A matching cut is a matching that is also an edge cut. In the problem Minimum Matching Cut, we ask for a matching cut with the minimum number of edges in the matching. We investigate the differences in complexity between Minimum Matching Cut, its counterpart Maximum Matching Cut, and the decision problem Matching Cut. Our polynomial-time algorithms for $P_8$-free, $S_{1,1,3}$-free and $(P_6 + P_4)$-free graphs extend the cases where Minimum Matching Cut and Maximum Matching Cut are known to differ in complexity. In addition, they solve open cases for the well-studied problem Matching Cut. The NP-hardness proof for $3P_3$-free graphs implies that Minimum Matching Cut and Matching Cut, which is polynomial-time solvable even for $sP_3$-free graphs, for any $s \geq 1$, differ in complexity on certain graph classes. Further, we give complexity dichotomies for both general and bipartite graphs of bounded radius and diameter.

翻译：匹配割是指同时作为边割的匹配。在最小匹配割问题中，我们要求寻找匹配边数最少的匹配割。本文研究了最小匹配割、其对应问题最大匹配割以及判定问题匹配割在计算复杂度上的差异。我们针对$P_8$-自由图、$S_{1,1,3}$-自由图和$(P_6 + P_4)$-自由图提出的多项式时间算法，扩展了最小匹配割与最大匹配割已知存在复杂度差异的图类范围。此外，这些算法解决了被深入研究的匹配割问题中若干未决情形。针对$3P_3$-自由图的NP困难性证明表明：对于任意$s \geq 1$，即使在$sP_3$-自由图上可在多项式时间求解的匹配割问题，与最小匹配割问题在特定图类上具有不同的计算复杂度。进一步地，我们给出了有界半径与直径的一般图及二部图的复杂度二分性结论。