Few induced disjoint paths for H-free graphs

Paths P¹,…,P^k in a graph G=(V,E) are mutually induced if any two distinct Pⁱ 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 Pⁱ such that each Pⁱ 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.