The {-3}-reconstruction and the {-3}-self duality of tournaments

Abstract

Let T = (V, A) be a (finite) tournament and k be a non negative integer. For every subset X of V is associated the subtournament T [X] = (X, A ∩ (X × X)) of T , induced by X. The dual tournament of T , denoted by T * , is the tournament obtained from T by reversing all its arcs. The tournament T is self dual if it is isomorphic to its dual. T is {−k}-self dual if for each set X of k vertices, T [V \ X] is self dual. T is strongly self dual if each of its induced subtournaments is self dual. A subset I of V is an interval of T if for a, b ∈ I and for x ∈ V \ I, (a, x) ∈ A if and only if (b, x) ∈ A. For instance, ∅, V and {x}, where x ∈ V , are intervals of T called trivial intervals. T is indecomposable if all its intervals are trivial; otherwise, it is decomposableSuppose that T is decomposable and | V |≥ 9. In this paper, we begin by proving the equivalence between the {−3}-self duality and the strong self duality of T . Then we characterize each tournament {−3}-hypomorphic to T . As a consequence of this characterization, we prove that if there is no intervalFinally, we conclude by reducing the {−3}-reconstruction problem to the indecomposable case (between a tournament and its dual). In particular, we find and improve, in a less complicated way, the results of [6] found by Y. Boudabbous and A. Boussaïri.

Les paires de tournois{−3}-hypomorphes

“…Pour conclure, en utilisant le Lemme 3.5 et le résultat suivant, nous vérifions que pour tout X ∈ M(T ), le sous-tournoi T [X] est fortement autodual ou indécomposable : Théorème 4.1. (Voir [1].) Un tournoi décomposable T ayant n 9 sommets est {−3}-autodual si et seulement si T est fortement autodual.…”

“…Récemment, nous avons fait quelques progrès. Ils sont en partie inclus dans[1] et en partie dans la présente Note.…”