A matching is said to be disconnected if the saturated vertices induce a disconnected subgraph and induced if the saturated vertices induce a 1-regular graph. The disconnected and induced matching numbers are defined as the maximum cardinality of such matchings, respectively, and are known to be NP-hard to compute. In this paper, we study the relationship between these two parameters and the matching number. In particular, we discuss the complexity of two decision problems; first: deciding if the matching number and disconnected matching number are equal; second: deciding if the disconnected matching number and induced matching number are equal. We show that given a bipartite graph with diameter four, deciding if the matching number and disconnected matching number are equal is NP-complete; the same holds for bipartite graphs with maximum degree three. We characterize diameter three graphs with equal matching number and disconnected matching number, which yields a polynomial time recognition algorithm. Afterwards, we show that deciding if the induced and disconnected matching numbers are equal is co-NP-complete for bipartite graphs of diameter 3. When the induced matching number is large enough compared to the maximum degree, we characterize graphs where these parameters are equal, which results in a polynomial time algorithm for bounded degree graphs.
A hierarchy of edge-weight symmetries in perfect matchings
MTA-ELTE Matroid Optimization Research Group and HUN-REN–ELTE Egerváry Research Group, Department of Operations Research, ELTE Eötvös Loránd University, and HUN-REN Alfréd Rényi Institute of Mathematics（MTA-ELTE拟阵优化研究组和HUN-REN–ELTE埃格瓦里研究组，ELTE罗兰大学运营研究系，以及HUN-REN阿尔弗雷德·雷尼数学研究所）；Department of Operations Research, ELTE Eötvös Loránd University（ELTE罗兰大学运营研究系）；Department of Information and Physical Sciences, Graduate School of Information Science and Technology, Osaka University（大阪大学情报理工学研究科信息物理科学系）
Two-sided popular matchings in bipartite graphs with forbidden/forced elements and weights
二部图中的双边流行匹配是婚姻环境中稳定匹配的一个众所周知的推广，当偏好列表不完整时，它们尤其相关。在这种情况下，稳定匹配的基数可以小到最大匹配大小的一半。流行的匹配允许更大规模的分配，同时仍然保证一定的公平性条件。事实上，稳定匹配是最小大小的流行匹配，并且在给定实例中，最大大小的流行匹配可以是（ny）稳定匹配的大小的两倍。流行匹配的结构似乎比稳定匹配的结构更复杂，目前也更难理解。在本文中，我们专注于流行的匹配相关的三个优化问题。首先，我们对带有禁止和强制元素的流行匹配问题的复杂性进行了粒度分析，从而补充了[Cseh和Kavitha，2016]的结果。特别是，我们表明，决定是否存在一个流行的匹配（或没有）两个给定的边缘是NP-难。这意味着找到一个流行的匹配的最大值（或）。最小）权重是NP-难的，即使所有权重都是非负的，也不能近似到1/2（分别是任何因素）。[Cseh and Kavitha，2016]中的分解定理可以用于对具有非负权重的最大加权流行匹配问题进行1/2近似，从而完全解决这些问题的复杂性。
A complexity dichotomy for Matching Cut in (bipartite) graphs of fixed diameter
在图中，匹配割是匹配的边割。匹配割是判定给定图是否有匹配割的问题，即使仅限于二分图，也是NP完全的。证明了直径为2的图的匹配割是多项式可解的。本文证明了对任意固定的整数$d\ge 3$，匹配割在直径为$d$的图类中是NP-完全的.这解决了Borowiecki和Jesse-Józefczyk在[Matching cutsets in graphs of diameter $2$，Theoretical计算机科学407（2008）574-582]中提出的公开问题。然后，我们表明，对于任何固定的整数$d\ge 4$，匹配割是NP-完全的，即使限制在类的二部图的直径$d$。作为对硬度结果的补充，我们证明了匹配割在直径不超过3的二部图类中是多项式时间可解的，并提出了一种新的求解直径为$2$的图类中匹配割的简单多项式时间算法.
