Internal Closedness and von Neumann-Morgenstern Stability in Matching Theory: Structures and Complexity

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 ${\cal M}$ of $G$ is called internally stable if there are no matchings $M,M' \in {\cal M}$ such that an edge of $M$ blocks $M'$. The sets of stable (\`a la Gale and Shapley) matchings and of von Neumann-Morgenstern stable matchings are examples of internally stable sets of matching. In this paper, we 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 or von Neumann-Morgenstern stable, and of finding sets with those properties.