Isometric Path Complexity of Graphs
A set S of isometric paths of a graph G is "v-rooted", where v is a vertex of G, if v is one of the end-vertices of all the isometric paths in S. The isometric path complexity of a graph G, denoted by ipco (G), is the minimum integer k such that there exists a vertex v ∈ V(G) satisfying the following property: the vertices of any isometric path P of G can be covered by k many v-rooted isometric paths.
First, we provide an O(n² m)-time algorithm to compute the isometric path complexity of a graph with n vertices and m edges. Then we show that the isometric path complexity remains bounded for graphs in three seemingly unrelated graph classes, namely, hyperbolic graphs, (theta, prism, pyramid)-free graphs, and outerstring graphs. Hyperbolic graphs are extensively studied in Metric Graph Theory. The class of (theta, prism, pyramid)-free graphs are extensively studied in Structural Graph Theory, e.g. in the context of the Strong Perfect Graph Theorem. The class of outerstring graphs is studied in Geometric Graph Theory and Computational Geometry. Our results also show that the distance functions of these (structurally) different graph classes are more similar than previously thought.
There is a direct algorithmic consequence of having small isometric path complexity. Specifically, using a result of Chakraborty et al. [ISAAC 2022], we show that if the isometric path complexity of a graph G is bounded by a constant k, then there exists a k-factor approximation algorithm for Isometric Path Cover, whose objective is to cover all vertices of a graph with a minimum number of isometric paths.
Shortest paths
Isometric path complexity
Hyperbolic graphs
Truemper Configurations
Outerstring graphs
Isometric Path Cover
Theory of computation~Design and analysis of algorithms
32:1-32:14
Regular Paper
https://arxiv.org/abs/2301.00278
We thank Nicolas Trotignon for suggesting us to study the class of (t-theta, t-pyramid, t-prism)-free graphs.
Dibyayan
Chakraborty
Dibyayan Chakraborty
Univ Lyon, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR5668, France
Jérémie
Chalopin
Jérémie Chalopin
Laboratoire d'Informatique et Systèmes, Aix-Marseille Université and CNRS, Faculté des Sciences de Luminy, F-13288 Marseille, Cedex 9, France
https://orcid.org/0000-0002-2988-8969
This author was financed by the ANR projects DISTANCIA (ANR-17-CE40-0015) and DUCAT (ANR-20-CE48-0006).
Florent
Foucaud
Florent Foucaud
Université Clermont Auvergne, CNRS, Mines Saint-Étienne, Clermont Auvergne INP, LIMOS, 63000 Clermont-Ferrand, France
https://perso.limos.fr/ffoucaud/
https://orcid.org/0000-0001-8198-693X
This author was financed by the ANR project GRALMECO (ANR-21-CE48-0004-01) and the French government IDEX-ISITE initiative 16-IDEX-0001 (CAP 20-25).
Yann
Vaxès
Yann Vaxès
Laboratoire d'Informatique et Systèmes, Aix-Marseille Université and CNRS, Faculté des Sciences de Luminy, F-13288 Marseille, Cedex 9, France
This author was financed by the ANR project DISTANCIA (ANR-17-CE40-0015)
10.4230/LIPIcs.MFCS.2023.32
P. Aboulker, M. Chudnovsky, P. Seymour, and N. Trotignon. Wheel-free planar graphs. European Journal of Combinatorics, 49:57-67, 2015.
I. Abraham, C. Gavoille, A. Gupta, O. Neiman, and K. Talwar. Cops, robbers, and threatening skeletons: Padded decomposition for minor-free graphs. SIAM Journal on Computing, 48(3):1120-1145, 2019.
M. Aigner and M. Fromme. A game of cops and robbers. Discrete Applied Mathematics, 8(1):1-12, 1984.
T. Biedl, A. Biniaz, and M. Derka. On the size of outer-string representations. In 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018), 2018.
P. Bose, P. Carmi, J. M. Keil, A. Maheshwari, S. Mehrabi, D. Mondal, and M. Smid. Computing maximum independent set on outerstring graphs and their relatives. Computational Geometry, 103:101852, 2022.
J. Cardinal, S. Felsner, T. Miltzow, C. Tompkins, and Birgit Vogtenhuber. Intersection graphs of rays and grounded segments. In International Workshop on Graph-Theoretic Concepts in Computer Science, pages 153-166. Springer, 2017.
D. Chakraborty, A. Dailly, S. Das, F. Foucaud, H. Gahlawat, and S. K. Ghosh. Complexity and algorithms for ISOMETRIC PATH COVER on chordal graphs and beyond. In Proceedings of the 33rd International Symposium on Algorithms and Computation, ISAAC, volume 248 of LIPIcs, pages 12:1-12:17, 2022.
