A remark on the second neighborhood problem

Seymour’s second neighborhood conjecture states that every simple digraph (without digons) hasa vertex whose first out-neighborhood is at most as large as its second out-neighborhood. Such avertex is said to have the second neighborhood property (SNP). We define ”good” digraphs andprove a statement that implies that every feed vertex of a tournament has the SNP. In the case ofdigraphs missing a matching, we exhibit a feed vertex with the SNP by refining a proof due toFidler and Yuster and using good digraphs. Moreover, in some cases we exhibit two vertices withSNP.