The (Perfect) Matching Cut problem is to decide if a connected graph has a (perfect) matching that is also an edge cut. The Disconnected Perfect Matching problem is to decide if a connected 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 subcubic bipartite graphs of arbitrarily large fixed girth. We prove that Matching Cut and Disconnected Perfect Matching are also NP-complete for bipartite graphs of arbitrarily large fixed girth and bounded maximum degree. Our result for Matching Cut resolves a 20-year old open problem. We also show that the more general problem $d$-Cut, for every fixed $d\geq 1$, is NP-complete for graphs of arbitrarily large fixed girth. Furthermore, we show that Matching Cut, Perfect Matching Cut and Disconnected Perfect Matching 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 not only with each other, but also with a known and full classification of the Maximum Matching Cut problem, which is to determine a largest matching cut of a given graph $G$. Finally, by combining 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)匹配割问题是判断一个连通图是否具有一个既是匹配又是边割的(完美)匹配。不连通完美匹配问题是判断一个连通图是否具有一个包含匹配割的完美匹配。对于围长为5的平面图，匹配割与不连通完美匹配问题均为NP完全的，而完美匹配割问题对于任意大固定围长的次立方二分图甚至是NP完全的。我们证明了匹配割与不连通完美匹配对于任意大固定围长且有界最大度的二分图也是NP完全的。我们的匹配割结果解决了一个存在20年的开放问题。同时，我们证明了更一般的问题d-割（对于每个固定d≥1）对于任意大固定围长的图是NP完全的。此外，我们证明了当H包含一个具有至少两个度数≥3的顶点的连通分支时，匹配割、完美匹配割和不连通完美匹配对于H-free图是NP完全的。随后，我们更新了H-free图的最新研究总结，不仅将其相互比较，还与已知的"最大匹配割"问题的完整分类进行比较，该问题旨在确定给定图G的最大匹配割。最后，通过结合现有结果，我们获得了完美匹配割对于H-子图-free图（其中H为任意有限图集）的完整复杂性分类。