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 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.

翻译：（完美）匹配割问题旨在判定图$G$是否存在一个（完美）匹配割，即该（完美）匹配同时构成$G$的边割集。已知匹配割和完美匹配割问题均为NP完全的，这导致针对特殊图类上这两个问题的大量复杂性结果。完美匹配割也是具有最大边数的匹配割。为加深对两者关系的理解，我们引入最大匹配割问题，该问题旨在确定图中最大的匹配割。我们推广并统一了已知的限制于直径至多为$2$的图及$(P_6 + sP_2)$-自由图上的匹配割与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3、半径2的$2P_3$-自由四边形图与无三角形图的次立方线图中为NP困难，我们揭示了最大匹配割的复杂性区别于匹配割与完美匹配割。由此，我们获得了有界直径、有界半径及$H$-自由图上最大匹配割的完整二分性结果。