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 bipartite graphs of arbitrarily large fixed girth and bounded maximum degree. 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 with each other, and with a known and full classification of the Maximum Matching Cut problem, which is to determine a largest matching cut of a 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）匹配割问题旨在判断一个连通图是否包含一个既是匹配又是边割的（完美）匹配。不连通完美匹配问题则判断连通图是否存在一个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的平面图中是NP完全的，而完美匹配割已知即使对于任意大固定围长的次三次二部图也是NP完全的。我们证明了匹配割和不连通完美匹配对于任意大固定围长且最大度有界的二部图同样是NP完全的。其中关于匹配割的结果解决了一个存在20年的开放问题。我们还表明，对于任意固定d ≥ 1，更一般的d-割问题在任意大固定围长且最大度有界的二部图中是NP完全的。此外，我们证明了当H包含一个具有两个度数至少为3的顶点的连通分量时，匹配割、完美匹配割和不连通完美匹配在无H图中是NP完全的。随后，我们更新了无H图的最新研究总结，并相互比较，同时与已知且完整的最大匹配割问题分类进行对比，最大匹配割问题旨在确定图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。