Median orders of tournaments: a tool for the second neighbourhood problem and Sumner’s conjecture

We give a short constructive proof of a theorem of Fisher: every tournament contains a vertex whose second outneighbourhood is as large as its first outneighbourhood. Moreover, we exhibit two such vertices provided that the tournament has no dominated vertex. The proof makes use of median orders. A second application of median orders is that every tournament of order 2n − 2 contains every arborescence of order n > 1. This is a particular case of Sumner’s conjecture: every tournament of order 2n− 2 contains every oriented tree of order n > 1. Using our method, we prove that every tournament of order (7n−5)/2 contains every oriented tree of order n. A median order of a tournament T is a linear extension of an acyclic subdigraph of T , maximal with respect to its number of arcs. This concept arises naturally in voting theory, and many articles deal with the computation of such orders. Determining a median order of a digraph is NP-hard, and the complexity for tournaments is still unknown (see [1]). Surprisingly, the notion of median order, well-studied for its own sake, has been seldom