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We study some problems pertaining to the tournament equilibrium set (TEQ forshort). 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 threetournaments $H_1, H_2$ and $T$, we say $T$ is a $(H_1,H_2)$-TEQ-retentivetournament 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 showthat there are no $(H_1,H_2)$-retentive tournaments for $H_1$ and $H_2$ beingsmall tournaments. Our results imply that Schwartz's Conjecture holds in alltournaments of size at most 14. Finally, we study Schwartz's Conjecture inseveral classes of tournaments. To achieve these results, we study the relationbetween (directed) domination graphs of tournaments and TEQ-retentive sets, andderive a number of properties on minimal TEQ-retentive sets.

A Further Step Towards an Understanding of the Tournament Equilibrium Set