Strong isometric path complexity of graphs: Asymptotic minors, restricted holes, and graph operations

Abstract: The (strong) isometric path complexity is a recently introduced graph invariant that captures how arbitrary isometric paths (i.e. shortest paths) of a graph can be viewed as a union of few ``rooted" isometric paths (i.e. isometric paths with a common end-vertex). It is known that this parameter can be computed optimally in polynomial time. Seemingly unrelated graph classes studied in metric graph theory (e.g. hyperbolic graphs), geometric intersection graph theory (e.g. outerstring graphs), and structural graph theory (e.g. (theta, prism, pyramid)-free graphs) have been shown to have bounded strong isometric path complexity [Chakraborty et al., MFCS '23]. We show that important graph classes studied in \emph{coarse graph theory} (as introduced by [Georgakopoulos & Papasoglu '23]) have bounded strong isometric path complexity. We show that the strong isometric path complexity of $K_{2,t}$-asymptotic minor-free graphs is bounded. Let $U_t$ denote the graph obtained by adding a universal vertex to a path of $t-1$ edges. We show that the strong isometric path complexity of $U_t$-asymptotic minor-free graphs is bounded. This implies $K_4^-$-asymptotic minor-free graphs, i.e. graphs that are quasi-isometric to a cactus [Fujiwara & Papasoglu '23] have bounded strong isometric path complexity. On the other hand, $K_4$-minor-free graphs have unbounded strong isometric path complexity. We show that graphs whose all induced cycles of length at least 4 have the same length (also known as monoholed graphs as defined by [Cook et al., JCTB '24]) form a subclass of $U_4$-asymptotic minor-free graphs. Hence, the strong isometric path complexity of monoholed graphs is bounded. We show that even-hole free graphs have unbounded strong isometric path complexity. We show that the strong isometric path complexity is preserved under the fixed power, line graph, and clique-sums operators.