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 $s\geq 0$, which resolves an open problem of Bouquet and Picouleau (arXiv, 2020).

翻译：（完美）匹配割问题旨在判断一个图$G$是否具有（完美）匹配割，即同时为$G$的边割的（完美）匹配。已知匹配割和完美匹配割均为NP完全问题。完美匹配割也是一种边数最大的匹配割。为加深对两类问题关系的理解，我们引入最大匹配割问题，即确定图中边数最多的匹配割。我们推广并统一了针对直径至多为$2$的图及$(P_6+sP_2)$自由图的匹配割与完美匹配割的已知多项式时间算法。通过证明最大匹配割对直径为$3$、半径为$2$的$2P_3$自由四边形图以及无三角形图的次三次线图具有NP难度，我们揭示了最大匹配割的复杂度与匹配割及完美匹配割的差异。由此，我们获得了有界直径、有界半径图及$H$自由图上最大匹配割的完整二分性。最后，我们将技术应用于$H$自由图的最大不连通完美匹配问题的二分性判定。图$G$的不连通完美匹配是指包含$G$中某个匹配割的完美匹配。最大不连通完美匹配问题要求确定连通图$G$的所有不连通完美匹配中具有最大匹配割的完美匹配。我们的二分性结果表明：原始判定问题“不连通完美匹配”对每个$s\geq 0$的$(P_6+sP_2)$自由图均可在多项式时间内求解，从而解决了Bouquet与Picouleau（arXiv, 2020）提出的开放问题。