Finding contractions and induced minors in chordal graphs via disjoint paths

Authors:
Rémy Belmonte;Petr A. Golovach;Pinar Heggernes
Affiliations:
Department of Informatics, University of Bergen, Norway;School of Engineering and Computing Sciences, Durham University, UK;Department of Informatics, University of Bergen, Norway
Venue:
ISAAC'11 Proceedings of the 22nd international conference on Algorithms and Computation
Year:
2011

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Abstract

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 vertex-disjoint 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.