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）匹配割问题是判定一个图$G$是否存在（完美）匹配割，即一个（完美）匹配同时是$G$的边割。已知匹配割和完美匹配割问题均是NP完全的，这导致两者在特殊图类上产生了许多复杂性结论。完美匹配割也是边数最多的匹配割。为加深对这两个问题之间关系的理解，我们引入最大匹配割问题，即确定图中最大规模的匹配割。我们推广并统一了已知针对直径不超过$2$的图以及$(P_6 + sP_2)$-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为3、半径为2的$2P_3$-自由四边形图和三角形自由图的次三次线图中的NP难度性，我们还揭示了最大匹配割问题与匹配割、完美匹配割问题在复杂性上的差异。由此，我们获得了有界直径、有界半径以及$H$-自由图类中最大匹配割问题的完整二分性分类。