Let $G$ be a graph and suppose we are given, for each $v \in V(G)$, a strict ordering of the neighbors of $v$. A set of matchings $\mathcal{M}$ of $G$ is called internally stable if there are no matchings $M,M' \in \mathcal{M}$ such that an edge of $M$ blocks $M'$. The sets of stable matchings and of von Neumann-Morgenstern stable matchings are examples of internally stable sets of matching. In this paper, we introduce and study, in both the marriage and the roommate case, inclusionwise maximal internally stable sets of matchings. We call those sets internally closed. By building on known and newly developed algebraic structures associated to sets of matchings, we investigate the complexity of deciding if a set of matchings is internally closed, and if it is von Neumann-Morgenstern stable.
翻译:让$G$成为图表, 假设我们被给出了, 每1美元 $v\ in V( G) 的严格排序。 一组匹配 $mathcal{ M} $G$ 被称为内部稳定, 如果没有匹配 $M, M'\ in\ mathcal{ M} 美元, 其边缘为 $m美元。 稳定的匹配和 von Neumann- Morgenstern 稳定匹配是内部稳定的匹配组合的例子 。 在本文中, 我们引入并研究婚姻和室友的匹配组合, 包括最起码的内部稳定匹配组合。 我们将这些组合称为内部封闭 。 通过建立已知和新开发的匹配组合相关的代数结构, 我们调查确定一组匹配是否内部关闭的复杂性, 如果是冯 Neuumann- Morgenstern 稳定 。