Partially critical tournaments and partially critical supports

Given a tournament $T=(V,A)$, with each subset $X$ of $V$ is associated the subtournament $T[X]=(X,A\cap (X\times X))$ of $T$ induced by $X$. A subset $I$ of $V$ is an interval of $T$ provided that for any $a,b\in I$ and $x\in V\setminus I$, $(a,x)\in A$ if and only if $(b,x)\in A$. For example, $\emptyset$, $\{x\}$, where $x\in V$, 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 $x\in V$, $T[V\setminus\{x\}]$ 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 $x\in V\setminus X$, $T[V\setminus\{x\}]$ 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 $x\in V\setminus X$ such that $T[V\setminus\{x\}]$ is indecomposable and $T[V\setminus\{x,y\}]$ is decomposable for every $y\in (V\setminus X)\setminus\{x\}$. 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

Partially critical tournaments and partially critical supports

Given a tournament $T=(V,A)$, with each subset $X$ of $V$ is associated the subtournament $T[X]=(X,A\cap (X\times X))$ of $T$ induced by $X$. A subset $I$ of $V$ is an interval of $T$ provided that for any $a,b\in I$ and $x\in V\setminus I$, $(a,x)\in A$ if and only if $(b,x)\in A$. For example, $\emptyset$, $\{x\}$, where $x\in V$, 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 $x\in V$, $T[V\setminus\{x\}]$ 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 $x\in V\setminus X$, $T[V\setminus\{x\}]$ 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 $x\in V\setminus X$ such that $T[V\setminus\{x\}]$ is indecomposable and $T[V\setminus\{x,y\}]$ is decomposable for every $y\in (V\setminus X)\setminus\{x\}$. 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