The perfect matching cut problem revisited

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