Indecomposability and duality of tournaments Original Research Article

Let T=(V,A) be a tournament. A subset X of V is an interval of T provided that for a,b∈X and for x∈V−X, (a,x)∈A if and only if (b,x)∈A. For example, ∅,{x}, where x∈V, and V are intervals of T, called trivial intervals. A tournament is said to be indecomposable if all of its intervals are trivial. In another respect, with each tournament T=(V,A) is associated the dual tournament T★=(V,A★) defined as: for x,y∈V,(x,y)∈A★ if (y,x)∈A. A tournament T is said to be self-dual if T and T★ are isomorphic. The paper characterizes the finite tournaments T=(V,A) fulfilling: for every proper subset X of V, if the subtournament T(X) of T is indecomposable, then T(X) is self-dual. The corollary obtained is: given a finite and indecomposable tournament T=(V,A), if T is not self-dual, then there is a subset X of V such that 6⩽|X|⩽10 and such that T(X) is indecomposable without being self-dual. An analogous examination is made in the case of infinite tournaments. The paper concludes with an introduction of a new mode of reconstruction of tournaments from their proper and indecomposable subtournaments.