The prime tournaments T with |W5(T)|=|T |−2

Abstract

We consider a tournament T = (V,A). For X ⊆ V , the subtournament of T induced by X is T[X] = (X,A ∩ (X ×X)). A module of T is a subset X of V such that for a,b ∈X and x ∈ V X, (a,x) ∈A if and only if (b,x) ∈ A. The trivial modules of T are ∅, {x}(x ∈ V), and V . A tournament is prime if all its modules are trivial. For n ≥ 2, W2n+1 denotes the unique prime tournament defined on {0,...,2n} such that W2n+1[{0,...,2n−1}] is the usual total order. Given a prime tournament T , W5(T) denotes the set of v ∈ V such that there is W ⊆ V satisfying v ∈ W and T[W] is isomorphic to W5. B.J. Latka characterized the prime tournaments T such that W5(T) = ∅. The authors proved that if W5(T) ̸= ∅, then |W5(T)|≥|V | −2. In this article, we characterize the prime tournaments T such that |W5(T)|=|V | −2.