Isometric path complexity of graphs

We introduce and study a new graph parameter, called the \emph{isometric path complexity} of a graph. A path is \emph{isometric} if it is a shortest path between its endpoints. 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 end-vertices of all the isometric paths in $S$. The \emph{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 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, \emph{hyperbolic graphs}, \emph{(theta, prism, pyramid)-free graphs}, and \emph{outerstring graphs}. Hyperbolic graphs are extensively studied in \emph{Metric Graph Theory}. The class of (theta, prism, pyramid)-free graphs are extensively studied in \emph{Structural Graph Theory}, \textit{e.g.} in the context of the \emph{Strong Perfect Graph Theorem}. The class of outerstring graphs is studied in \emph{Geometric Graph Theory} and \emph{Computational Geometry}. Our results also show that the distance functions of these (structurally) different graph classes are more similar than previously thought. Finally, we apply this new concept to the ISOMETRIC PATH COVER problem, whose objective is to cover all vertices of a graph with a minimum number of isometric paths, to all the above graph classes. Indeed, 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.