The {-2,-1}-selfdual and decomposable tournaments

International audienceWe only consider finite tournaments. The dual of a tournament is obtained by reversing all the arcs. A tournament is selfdual if it is isomorphic to its dual. Given a tournament T, a subset X of V(T) is a module of T if each vertex outside X dominates all the elements of X or is dominated by all the elements of X. A tournament T is decomposable if it admits a module X such that 1 < vertical bar X vertical bar < vertical bar V(T)vertical bar. We characterize the decomposable tournaments whose subtournaments obtained by removing one or two vertices are selfdual. We deduce the following result. Let T be a non decomposable tournament. If the subtournaments of T obtained by removing two or three vertices are selfdual, then the subtournaments of T obtained by removing a single vertex are not decomposable. Lastly, we provide two applications to tournaments reconstruction

The morphology of infinite tournaments. application to the growth of their profile

A tournament is acyclically indecomposable if no acyclic autonomous set of vertices has more than one element. We identify twelve infinite acyclically indecomposable tournaments and prove that every infinite acyclically indecomposable tournament contains a subtournament isomorphic to one of these tournaments. The profile of a tournament T is the function ϕT which counts for each integer n the number ϕT(n) of tournaments induced by T on the n-element subsets of T, isomorphic tournaments being identified. As a corollary of the result above we deduce that the growth of ϕT is either polynomial, in which case ϕT(n)≃an k, for some positive real a, some non-negative integer k, or as fast as some exponential