Induced Disjoint Paths in AT-free graphs

Paths P1,…,Pk are mutually induced if any two distinct Pi and Pj have neither common vertices nor adjacent vertices (except perhaps their end-vertices). The INDUCED DISJOINT PATHS problem is to decide if a graph G with k pairs of specified vertices (si,ti) contains k mutually induced paths Pi such that each Pi connects si and ti. This problem is NP-complete even 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. Consequently, the problem of deciding if an AT-free graph contains a fixed graph H as an induced topological minor admits a polynomial-time algorithm. We show that such an algorithm is essentially optimal by proving that the problem is W[1]-hard with parameter |VH|, even for a subclass of AT-free graphs, namely cobipartite graphs. We also show that the problems k-IN-A-PATH and k-IN-A-TREE are polynomial-time solvable on AT-free graphs even if k is part of the input