  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.
