Induced Disjoint Paths in AT-Free Graphs

Abstract

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. 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 |VH|. 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.