The indecomposable tournaments $T$ with $\mid W_5(T)\mid =\mid T\mid -2$

Abstract

We consider a tournament $T=(V,A)$. For $X\subseteq V$, the subtournament of $T$ induced by $X$ is $T[X]=(X,A\cap (X\times X))$. An interval of $T$ is a subset $X$ of $V$ such that for $a,b\in X$ and $x\in V\setminus X$, $(a,x)\in A$ if and only if $(b,x)\in A$. The trivial intervals of $T$ are $\varnothing$, $\{x\}(x\in V)$ and $V$. A tournament is indecomposable if all its intervals are trivial. For $n\ge 2$, $W_{2n+1}$ denotes the unique indecomposable tournament defined on $\{0,\dots ,2n\}$ such that $W_{2n+1}[\{0,\dots ,2n-1\}]$ is the usual total order. Given an indecomposable tournament $T$, $W_5(T)$ denotes the set of $v\in V$ such that there is $W\subseteq V$ satisfying $v\in W$ and $T[W]$ is isomorphic to $W_5$. Latka \cite{BJL} characterized the indecomposable tournaments $T$ such that $W_5(T)=\varnothing$. The authors \cite{HIK} proved that if $W_5(T)\ne \varnothing$, then $\mid W_5(T)\mid \ge \mid V\mid -2$. In this article, we characterize the indecomposable tournaments $T$ such that $\mid W_5(T)\mid =\mid V\mid -2$.