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.