In a graph, a (perfect) matching cut is an edge cut that is a (perfect) matching. Matching Cut (MC), respectively, Perfect Matching Cut (PMC), is the problem of deciding whether a given graph has a matching cut, respectively, a perfect matching cut. The Disconnected Perfect Matching problem (DPM) is to decide if a graph has a perfect matching that contains a matching cut. Solving an open problem posed in [Lucke, Paulusma, Ries (ISAAC 2022, Algorithmica 2023) & Feghali, Lucke, Paulusma, Ries (arXiv:2212.12317)], we show that PMC is NP-complete in graphs without induced $14$-vertex path $P_{14}$. Our reduction also works simultaneously for MC and DPM, improving the previous hardness results of MC on $P_{15}$-free graphs and of DPM on $P_{19}$-free graphs to $P_{14}$-free graphs for both problems. Actually, we prove a slightly stronger result: within $P_{14}$-free $8$-chordal graphs (graphs without chordless cycles of length at least $9$), it is hard to distinguish between those without matching cuts (respectively, perfect matching cuts, disconnected perfect matchings) and those in which every matching cut is a perfect matching cut. Moreover, assuming the Exponential Time Hypothesis, none of these problems can be solved in $2^{o(n)}$ time for $n$-vertex $P_{14}$-free $8$-chordal graphs. On the positive side, we show that, as for MC [Moshi (JGT 1989)], DPM and PMC are polynomially solvable when restricted to $4$-chordal graphs. Together with the negative results, this partly answers an open question on the complexity of PMC in $k$-chordal graphs asked in [Le, Telle (TCS 2022) & Lucke, Paulusma, Ries (MFCS 2023)].

翻译：在图中，（完美）匹配割是一个（完美）匹配的边割。匹配割（MC）与完美匹配割（PMC）问题分别判定给定图是否含有匹配割或完美匹配割。不连通完美匹配问题（DPM）则判定图是否存在包含匹配割的完美匹配。针对[Lucke, Paulusma, Ries (ISAAC 2022, Algorithmica 2023) 及 Feghali, Lucke, Paulusma, Ries (arXiv:2212.12317)]中提出的未解决问题，我们证明PMC在无诱导14-顶点路径$P_{14}$的图中是NP完全的。我们的归约同时适用于MC与DPM，将MC在$P_{15}$-free图及DPM在$P_{19}$-free图上的先前困难性结果改进为两个问题在$P_{14}$-free图上均成立。实际上，我们证明了稍强结论：在$P_{14}$-free 8-弦图（不含长度至少为9的无弦环的图）中，难以区分不存在匹配割（或完美匹配割、不连通完美匹配）的图与所有匹配割均为完美匹配割的图。此外，假设指数时间假说，对于n顶点$P_{14}$-free 8-弦图，上述问题均不能在$2^{o(n)}$时间内求解。在正面结果方面，我们证明如MC [Moshi (JGT 1989)]所示，DPM与PMC在限制于4-弦图时可在多项式时间内求解。结合否定结果，这一结论部分回答了[Le, Telle (TCS 2022) 及 Lucke, Paulusma, Ries (MFCS 2023)]提出的关于k-弦图中PMC复杂度的开放问题。