  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 $\emptyset$, $\{u\}$ $(u\in
V(T))$ and $V(T)$. The tournament $T$ 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 $n
\geq 5$, we have $\delta(n) = \left\lceil \frac{n+1}{4} \right\rceil$, where
$\delta(n)$ is the maximum of $\delta(T)$ over the tournaments $T$ with $n$
vertices. In this paper, we characterize the tournaments $T$ with
$\delta$-maximal decomposability, i.e., such that $\delta(T)=\delta(\vert
T\vert)$.
