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, leading to many complexity results for both problems on special graph classes. 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 graphs of diameter 3 and radius 2 and for line graphs. In this way, we obtain full dichotomies of Maximum Matching Cut for graphs of bounded diameter, bounded radius and $H$-free graphs.

翻译：（完美）匹配割问题是判断一个图$G$是否具有（完美）匹配割，即同时作为$G$边割的（完美）匹配。已知匹配割和完美匹配割均为NP完全问题，这导致两类问题在众多特殊图类上具有复杂性结果。完美匹配割还是边数最大的匹配割。为加深对两类问题间关系的理解，我们引入最大匹配割问题。该问题旨在确定图中的最大匹配割。我们归纳并统一了已知关于直径不超过$2$的图与$(P_6 + sP_2)$-自由图上的匹配割和完美匹配割的多项式时间算法。通过证明最大匹配割在直径为3、半径为2的$2P_3$-自由图及线图上具有NP难度性，我们揭示了最大匹配割与匹配割和完美匹配割在复杂性上的差异。由此，我们获得了有界直径、有界半径及$H$-自由图类上最大匹配割问题的完整二分性结果。