Induced Disjoint Paths in Claw-Free Graphs

Paths P1,...,Pk in a graph G=(V,E) are said to be mutually induced if for any 1 <= i < j <= k, Pi and Pj have neither common vertices nor adjacent vertices (except perhaps their end-vertices). The Induced Disjoint Paths problem is to test whether a graph G with k pairs of specified vertices (si,ti) contains k mutually induced paths Pi such that Pi connects si and ti for i=1,...,k. We show that this problem is fixed-parameter tractable for claw-free graphs when parameterized by k. Several related problems, such as the k-in-a-Path problem, are proven to be fixed-parameter tractable for claw-free graphs as well. We show that an improvement of these results in certain directions is unlikely, for example by noting that the Induced Disjoint Paths problem cannot have a polynomial kernel for line graphs (a type of claw-free graphs), unless NP \subseteq coNP/poly. Moreover, the problem becomes NP-complete, even when k=2, for the more general class of K_1,4-free graphs. Finally, we show that the n^O(k)-time algorithm of Fiala et al. for testing whether a claw-free graph contains some k-vertex graph H as a topological induced minor is essentially optimal by proving that this problem is W[1]-hard even if G and H are line graphs.