The complexity of the perfect matching-cut problem

PERFECT MATCHING-CUT is the problem of deciding whether a graph has a perfect matching that contains an edge-cut. We show that this problem is NP-complete for planar graphs with maximum degree four, for planar graphs with girth five, for bipartite five-regular graphs, for graphs of diameter three, and for bipartite graphs of diameter four. We show that there exist polynomial-time algorithms for the following classes of graphs: claw-free, P5 ${P}_{5}$-free, diameter two, bipartite with diameter three, and graphs with bounded treewidth.