V. Chepoi, F. Dragan, B. Estellon, M. Habib, and Y. Vaxès. Diameters, centers, and approximating trees of δ-hyperbolic geodesic spaces and graphs. In Proceedings of the twenty-fourth annual symposium on Computational geometry, pages 59-68, 2008.
V. Chepoi, F. Dragan, M. Habib, Y. Vaxès, and H. Alrasheed. Fast approximation of eccentricities and distances in hyperbolic graphs. Journal of Graph Algorithms and Applications, 23(2):393-433, 2019.
V. Chepoi, F. F. Dragan, and Y. Vaxes. Core congestion is inherent in hyperbolic networks. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 2264-2279. SIAM, 2017.
M. Chudnovsky, N. Robertson, P. Seymour, and R. Thomas. The strong perfect graph theorem. Annals of mathematics, pages 51-229, 2006.
M. Conforti, G. Cornuéjols, A. Kapoor, and K. Vušković. Universally signable graphs. Combinatorica, 17(1):67-77, 1997.
D. G. Corneil, B. Dalton, and M. Habib. Ldfs-based certifying algorithm for the minimum path cover problem on cocomparability graphs. SIAM Journal on Computing, 42(3):792-807, 2013.
D. G. Corneil, S. Olariu, and L. Stewart. Asteroidal triple-free graphs. SIAM Journal on Discrete Mathematics, 10(3):399-430, 1997.
D. Coudert, A. Nusser, and L. Viennot. Enumeration of far-apart pairs by decreasing distance for faster hyperbolicity computation. arXiv preprint, 2021. URL: https://arxiv.org/abs/2104.12523.
https://arxiv.org/abs/2104.12523
B. Das Gupta, M. Karpinski, N. Mobasheri, and F. Yahyanejad. Effect of Gromov-hyperbolicity parameter on cuts and expansions in graphs and some algorithmic implications. Algorithmica, 80(2):772-800, 2018.
J. Davies and R. McCarty. Circle graphs are quadratically χ-bounded. Bulletin of the London Mathematical Society, 53(3):673-679, 2021.
É. Diot, M. Radovanović, N. Trotignon, and K. Vušković. The (theta, wheel)-free graphs Part I: only-prism and only-pyramid graphs. Journal of Combinatorial Theory, Series B, 143:123-147, 2020.
M. Francis, P. Hell, and J. Stacho. Forbidden structure characterization of circular-arc graphs and a certifying recognition algorithm. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1708-1727. SIAM, 2014.
M. Gromov. Hyperbolic groups. In Essays in group theory, pages 75-263. Springer, 1987.
J. M. Keil, J.S.B Mitchell, D. Pradhan, and M. Vatshelle. An algorithm for the maximum weight independent set problem on outerstring graphs. Computational Geometry, 60:19-25, 2017.
A. Kosowski, B. Li, N. Nisse, and K. Suchan. k-chordal graphs: From cops and robber to compact routing via treewidth. Algorithmica, 72(3):758-777, 2015.
J. Kratochvíl. String graphs. I. the number of critical nonstring graphs is infinite. Journal of Combinatorial Theory, Series B, 52(1):53-66, 1991.
A. Rok and B. Walczak. Outerstring graphs are χ-bounded. SIAM Journal on Discrete Mathematics, 33(4):2181-2199, 2019.
Y. Shavitt and T. Tankel. On the curvature of the internet and its usage for overlay construction and distance estimation. In IEEE INFOCOM 2004, volume 1. IEEE, 2004.
M. Thiessen and T. Gaertner. Active learning of convex halfspaces on graphs. In Proceedings of the 35th Conference on Neural Information Processing Systems, NeurIPS 2021, volume 34, pages 23413-23425. Curran Associates, Inc., 2021. URL: https://proceedings.neurips.cc/paper/2021/file/c4bf1e24f3e6f92ca9dfd9a7a1a1049c-Paper.pdf.
https://proceedings.neurips.cc/paper/2021/file/c4bf1e24f3e6f92ca9dfd9a7a1a1049c-Paper.pdf
N. Trotignon. Perfect graphs: a survey. arXiv preprint, 2013. URL: https://arxiv.org/abs/1301.5149.
https://arxiv.org/abs/1301.5149
N. Trotignon. Private communication, 2022.
K. Vušković. The world of hereditary graph classes viewed through truemper configurations. Surveys in Combinatorics 2013, 409:265, 2013.
J. A. Walter and H. Ritter. On interactive visualization of high-dimensional data using the hyperbolic plane. In Proceedings of the eighth ACM SIGKDD international conference on Knowledge discovery and data mining, pages 123-132, 2002.
M. E. Watkins and D. M. Mesner. Cycles and connectivity in graphs. Canadian Journal of Mathematics, 19:1319-1328, 1967.
Dibyayan Chakraborty, Jérémie Chalopin, Florent Foucaud, and Yann Vaxès
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