The C3-structure of the tournaments Original Research Article

Let T=(V,E) be a tournament. The C3-structure of T is the family C3(T) of the subsets {x,y,z} of V such that the subtournament T({x,y,z}) is a cycle on three vertices. In another respect, a subset X of V is an interval of T provided that for a,b∈X and x∈V−X, (a,x)∈E if and only if (b,x)∈E. For example, ∅, {x}, where x∈V, and V are intervals of T, called trivial intervals. A tournament is indecomposable if all its intervals are trivial. Lastly, with each tournament T=(V,E) is associated the dual tournament T★=(V,E★), defined as: for x,y∈V, (x,y)∈E★ if (y,x)∈E. The following theorem is proved. Given tournaments T=(V,E) and T=(V,E′) such that C3(T)=C3(T′), if T is indecomposable, then T′=T or T′=T★. In order to treat the nonindecomposable case, the interval inversion is introduced. The paper concludes with an extension of this result to the digraphs which do not admit as subdigraphs ({0,1,2},{(0,1),(1,0),(1,2)}) and ({0,1,2},{(0,1),(1,0),(2,1)}), and with a brief consideration of the infinite case.