Tournaments with maximal decomposability

Given a tournament $T$, a module of $T$ is a subset $M$ of $V(T)$ such that for $x,y\in M$ and $v\in V(T)\setminus M$, $(x,v)\in A(T)$ if and only if $(y,v)\in A(T)$. The trivial modules of $T$ are $\mathrm{\varnothing }$, $\{u\}$ $(u\in V(T))$and$V(T)$\mathrm{.}Thetournament$T$ is indecomposable if all its modules are trivial; otherwise it is decomposable. The decomposability index of $T$, denoted by $\delta (T)$, is the smallest number of arcs of $T$ that must be reversed to make $T$ indecomposable. In a previous paper, we proved that for $n \geq 5$,wehave$\delta(n) = \left\lceil \frac{n+1}{4} \right\rceil$,where$\delta(n)$isthemaximumof$\delta(T)$overthetournaments$T$with$n$vertices\mathrm{.}Inthispaper,wecharacterizethetournaments$T$with$\delta$-maximaldecomposability,i\mathrm{.}e\mathrm{.},suchthat$\delta(T)=\delta(\vert T\vert)$.