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 perform a complexity study for the Maximum Matching Cut problem, which is to determine a largest matching cut in a graph. Our results yield full dichotomies of Maximum Matching Cut for graphs of bounded diameter, bounded radius and $H$-free graphs. A disconnected perfect matching of a graph $G$ is a perfect matching that contains a matching cut of $G$. We also show how our new techniques can be used for finding a disconnected perfect matching with a largest matching cut for special graph classes. In this way we can prove that the decision problem Disconnected Perfect Matching is polynomial-time solvable for $(P_6+sP_2)$-free graphs for every $s\geq 0$, extending a known result for $P_5$-free graphs (Bouquet and Picouleau, 2020).

翻译：（完美）匹配割问题旨在判定图$G$是否具有（完美）匹配割，即一个同时是$G$边割的（完美）匹配。已知匹配割问题与完美匹配割问题均为NP完全问题。完美匹配割也是包含最多边数的匹配割。为深入理解这两个问题之间的关系，我们对最大匹配割问题——即确定图中最大规模匹配割的问题——进行了复杂性研究。我们的研究结果针对有界直径图、有界半径图及$H$-自由图，建立了最大匹配割问题的完整二分性分类。图$G$的不连通完美匹配是指包含$G$某个匹配割的完美匹配。我们还展示了如何将新技术应用于特殊图类中寻找具有最大匹配割的不连通完美匹配。通过这种方法，我们证明了对于任意$s\geq 0$，决策问题“不连通完美匹配”在$(P_6+sP_2)$-自由图上具有多项式时间解法，这扩展了$P_5$-自由图上的已知结果（Bouquet与Picouleau，2020）。