Detecting induced minors in AT-free graphs

The problem Induced Minor is to test whether a graph G can be modified into a graph H by a sequence of vertex deletions and edge contractions. We prove that Induced Minor is polynomial-time solvable when G is AT-free, and H is fixed, i.e., not part of the input. Our result can be considered to be optimal in some sense as we also prove that Induced Minor is W[1]-hard on AT-free graphs, when parameterized by |VH|. In order to obtain it we prove that the Set-Restricted k-Disjoint Paths problem can be solved in polynomial time on AT-free graphs for any fixed k. We also use the latter result to prove that the Set-Restricted k-Disjoint Connected Subgraphs problem is polynomial-time solvable on AT-free graphs for any fixed k