Abstract
We consider the transformation reversing all arcs of a subset X of the vertex set of a tournament T. The index of T, denoted by i(T), is the smallest number of subsets that must be reversed to make T acyclic. It turns out that critical tournaments and (-1)-critical tournaments can be defined in terms of inversions (at most two for the former, at most four for the latter). We interpret i(T) as the minimum distance of T to the transitive tournaments on the same vertex set, and we interpret the distance between two tournaments T and T' as the Boolean dimension of a graph, namely the Boolean sum of T and T'. On n vertices, the maximum distance is at most n - 1, whereas i(n), the maximum of i(T) over the tournaments on n vertices, satisfies n-1/2 - log(2) n <= i(n) <= n - 3, for n >= 4. Let I(m)(<w) (resp. I(m)(<= w)) be the class of finite (resp. at most countable) tournaments T such that i(T) <= m. The class I(m)(<omega) is determined by finitely many obstructions. Wegive a morphological description of the members of I(1)(<omega) and a description of the critical obstructions. We give an explicit description of a universal tournament of the class I(m)(<= w) (C) 2010 Publie par Elsevier Masson SAS pour l'Academie des sciences.