The (Perfect) Matching Cut is to decide if a graph has a (perfect) matching that is also an edge cut. The Disconnected Perfect Matching problem is to decide if a graph has a perfect matching that contains a matching cut. Both Matching Cut and Disconnected Perfect Matching are NP-complete for planar graphs of girth 5, whereas Perfect Matching Cut is known to be NP-complete even for graphs of arbitrarily large fixed girth. We prove the last result also for the other two problems, solving a 20-year old problem for Matching Cut. Moreover, we give three new general hardness constructions, which imply that all three problems are NP-complete for H-free graphs whenever H contains a connected component with two vertices of degree at least 3. Afterwards, we update the state-of-the-art summaries for H-free graphs and compare them with each other. Finally, by combining our new hardness construction for Perfect Matching Cut with two existing results, we obtain a complete complexity classification of Perfect Matching Cut for H-subgraph-free graphs where H is any finite set of graphs.

翻译：（完美）匹配割问题旨在判定图是否存在一个（完美）匹配同时构成边割。不连通完美匹配问题则判定图是否存在包含匹配割的完美匹配。匹配割与不连通完美匹配在围长为5的平面图中均为NP完全问题，而完美匹配割即使在任意大固定围长的图中也已知为NP完全。我们证明了该结论对另外两个问题同样成立，由此解决了困扰匹配割问题20年之久的难题。此外，我们给出了三个新的通用归约构造，表明当H包含一个具有两个度数至少为3的顶点的连通分支时，所有三个问题在H-自由图中均为NP完全。随后，我们更新了H-自由图的最新研究进展总结并进行横向对比。最终，通过将我们关于完美匹配割的新归约构造与两个现有结果相结合，得到了完美匹配割在H-子图自由图中的完全复杂性分类，其中H为任意有限图集合。