The Second Neighborhood Conjecture for Oriented Graphs Missing Combs

TL;DR

This paper proves Seymour's Second Neighborhood Conjecture for oriented graphs missing combs, characterized by specific forbidden induced subgraphs, and extends the result to oriented combs and threshold graphs.

Contribution

It characterizes combs and related graphs using dependency digraphs and proves the conjecture holds for these classes of oriented graphs.

Findings

Seymour's conjecture holds for oriented graphs missing combs.

Oriented combs and threshold graphs satisfy the conjecture.

Dependency digraphs characterize combs and related graphs.

Abstract

Seymour's Second Neighborhood Conjecture asserts that every oriented graph has a vertex whose first out-neighborhood is at most as large as its second out-neighborhood. Combs are the graphs having no induced $C_4$, $\overline{C_4}$, $C_5$, chair or $\overline{chair}$. We characterize combs using dependency digraphs. We characterize the graphs having no induced $C_4$, $\overline{C_4}$, chair or $\overline{chair}$ using dependency digraphs. Then we prove that every oriented graph missing a comb satisfies this conjecture. We then deduce that every oriented comb and every oriented threshold graph satisfies Seymour's conjecture.