Few Induced Disjoint Paths for $H$-Free Graphs

Abstract: Paths $P<sup>1,\ldots,P<sup>k$ in a graph $G=(V,E)$ are mutually induced if any two distinct $P<sup>i$ and $P<sup>j$ have neither common vertices nor adjacent vertices. For a fixed integer $k$, the $k$-Induced Disjoint Paths problem is to decide if a graph $G$ with $k$ pairs of specified vertices $(s_i,t_i)$ contains $k$ mutually induced paths $P<sup>i$ such that each $P<sup>i$ starts from $s_i$ and ends at $t_i$. Whereas the non-induced version is well-known to be polynomial-time solvable for every fixed integer $k$, a classical result from the literature states that even $2$-Induced Disjoint Paths is NP-complete. We prove new complexity results for $k$-Induced Disjoint Paths if the input is restricted to $H$-free graphs, that is, graphs without a fixed graph $H$ as an induced subgraph. We compare our results with a complexity dichotomy for Induced Disjoint Paths, the variant where $k$ is part of the input.