Editor′s Foreword

Abstract

Given a graph G=(V,E) and a set S⊆V, a set U⊆V is a subset feedback vertex set of (G,S) if no cycle in G[V∖U] contains a vertex of S. The Subset Feedback Vertex Set problem takes as input G, S, and an integer k, and the question is whether (G,S) has a subset feedback vertex set of cardinality or weight at most k. Both the weighted and the unweighted versions of this problem are NP-complete on chordal graphs, even on their subclass split graphs. We give an algorithm with running time O(1.6708n) that enumerates all minimal subset feedback vertex sets on chordal graphs on n vertices. As a consequence, Subset Feedback Vertex Set can be solved in time O(1.6708n) on chordal graphs, both in the weighted and in the unweighted case. As a comparison, on arbitrary graphs the fastest known algorithm for these problems has O(1.8638n) running time. We also obtain that a chordal graph G has at most 1.6708n minimal subset feedback vertex sets, regardless of S. This narrows the gap with respect to the best known lower bound of 1.5848n on this graph class. For arbitrary graphs, the gap is substantially wider, as the best known upper and lower bounds are 1.8638n and 1.5927n, respectively.