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(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 kfactor approximation algorithm for Isometric Path Cover, whose objective is to cover all vertices of a graph with a minimum number of isometric paths.
BibTeX  Entry
@InProceedings{chakraborty_et_al:LIPIcs.MFCS.2023.32,
author = {Chakraborty, Dibyayan and Chalopin, J\'{e}r\'{e}mie and Foucaud, Florent and Vax\`{e}s, Yann},
title = {{Isometric Path Complexity of Graphs}},
booktitle = {48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)},
pages = {32:132:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959772921},
ISSN = {18688969},
year = {2023},
volume = {272},
editor = {Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2023/18566},
URN = {urn:nbn:de:0030drops185666},
doi = {10.4230/LIPIcs.MFCS.2023.32},
annote = {Keywords: Shortest paths, Isometric path complexity, Hyperbolic graphs, Truemper Configurations, Outerstring graphs, Isometric Path Cover}
}
Keywords: 

Shortest paths, Isometric path complexity, Hyperbolic graphs, Truemper Configurations, Outerstring graphs, Isometric Path Cover 
Collection: 

48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023) 
Issue Date: 

2023 
Date of publication: 

21.08.2023 