Decomposability index of tournaments

Given a tournament T, a module of T is a subset X of V(T) such that for x,y∈X and v∈V(T)∖X, (x,v)∈A(T) if and only if (y,v)∈A(T). The trivial modules of T are ∅, {u}(u∈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 δ(T), is the smallest number of arcs of T that must be reversed to make T indecomposable. For n≥5, let δ(n) be the maximum of δ(T) over the tournaments T with n vertices. We prove that n+14≤δ(n)≤n−13 and that the lower bound is reached by the transitive tournaments.