Isometric path complexity of graphs

TL;DR

This paper introduces the concept of isometric path complexity in graphs, provides an algorithm to compute it, and shows it remains bounded in certain graph classes, enabling efficient approximation algorithms for isometric path cover problems.

Contribution

It defines isometric path complexity, develops an $O(n^2 m)$-time algorithm to compute it, and demonstrates its boundedness in specific graph classes, leading to efficient approximation algorithms.

Findings

Algorithm computes isometric path complexity in $O(n^2 m)$ time.

Isometric path complexity is bounded in hyperbolic, (theta, prism, pyramid)-free, and outerstring graphs.

Bounded isometric path complexity yields polynomial-time constant-factor approximation for ISOMETRIC PATH COVER.

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.