CR tournaments

The determinant of a tournament T is defined as the determinant of the skew-adjacency matrix of T. For a positive odd integer k, let Dₖ be the set of tournaments whose all subtournaments have determinant at most k². Some existing results show that, for k ∈ \1,3,5\, a tournament T ∈ Dₖ Dₖ₋₂ (T ∈ D₁ when $k=1$) if and only if T is switching equivalent to a transitive blowup of Lₖ₊₁, where Lₖ₊₁ is a tournament of order $k+1$ with a specific structure. There exist some tournaments with the special property that adding any vertex that does not conform to their structure increases the maximum value of determinants among their subtournaments. We define these tournaments as CR tournaments. In this paper, we introduce CR tournaments, strong CR tournaments and basic tournaments, and show some properties and conclusions on these tournaments. For a basic strong CR tournament H ∈ Dₖ Dₖ₋₂, we show that if T contains a subtournament which is switching isomorphic to H, then T ∈ Dₖ Dₖ₋₂ if and only if T is switching equivalent to a transitive blowup of H. Moreover, we demonstrate that all Lₙ are strong CR tournaments, and based on this conclusion, we answer a question posed in [J. Zeng, L. You, On determinants of tournaments and Dₖ, arXiv:2408.06992, 2024.], and propose some questions for further research.