Decomposability and co-modular indices of tournaments

Given a tournament T, module of T is subset X V(T) such that for x,y?X and v?V(T)?X, (x,v)?A(T) if only (y,v)?A(T). The trivial modules are 0?, {u} (u?V(T)) V(T). indecomposable all its trivial; otherwise it decomposable. decomposability index denoted by ?(T), the smallest number arcs must be reversed to make indecomposable. first author conjectured n?5, we have ?(n)=n+14, where ?(n) maximum ?(T) over tournaments with n vertices. In this paper prove conjecture introducing co-modular ?(T), as largest disjoint co-modules co-module M or V(T)?M nontrivial T. We n?3, ?(n)=n+12, ?(n) ?(T) Our main result following close relationship between above two indices: every at least 5 vertices, ?(T)=?(T)2. As consequence, obtain ?(n)=?(n)2=n+14 answer some further related questions.