Decomposability index of tournaments

Given a tournament $T$, a module of $T$ is a subset $X$ of $V\left(T\right)$ such that for $x,y\in X$ and $v\in V\left(T\right)?X$, $(x,v)\in A\left(T\right)$ if and only if $(y,v)\in A\left(T\right)$. The trivial modules of $T$ are $?$, $\left\{u\right\}$ $(u\in V\left(T\right))$ and $V\left(T\right)$. The tournament $T$ is indecomposable if all its modules are trivial; otherwise it is decomposable. The decomposability index of $T$, denoted by $\delta \left(T\right)$, is the smallest number of arcs of $T$ that must be reversed to make $T$ indecomposable. For $n\ge 5$, let $\delta \left(n\right)$ be the maximum of $\delta \left(T\right)$ over the tournaments $T$ with $n$ vertices. We prove that $?\frac{n+1}{4}?\le \delta \left(n\right)\le ?\frac{n?1}{3}?$ and that the lower bound is reached by the transitive tournaments.     

The indecomposable tournaments T with |W5(T)|=|T|−2

We consider a tournament $T=(V,A)$. For $X?V$, 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 $x\in V?X$, $(a,x)\in A$ if and only if $(b,x)\in A$. The trivial intervals of T are ?, $\{x\}(x\in V)$ and V. A tournament is indecomposable if all its intervals are trivial. For $n?2$, $W_{2n+1}$ 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 $W?V$ satisfying $v\in W$ and $T[W]$ is isomorphic to $W_5$. Latka [6] characterized the indecomposable tournaments T such that $W_5(T)=?$. The authors [1] proved that if $W_5(T)\ne ?$, then $|W_5(T)|?|V|?2$. In this note, we characterize the indecomposable tournaments T such that $|W_5(T)|=|V|?2$.