A 16-vertex tournament for which Banks set and Slater set are disjoint

摘要

Given a tournament T, a Banks winner of T is the first vertex of any maximal (with respect to inclusion) transitive subtournament of T; a Slater winner of T is the first vertex of any transitive tournament at minimum distance of T (the distance being the number of arcs to reverse in T to make T transitive). In this note, we show that there exists a tournament with 16 vertices for which no Slater winner is a Banks winner. This counterexample improves the previous one, due to G. Laffond and J.-F. Laslier, which has 75 vertices.