A Further Step Towards an Understanding of the Tournament Equilibrium Set

We study some problems pertaining to the tournament equilibrium set (τ for short). A tournament H is a τ-retentive tournament if there is a tournament T which has a minimal τ-retentive set R such that T [R] is isomorphic to H. We study τ-retentive tournaments and achieve many significant results. In particular, we prove that there are no τretentive tournaments of size 4, only 2 non-isomorphic τ-retentive tournaments of sizes 5 and 6, respectively, and 26 non-isomorphic τ-retentive tournaments of size 7. For three tournaments H1,H2 and T , we say T is a (H1,H2)τ-retentive tournament if T has two minimal τ-retentive sets R1 and R2 such that T [R1] and T [R2] are isomorphic to H1 and H2, respectively. We show that there are no (H1,H2)-retentive tournaments for H1 and H2 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 τ-retentive sets, and derive a number of properties on minimal τretentive sets.