A matching cut in a graph G is an edge cut of G that is also a matching. This short survey gives an overview of old and new results and open problems for Maximum Matching Cut, which is to determine the size of a largest matching cut in a graph. We also compare this problem with the related problems Matching Cut, Minimum Matching Cut, and Perfect Matching Cut, which are to determine if a graph has a matching cut; the size of a smallest matching cut in a graph; and if a graph has a matching cut that is a perfect matching, respectively. Moreover, we discuss a relationship between Maximum Matching Cut and Max Cut, which is to determine the size of a largest edge cut in a graph, as well as a relationship between Minimum Matching Cut and Min Cut, which is to determine the size of a smallest edge cut in a graph.

翻译：本文对图G中的匹配割（即同时为匹配的边割）这一概念进行了综述。我们系统梳理了关于最大匹配割问题（即确定图中最大匹配割的规模）的新旧研究成果与开放问题。此外，将该问题与相关变体进行对比：匹配割判定问题（判断图是否存在匹配割）、最小匹配割问题（确定图中最小匹配割的规模）以及完美匹配割问题（判断图是否存在完美匹配的匹配割）。同时，本文探讨了最大匹配割与最大割问题（确定图中最大边割的规模）之间的关联，并分析了最小匹配割与最小割问题（确定图中最小边割的规模）之间的联系。