Finding contractions and induced minors in chordal graphs via disjoint paths

The k-Disjoint Paths problem, which takes as input a graph G and k pairs of specified vertices (s i ,t i ), asks whether G contains k mutually vertex-disjoint paths P i such that P i connects s i and t i , for i = 1,…,k. We study a natural variant of this problem, where the vertices of P i must belong to a specified vertex subset U i for i = 1,…,k. In contrast to the original problem, which is polynomial-time solvable for any fixed integer k, we show that this variant is NP-complete even for k = 2. On the positive side, we prove that the problem becomes polynomial-time solvable for any fixed integer k if the input graph is chordal. We use this result to show that, for any fixed graph H, the problems H-Contractibility and H-Induced Minor can be solved in polynomial time on chordal graphs. These problems are to decide whether an input graph G contains H as a contraction or as an induced minor, respectively