Induced Disjoint Paths in Claw-Free Graphs
Paths P 1,…,P k in a graph G = (V,E) are said to be mutually induced if for any 1 ≤ i < j ≤ k, P i and P j 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 (s i ,t i ) contains k mutually induced paths P i such that P i connects s i and t i 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 O(k)-time algorithm. We improve the latter result by showing that the 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 shown to be fixed-parameter tractable for claw-free graphs as well. We prove 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 ⊆ 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-hard even if G and H are line graphs.
KeywordsLine Graph Interval Graph Polynomial Kernel Disjoint Path Graph Class
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