Median orders of tournaments: A tool for the second neighborhood problem and Sumner's conjecture

We give a short constructive proof of a theorem of Fisher: every tournament contains a vertex whose second outneighborhood is as large as its ®rst outneighborhood. 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 2nÿ 2 contains every arborescence of order n> 1. This is a particular case of Sumner's conjecture: every tournament of order 2nÿ 2 contains every oriented tree of order n> 1. Using our method, we prove that every tournament of order (7nÿ 5)/2 contains every oriented tree of order n. ß 2000 John Wiley & Sons, Inc. J Graph Theory 35: 244±256, 2000