Indecomposable Tournaments and Their Indecomposable Subtournaments on \(5\) and \(7\) Vertices

Abstract

Given a tournament \(T = (V, A)\), a subset \(X\) of \(V\) is an interval of \(T\) provided that for every \(a, b \in X\) and \(x \in V – X\), \((a, x) \in A\) if and only if \((b, x) \in A\). For example, \(\emptyset\), \(\{x\}\) (\(x \in V\)), and \(V\) are intervals of \(T\), called trivial intervals. A tournament, all the intervals of which are trivial, is indecomposable; otherwise, it is decomposable. A critical tournament is an indecomposable tournament \(T\) of cardinality \(\geq 5\) such that for any vertex \(x\) of \(T\), the tournament \(T – x\) is decomposable. The critical tournaments are of odd cardinality and for all \(n \geq 2\) there are exactly three critical tournaments on \(2n + 1\) vertices denoted by \(T_{2n+1}\), \(U_{2n+1}\), and \(W_{2n+1}\). The tournaments \(T_5\), \(U_5\), and \(W_5\) are the unique indecomposable tournaments on 5 vertices. We say that a tournament \(T\) embeds into a tournament \(T’\) when \(T\) is isomorphic to a subtournament of \(T’\). A diamond is a tournament on 4 vertices admitting only one interval of cardinality 3. We prove the following theorem: if a diamond and \(T_5\) embed into an indecomposable tournament \(T\), then \(W_5\) and \(U_5\) embed into \(T’\). To conclude, we prove the following: given an indecomposable tournament \(T\) with \(|V(T)| \geq 7\), \(T\) is critical if and only if only one of the tournaments \(T_7\), \(U_7\), or \(W_7\) embeds into \(T\).