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arXiv · 2301.00278

Isometric path complexity of graphs

Dibyayan Chakraborty·Jérémie Chalopin·Florent Foucaud·Yann Vaxès
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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 endpoints 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 $v\in V(G)$ satisfying the following property: the vertices of any single isometric path $P$ of $G$ can be covered by $k$ many $v$-rooted isometric paths. First, we provide an $O(n^2 m)$-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. There is a direct algorithmic consequence of having small isometric path complexity. Specifically, we show that if the isometric path complexity of a graph $G$ is bounded by a constant, then there exists a polynomial-time constant-factor approximation algorithm for ISOMETRIC PATH COVER, whose objective is to cover all vertices of a graph with a minimum number of isometric paths. This applies to all the above graph classes.

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BibTeXRIS

Dibyayan Chakraborty, Jérémie Chalopin, Florent Foucaud, Yann Vaxès. 2022-12-31. Isometric path complexity of graphs. https://doi.org/10.1016/j.disc.2025.114743

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