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. This problem is known to be NP-complete already for k = 2, but for n-vertex claw-free graphs, Fiala et al. gave an n-time algorithm. We improve the latter result by showing that the problem is fixed-parameter tractable for clawfree graphs when parameterized by k. Several related problems, such as the k-in-a-Path problem, are shown to be fixed-parameter tractable for claw-free graphs as well. We also show that an improvement of these results in certain directions is unlikely, for example by observing that the Induced Disjoint Paths problem cannot have a polynomial kernel for line graphs (a type of claw-free graphs), unless NP ⊆ coNP/poly. Moreover, the problem becomes NP-complete, even when k = 2, for the more general class of K1,4-free graphs. Finally, we show that the n -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