Partially critical tournaments and partially critical supports

Given a tournament $T=(V,A)$, with each subset $X$ of $V$ is associated the subtournament $T[X]=(X,Acap (X imes X))$ of $T$ induced by $X$. A subset $I$ of $V$ is an interval of $T$ provided that for any $a,bin I$ and $xin Vsetminus I$, $(a,x)in A$ if and only if $(b,x)in A$. For example, $emptyset$, ${x}$, where $xin V$, and $V$ are intervals of $T$ called emph{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 emph{critical} if for every $xin V$, $T[Vsetminus{x}]$ is decomposable. It is emph{partially critical} if there exists a proper subset $X$ of $V$ such that $| X| geq 3$, $T[X]$ is indecomposable and for every $xin Vsetminus X$, $T[Vsetminus{x}]$ 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 $|X|geq 3$ and $T[X]$ is indecomposable. The partially critical support of $T$ according to $T[X]$ is the family of $xin Vsetminus X$ such that $T[Vsetminus{x}]$ is indecomposable and $T[Vsetminus{x,y}]$ is decomposable for every $yin (Vsetminus X)setminus{x}$. 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.