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Isometric Path Complexity of Graphs

Authors Dibyayan Chakraborty, Jérémie Chalopin , Florent Foucaud , Yann Vaxès

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Author Details

Dibyayan Chakraborty
  • Univ Lyon, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR5668, France
Jérémie Chalopin
  • Laboratoire d'Informatique et Systèmes, Aix-Marseille Université and CNRS, Faculté des Sciences de Luminy, F-13288 Marseille, Cedex 9, France
Florent Foucaud
  • Université Clermont Auvergne, CNRS, Mines Saint-Étienne, Clermont Auvergne INP, LIMOS, 63000 Clermont-Ferrand, France
Yann Vaxès
  • Laboratoire d'Informatique et Systèmes, Aix-Marseille Université and CNRS, Faculté des Sciences de Luminy, F-13288 Marseille, Cedex 9, France

Funding

  • Chalopin, Jérémie: This author was financed by the ANR projects DISTANCIA (ANR-17-CE40-0015) and DUCAT (ANR-20-CE48-0006).
  • Foucaud, Florent: 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).
  • Vaxès, Yann: This author was financed by the ANR project DISTANCIA (ANR-17-CE40-0015)

Acknowledgements

We thank Nicolas Trotignon for suggesting us to study the class of (t-theta, t-pyramid, t-prism)-free graphs.

Cite As Get BibTex

Dibyayan Chakraborty, Jérémie Chalopin, Florent Foucaud, and Yann Vaxès. Isometric Path Complexity of Graphs. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 32:1-32:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.MFCS.2023.32

Abstract

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Shortest paths
  • Isometric path complexity
  • Hyperbolic graphs
  • Truemper Configurations
  • Outerstring graphs
  • Isometric Path Cover

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