Induced disjoint paths in AT-free graphs

Abstract

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. We prove that it can be solved in polynomial time for AT-free graphs even when k is part of the input. As a consequence, the problem of testing whether a given AT-free graph contains some graph H as an induced topological minor admits a polynomial-time algorithm, as long as H is fixed; we show that such an algorithm is essentially optimal by proving that the problem is W[1]-hard, even on a subclass of AT-free graphs, namely cobipartite graphs, when parameterized by |V H |. We also show that the problems k -in-a-Path and k -in-a-Tree can be solved in polynomial time, even when k is part of the input. These problems are to test whether a graph contains an induced path or induced tree, respectively, spanning k given vertices. © 2012 Springer-Verlag.