Isometric path antichain covers: beyond hyperbolic graphs

The \emph{isometric path antichain cover number} of a graph $G$, denoted by $ipacc(G)$, is a graph parameter that was recently introduced to provide a constant factor approximation algorithm for \textsc{Isometric Path Cover}, whose objective is to cover all vertices of a graph with a minimum number of isometric paths (i.e. shortest paths between their end-vertices). This parameter was previously shown to be bounded for chordal graphs and, more generally, for graphs of bounded \emph{chordality} and bounded \emph{treelength}. In this paper, we show that the isometric path antichain cover number 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 imply a constant factor approximation algorithm for \textsc{Isometric Path Cover} on all the above graph classes. Our results also show that the distance functions of these (structurally) different graph classes are more similar than previously thought.