The Second Neighborhood Conjecture for Oriented Graphs Missing $\{C_{4}, \overline{C_{4}}, S_{3},$ chair and co-chair$\}$-Free Graph

TL;DR

This paper proves Seymour's Second Neighborhood Conjecture for a class of oriented graphs missing specific induced subgraphs, expanding the classes of graphs for which the conjecture holds.

Contribution

It establishes the conjecture for oriented graphs missing certain forbidden induced subgraphs, including threshold graphs, generalized combs, and stars.

Findings

The conjecture holds for graphs missing $C_4$, $ar{C}_4$, $S_3$, chair, and co-chair.

It extends the validity of the conjecture to graphs missing threshold graphs.

It confirms the conjecture for graphs missing generalized combs and star graphs.

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.