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 (si, ti), asks whether G contains k mutually vertexdisjoint paths Pi such that Pi connects si and ti, for i = 1, . . . , k. We study a natural variant of this problem, where the vertices of Pi must belong to a specified vertex subset Ui 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.