Finding Matching Cuts in H -Free Graphs

The well-known NP-complete problem Matching Cut is to decide if a graph has a matching that is also an edge cut of the graph. We prove new complexity results for Matching Cut restricted to H -free graphs, that is, graphs that do not contain some fixed graph H as an induced subgraph. We also prove new complexity results for two recently studied variants of Matching Cut , on H -free graphs. The first variant requires that the matching cut must be extendable to a perfect matching of the graph. The second variant requires the matching cut to be a perfect matching. In particular, we prove that there exists a small constant r>0 such that the first variant is NP-complete for P_r -free graphs. This addresses a question of Bouquet and Picouleau (The complexity of the Perfect Matching-Cut problem. CoRR, arXiv:2011.03318 , (2020)). For all three problems, we give state-of-the-art summaries of their computational complexity for H -free graphs.