La \(\{-k\}\)-autodualité des Sommes Lexicographiques Finies de Tournois Suivant un \(3\)-cycle ou un Tournoi.Critique

Abstract

Let \(T = (V, A)\) be a finite tournament with \(n \geq 2\) vertices. The dual of T is the tournament \(T^* = (V, A^*)\) defined by: for all \(x,y \in V, (x,y) \in A^*\) if and only if \((y,x) \in A\). The tournament \(T\) is critical if \(T\) is indecomposable and if for all \(x \in V\), the subtournament \(T(V – \{x\})\) is decomposable. A \(3\)-cycle is a tournament isomorphic to the tournament \(T, = ({0,1,2}, {(0, 1), (1, 2), (2, 0)})\). Let \(F\) be a set of non negative integers \(k < n\). The tournament \(T\) is \(F\)-selfdual if for every subset \(X\) of \(V\) such that \(|X |\in F\), the subtournaments \(T(X)\) and \(T^*(X)\) are isomorphic. In this paper, we study, for each integer \(k \geq 1\), the \(\{n – k\}\)-selfduality of the tournaments, with \(n \geq 4+k\) vertices, that are lexicographical sums of tournaments under a \(3\)-cycle or a critical tournament. As application, we determine for each integer \(k \geq 1\), the tournaments, with \(n \geq 4+ k\) vertices, that are \(\{4,n – k\}\)-selfdual.