Chakraborty, D. orcid.org/0000000305346417, Chalopin, J, Foucaud, F et al. (1 more author) (2023) Isometric Path Complexity of Graphs. In: Leroux, J, Lombardy, S and Peleg, D, (eds.) Proceedings of 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023), 28 Aug  01 Sep 2023, Bordeaux, France. Leibniz International Proceedings in Informatics (LIPIcs), 272 . Schloss Dagstuhl  LeibnizZentrum fur Informatik , Dagstuhl, Germany , 32:132:14. ISBN 9783959772921
Abstract
A set S of isometric paths of a graph G is “vrooted”, where v is a vertex of G, if v is one of the endvertices 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 vrooted isometric paths. First, we provide an O(n 2m)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 kfactor approximation algorithm for Isometric Path Cover, whose objective is to cover all vertices of a graph with a minimum number of isometric paths.
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Authors/Creators: 


Copyright, Publisher and Additional Information:  © Dibyayan Chakraborty, Jérémie Chalopin, Florent Foucaud, and Yann Vaxès. This is an open access conference paper under the terms of the Creative Commons Attribution License (CCBY 4.0), which permits unrestricted use, distribution and reproduction in any medium, provided the original work is properly cited. 
Keywords:  Shortest paths, Isometric path complexity, Hyperbolic graphs, Truemper Configurations, Outerstring graphs, Isometric Path Cover 
Dates: 

Institution:  The University of Leeds 
Academic Units:  The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Computing (Leeds) 
Depositing User:  Symplectic Publications 
Date Deposited:  21 Sep 2023 10:21 
Last Modified:  21 Sep 2023 10:21 
Status:  Published 
Publisher:  Schloss Dagstuhl  LeibnizZentrum fur Informatik 
Series Name:  Leibniz International Proceedings in Informatics (LIPIcs) 
Identification Number:  https://doi.org/10.4230/LIPIcs.MFCS.2023.32 