The complexity of the matching-cut problem

Abstract

Finding a cut or finding a matching in a graph are so simple problems that they are hardly considered problems at all. In this paper, by means of a reduction from the NAE3SAT problem, we prove that combining these two problems together, i.e., finding a cut whose split edges are a matching is an NP-complete problem. It remains intractable even if we impose the graph to be simple (no multiple edges allowed) or its maximum degree to be k, with k ≥ 4. On the contrary, we give a linear time algorithm that computes a matching-cut of a series-parallel graph. It’s open whether the problem is tractable or not for planar graphs.