eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-08-21
32:1
32:14
10.4230/LIPIcs.MFCS.2023.32
article
Isometric Path Complexity of Graphs
Chakraborty, Dibyayan
1
Chalopin, Jérémie
2
https://orcid.org/0000-0002-2988-8969
Foucaud, Florent
3
https://orcid.org/0000-0001-8198-693X
Vaxès, Yann
2
Univ Lyon, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR5668, France
Laboratoire d'Informatique et Systèmes, Aix-Marseille Université and CNRS, Faculté des Sciences de Luminy, F-13288 Marseille, Cedex 9, France
Université Clermont Auvergne, CNRS, Mines Saint-Étienne, Clermont Auvergne INP, LIMOS, 63000 Clermont-Ferrand, France
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol272-mfcs2023/LIPIcs.MFCS.2023.32/LIPIcs.MFCS.2023.32.pdf
Shortest paths
Isometric path complexity
Hyperbolic graphs
Truemper Configurations
Outerstring graphs
Isometric Path Cover