Few Induced Disjoint Paths for H-Free Graphs

Abstract

AbstractPaths \(P^1,\ldots ,P^k\) in a graph \(G=(V,E)\) are mutually induced if any two distinct \(P^i\) and \(P^j\) 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 \(P^i\) such that each \(P^i\) 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.KeywordsInduced disjoint pathsH-free graphComplexity dichotomy