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）匹配割问题要求判定图$G$是否存在一个（完美）匹配割，即该（完美）匹配同时是$G$的一个边割。已知匹配割与完美匹配割均属于NP完全问题。完美匹配割也是边数最大的匹配割。为加深对两者关系的理解，我们引入了最大匹配割问题，旨在确定图中边数最多的匹配割。我们推广并统一了已有针对直径至多为$2$的图以及$(P_6+sP_2)$-自由图的多项式时间算法。通过证明最大匹配割在直径为$3$、半径为$2$的$2P_3$-自由四边形图和无三角形图的次三次线图中为NP难问题，我们揭示了最大匹配割与匹配割、完美匹配割在复杂度上的差异。由此，我们获得了最大匹配割问题针对有界直径、有界半径图和$H$-自由图的完整二分性结果。最后，我们应用相关技术得到了$H$-自由图的最大不连通完美匹配问题的二分性。图$G$的不连通完美匹配是指包含$G$的一个匹配割的完美匹配。最大不连通完美匹配问题要求对连通图$G$，在所有不连通完美匹配中确定具有最大匹配割的不连通完美匹配。我们的二分性结果表明，原判定问题“不连通完美匹配”对每个$s\geq 0$的$(P_6+sP_2)$-自由图可在多项式时间内求解，从而解决了Bouquet与Picouleau（arXiv, 2020）提出的一个开放问题。