About the second neighborhood problem in tournaments missing disjoint stars

Salman Ghazal (Department of Mathematics Faculty of Sciences I, Lebanese University, Lebanon, and Institute Camille Jordan Département de Mathématiques, Université Claude Bernard Lyon 1, France.)

Electronic Journal of Graph Theory and Applications (EJGTA)
Vol 4, No 2 (2016): Electronic Journal of Graph Theory and Applications

Let $D$ be a digraph without digons. Seymour's second neighborhood conjecture states that $D$ has a vertex $v$ such that $d^+(v) \leq d^{++}(v)$. Under some conditions, we prove this conjecture for digraphs missing $n$ disjoint stars. Weaker conditions are required when $n = 2$ or $3$. In some cases we exhibit two such vertices.