Few Induced Disjoint Paths for H-Free Graphs
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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.
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