Few induced disjoint paths for

Paths P 1 , … , 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.