Inversion dans les tournois

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 < ω (resp. I m ⩽ ω ) be the class of finite (resp. at most countable) tournaments T such that i ( T ) ⩽ m . The class I m < ω is determined by finitely many obstructions. We give a morphological description of the members of I 1 < ω and a description of the critical obstructions. We give an explicit description of a universal tournament of the class I m ⩽ ω .