Partially critical tournaments and partially critical supports

Given a tournament T=(V,A), with each subset X of V is associated the subtournament T[X]=(X,A∩(X×X)) of T induced by X. A subset I of V is an interval of T provided that for any a,b∈I and x∈V∖I, (a,x)∈A if and only if (b,x)∈A. For example, ∅, {x}, where x∈V, and V are intervals of T called trivial. A tournament is indecomposable if all its intervals are trivial; otherwise, it is decomposable. Let T=(V,A) be an indecomposable tournament. The tournament T is critical if for every x∈V, T[V∖{x}] is decomposable. It is partially critical if there exists a proper subset X of V such that |X|≥3, T[X] is indecomposable and for every x∈V∖X, T[V∖{x}] is decomposable. The partially critical tournaments are characterized. Lastly, given an indecomposable tournament T=(V,A), consider a proper subset X of V such that |X|≥3 and T[X] is indecomposable. The partially critical support of T according to T[X] is the family of x∈V∖X such that T[V∖{x}] is indecomposable and T[V∖{x,y}] is decomposable for every y∈(V∖X)∖{x}. It is shown that the partially critical support contains at most three vertices. The indecomposable tournaments whose partially critical supports contain at least two vertices are characterized.