On The Seymours’s Second Neighborhood Conjecure

In 1990, Paul Seymour proposed the following conjecture: Every oriented finite simple graph has a vertex whose second out-neighborhood is at least as large as its first out-neighborhood. This conjecture is known as Seymour’s Second Neighborhood Conjecture (SNC). It is proved for tournaments[1, 2], tournaments minus a matching[3], tournaments minus a generalized star[4]. WehaveprovedthattheSNCholdsfortournamentsmissingdisjointpaths of length at most two under some conditions on the dependency digraph. Recently, we present an approach of the SNC. We define seymour set and the SNC becomes: Every simple digraph has a singleton seymour set. We prove that there is a seymour set S with |S| = 2. Moreover, we prove that every simple digraph contains seymour sets of any size ≥ 2