📝 SummarXiv

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 a few ``rooted" isometric paths (i.e., isometric paths with a common end-vertex). We show that important graph classes studied in \emph{coarse graph theory} have bounded strong isometric path complexity. 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 that $K_4^-$-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. Hence, for a graph $H$ on at most four vertices, $H$-asymptotic minor-free graphs have bounded (strong) isometric path complexity if and only if $H\not=K_4$. 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., \textsc{JCTB '24}]) form a subclass of $U_4$-asymptotic minor-free graphs. Hence, the strong isometric path complexity of monoholed graphs is bounded. On the other hand, we show that even-hole free graphs have unbounded strong isometric path complexity. We investigate which graph operations preserve strong isometric path complexity. We show that the strong isometric path complexity is preserved under the \emph{fixed power} and \emph{line graph} operators, two important graph operations. We also show that the \emph{clique-sums} of finitely many graphs with strong isometric path complexity at most $k$, yield a graph with strong isometric path complexity at most $3k+18$.