Description of the Tournaments Which are Reconstructible from Their k-cycle Partial Digraphs for $$k \in \{3, 4\}$$ k ∈ { 3 , 4 }

International audienceIf T is a tournament on n vertices and k is an integer with , then the k-cycle partial digraph of T, denoted by T(k), is the spanning subdigraph of T for which the arcs are those of the k-cycles of T. In 1989, Thomassen proved that given two irreducible tournaments T and on the same vertex set with at least 6 vertices, if , then . This result allows us to introduce the following notion of reconstruction. A tournament T is reconstructible from its k-cycle partial digraph whenever is isomorphic to T for each tournament such that is isomorphic to T(k). In this paper, we give a complete description of the tournaments which are reconstructible from their k-cycle partial digraphs where . For , our proof is based on the above result of Thomassen. For , we introduce and study the tournaments for which all modules are irreducible. We use properties of the dilatation operator to describe such tournaments, and then we obtain a new modular decomposition of tournaments sometimes finer than that of Gallai