The perfect matching cut problem revisited

In a graph, a perfect matching cut is an edge cut that is a perfect matching. PERFECT MATCHING CUT (PMC) is the problem of deciding whether a given graph has a perfect matching cut, and is known to be NP-complete. We revisit the problem and show that PMC remains NP-complete when restricted to bipartite graphs of maximum degree 3 and arbitrarily large girth. Complementing this hardness result, we give two graph classes in which PMC is polynomial-time solvable. The first one includes claw-free graphs and graphs without an induced path on five vertices, the second one properly contains all chordal graphs. Assuming the Exponential Time Hypothesis, we show there is no O⁎(2o(n))-time algorithm for PMC even when restricted to n-vertex bipartite graphs, and also show that PMC can be solved in O⁎(1.2721n) time by means of an exact branching algorithm. © 2022 Elsevier B.V.