The Second Neighborhood Conjecture for Oriented Graphs Missing $\{C_4,\overline{C_4},S_3,$ chair and co-chair$\}$-Free Graph

Abstract

Seymour's Second Neighborhood Conjecture (SNC) asserts that every oriented graph has a vertex whose first out-neighborhood is at most as large as its second out-neighborhood. In this paper, we prove that if $G$ is a graph containing no induced $C_4$, $\overline{C_4}$, $S_3$, chair and $\overline{chair}$, then every oriented graph missing $G$ satisfies this conjecture. As a consequence, we deduce that the conjecture holds for every oriented graph missing a threshold graph, a generalized comb or a star.

arXiv admin note: text overlap with arXiv:1602.08631