Sommets fortement critiques d'un tournoi ind\'ecomposable

Abstract

Let $T=(V,A)$ be a tournament. For $X\subseteq V$, the subtournament of $T$ induced by $X$ is denoted by $T[X]$. A subset $I$ of $V$ is an interval of $T$ provided that for every $a,b\in I$ and $x\in V\setminus I$, $(a,x)\in A$ if and only if $(b,x)\in A$. For example, $\varnothing $, ${x}$ ($x \in V$) and $V$ are intervals of $T$, called trivial intervals. The tournament $T$ is indecomposable if all its intervals are trivial, otherwise, it is decomposable. A critical tournament is an indecomposable tournament $T$ of cardinality $\geqslant 5$ such that every vertex $x$ of $T$ is critical, i.e., the subtournament $T[V(T)\setminus\{x\}]$ is decomposable. Given an indecomposable tournament $T$, a vertex $x$ of $T$ is strongly critical, if for every $X\subseteq V(T)$ such that $x\in X$, $\vert X\vert \geqslant 5$ and $T[X]$ is indecomposable, $x$ is a critical vertex of $T[X]$. Let $T$ be an indecomposable tournament and let $\mathscr{C}(T)$ be the set of the strongly critical vertices of $T$. We prove that, if $T$ is non-critical, then $f(T):=\vert \mathscr{C}(T)\vert \leqslant 4$, and that the correspondence $f(T)$ is decreasing from the class of indecomposable and non-critical tournaments (defined by means of embedding) to $\{0,1,2,3,4\}$. By giving examples, we also verify that the bounds 0 and 4 are optimal. This article is an extract from my master's thesis \cite{mon mast\`ere}.