Abstract
Let T = (V, A) be a finite tournament with n vertices and let F be a set of non negative integers less than or equal to n. The dual of T is the tournament T* = (V, A*) defined by: for all x, y is an element of V, (y, x) is an element of A* if and only if (x, y) is an element of A. To every subset X of V is associated the subtournament T(X) = (X, A boolean AND (X x X)) of T induced by X. The tournament T is strongly connected if for all x, y is an element of V, with x not equal y, there is a sequence x(0) =x..., x(p) = y such that for i is an element of {0,..., p - 1}, (x(i), x(i+1)) is an element of A. An half-isomorphism from T onto a tournament T' is either an isomorphism from T onto T' or an isomorphism from T* onto T'. A tournament T', with the same set of vertices V than T, is F-half-isomorphic to T if for every subset X of V such that \X\ is an element of F, the subtournaments T(X) and T'(X) are half-isomorphic. We study the {3, n - 2}-half-isomorphy and the {n - 3}-half-isomorphy between two tournaments with n vertices, one of which is non strongly connected. (C) 2002 Academie des sciences Editions scientifiques et medicales Elsevier SAS.