Dichotomies for Maximum Matching Cut: H-Freeness, Bounded Diameter, Bounded Radius

The (Perfect) Matching Cut problem is to decide if a graph G has a (perfect) matching cut, i.e., a (perfect) matching that is also an edge cut of G. Both Matching Cut and Perfect Matching Cut are known to be NP-complete. A perfect matching cut is also a matching cut with maximum number of edges. To increase our understanding of the relationship between the two problems, we introduce the Maximum Matching Cut problem. This problem is to determine a largest matching cut in a graph. We generalize and unify known polynomial-time algorithms for Matching Cut and Perfect Matching Cut restricted to graphs of diameter at most 2 and to (P_6+sP_2)-free graphs. We also show that the complexity of Maximum Matching Cut differs from the complexities of Matching Cut and Perfect Matching Cut by proving NP-hardness of Maximum Matching Cut for 2P_3-free quadrangulated graphs of diameter 3 and radius 2 and for subcubic line graphs of triangle-free graphs. In this way, we obtain full dichotomies of Maximum Matching Cut for graphs of bounded diameter, bounded radius and H-free graphs. Finally, we apply our techniques to get a dichotomy for the Maximum Disconnected Perfect Matching problem for H-free graphs. A disconnected perfect matching of a graph G is a perfect matching that contains a matching cut of G. The Maximum Disconnected Perfect Matching problem asks to determine for a connected graph G, a disconnected perfect matching with a largest matching cut over all disconnected perfect matchings of G. Our dichotomy result implies that the original decision problem Disconnected Perfect Matching is polynomial-time solvable for (P_6+sP_2)-free graphs for every s≥ 0, which resolves an open problem of Bouquet and Picouleau (arXiv, 2020).