Isometric path complexity of graphs

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 ∈ 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(n²m)-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.