=w_3w_2=1.$ Putting together, we get\n$a_iu+yd_j=uw_5+1<uw_5+2=uw_5+w_5y=uy$, contradicting Eq.\n\\eqref{eq23}.\n\n\n\n\n \\paragraph {\\sc Case 4.2.2.2:} Neither $w_3$  nor  $w_5$ belongs to $P_c.$\n\n\nBecause there is just one $\\mathbb{H}$-edge between $P_a$ and $P_d$,\nwe see that $w_3,w_4,w_5$ are ordinary vertices of  $P_d, P_c$ and\n$P_a$, respectively.  By Lemma   \\ref{lem2.4} (iii), it occurs\n that  $a_iu=1+w_5u=1+w_4u$  and  $d_jy=1+w_3y=1+w_4y$. Consequently, we arrive at\n $a_iu+d_jy=2+uy,$ which is contrary to Eq.    \\eqref{eq23}.\n\n\n\n  \\paragraph {\\sc Case 4.2.2.3:}  One of  $w_3$  and  $w_5$ is outside of\n  $P_c$ and the other lies inside $P_c.$\n\n\n\nIncurring no loss of generality,  we make the assumption that\n$w_5\\notin P_c$ and $w_3\\in P_c$.  As there exists only one\n$\\mathbb{H}$-edge between $P_a$ and $P_d$, we know that\n$w_5=a_{i+1}$ and either $w_4=a_{i+2},w_3\\not=y$ or $w_4\\in\nP_c\\setminus \\{u\\}$. By Lemma \\ref{lem2.4} (iii), the former implies\nthat $uy=uw_3+w_3y= uw_4+w_3y= uw_4+yd_j=ua_i+yd_j-2$, violating Eq.\n\\eqref{eq23}.\n In consequence, we must have $w_4\\in P_c\\setminus \\{u\\}$ and hence it\nholds either $w_3=c_t,w_4=c_{t+1}$\n or $w_3=c_{t+1},w_4=c_{t}$  for some $t<yu-1$.\nIf it happens the latter case, we deduce from  Lemma \\ref{lem2.4}\n(iii) that $uy=uw_4+w_3y-1=uw_5+d_jy-1=ua_i+d_jy-2, $ yielding a\ncontradiction with the first part of   Eq.  \\eqref{eq23}. At this\npoint, our object is to\n exclude the first possibility as well.\nBy way of contradiction, let us assume that this case happens and\nturn to  the quartet $(w_1,w_2,w_3,u)$. The following calculation\ncan be trivially verified:\n$$uw_1+w_2w_3=uw_1+1;$$\n$$\\begin{array}{cll} uw_2+w_1w_3 & = &  (uw_1+1) +2  \\ \\ \\ \\ \\text{(By the first part of Eq.  \\eqref{eqn33})}\n\\\\\n& = & uw_1+3;\n\\end{array} $$\n$$\\begin{array}{cll} uw_3+w_1w_2 & = &  (uw_4+1)+1\n\\\\\n& = & (uw_5+1)+1     \\ \\ \\ \\ \\text{(By Lemma  \\ref{lem2.4} (iii))}\\\\\n& = &  uw_1+1.\n\\end{array} $$\n  This gives $\\delta\n(u, w_1,w_2,w_3)=1=\\delta^*(G)$ and $\\max (uw_2+w_1w_3, uw_1+w_2w_3,\nuw_3+w_1w_2 )=uw_1+3=(uw_1+1)+2\\leq ux+yv\\leq\nxy+uv-2\\delta^*(G)=xy+uv-2$, which is the desired contradiction to\nAssumption II  on $\\mathcal\n {Q}(x,u,y,v)$.\n {\\QED\\par \\bigskip \\par}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}