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

We consider a tournament T=(V,A)T=(V,A). For X⊆VX⊆V, the subtournament of T induced by X is T[X]=(X,A∩(X×X))T[X]=(X,A∩(X×X)). An interval of T is a subset X of V such that, for a,b∈Xa,b∈X and x∈V∖Xx∈V∖X, (a,x)∈A(a,x)∈A if and only if (b,x)∈A(b,x)∈A. The trivial intervals of T are ∅, {x}(x∈V) and V  . A tournament is indecomposable if all its intervals are trivial. For n⩾2n⩾2, W2n+1W2n+1 denotes the unique indecomposable tournament defined on {0,…,2n}{0,…,2n} such that W2n+1[{0,…,2n−1}]W2n+1[{0,…,2n−1}] is the usual total order. Given an indecomposable tournament T  , W5(T)W5(T) denotes the set of v∈Vv∈V such that there is W⊆VW⊆V satisfying v∈Wv∈W and T[W]T[W] is isomorphic to W5W5. Latka [6] characterized the indecomposable tournaments T such that W5(T)=∅W5(T)=∅. The authors [1] proved that if W5(T)≠∅W5(T)≠∅, then |W5(T)|⩾|V|−2|W5(T)|⩾|V|−2. In this note, we characterize the indecomposable tournaments T such that |W5(T)|=|V|−2|W5(T)|=|V|−2.