Abstract
Let T = (V, A) be a tournament. With every subset X of V is associated the subtournament T[X] = (X, A boolean AND (X x X)) of T, induced by X. The dual of T*, denoted by T*, is the tournament obtained from T by reversing all its arcs. Given a tournament T' = (V, A') and a non-negative integer k, T and T' are {-k}-hemimorphic provided that for all X subset of V, with vertical bar X vertical bar = k, T[V-X] and T' [V-X] or T* [V-X] and T' [V-X] are isomorphic. The tournaments T and T' are said to be hereditarily hemimorphic if for all subset X of V, the subtournaments T[X] and T' [X] are hemimorphic. The purpose of this paper is to establish the hereditary hemimorphy of the {-k}-hemimorphic tournaments on at least k + 7 vertices, for every k >= 5.