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 \(V\)\), the subtournament \(T[X] = (X, A \cap (X \times X))\) of \(T\), induced by \(X\), is associated. 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, \(T[V \setminus X]\) 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 \(a,b \in I\) and for \(x \in V \setminus I\), \((a,x) \in A\) if and only if \((b,x) \in A\). For instance, \(\emptyset\), \(V\), and \(\{x\}\), where \(x \in V\), are intervals of \(T\) called trivial intervals. \(T\) is indecomposable if all its intervals are trivial; otherwise, it is decomposable. A tournament \(T’\), on the set \(V\), is \((-k)\)-hypomorphic to \(T\) if for each set \(X\) on \(k\) vertices, \(T[V \setminus X]\) and \(T'[V \setminus X]\) are isomorphic. The tournament \(T\) is \((-k)\)-reconstructible if each tournament \((-k)\)-hypomorphic to \(T\) is isomorphic to it.

Suppose that \(T\) is decomposable and \(|V| \geq 9\). 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 interval \(X\) of \(T\) such that \(T[X]\) is indecomposable and \(|V \setminus X| \leq 2\), then \(T\) is \((-3)\)-reconstructible. Finally, we conclude by reducing the \((-3)\)-reconstruction problem.