A Further Step Towards an Understanding of the Tournament Equilibrium Set

Abstract: We study some problems pertaining to the tournament equilibrium set (TEQ for short). A tournament $H$ is a TEQ-retentive tournament if there is a tournament $T$ which has a minimal TEQ-retentive set $R$ such that $T[R]$ is isomorphic to $H$. We study TEQ-retentive tournaments and achieve many significant results. In particular, we prove that there are no TEQ-retentive tournaments of size 4, only 2 non-isomorphic TEQ-retentive tournaments of sizes 5 and 6, respectively, and 26 non-isomorphic TEQ-retentive tournaments of size 7. For three tournaments $H_1, H_2$ and $T$, we say $T$ is a $(H_1,H_2)$-TEQ-retentive tournament if $T$ has two minimal TEQ-retentive sets $R_1$ and $R_2$ such that $T[R_1]$ and $T[R_2]$ are isomorphic to $H_1$ and $H_2$, respectively. We show that there are no $(H_1,H_2)$-retentive tournaments for $H_1$ and $H_2$ being small tournaments. Our results imply that Schwartz's Conjecture holds in all tournaments of size at most 14. Finally, we study Schwartz's Conjecture in several classes of tournaments. To achieve these results, we study the relation between (directed) domination graphs of tournaments and TEQ-retentive sets, and derive a number of properties on minimal TEQ-retentive sets.