A contribution to the second neighborhood problem

TL;DR

This paper advances the understanding of Seymour's Second Neighborhood Conjecture by proving it for new classes of digraphs with specific missing graph structures, expanding the cases where the conjecture holds.

Contribution

The paper proves Seymour's Second Neighborhood Conjecture for digraphs whose missing graph is a comb, a complete graph minus two independent edges, or minus a 5-cycle.

Findings

Proved the conjecture for digraphs with a comb missing graph.

Established the conjecture for digraphs missing a complete graph minus two independent edges.

Confirmed the conjecture for digraphs missing a cycle of length five.

Abstract

Seymour's Second Neighborhood Conjecture asserts that every digraph (without digons) has a vertex whose first out-neighborhood is at most as large as its second out-neighborhood. It is proved for tournaments, tournaments missing a matching and tournaments missing a generalized star. We prove this conjecture for classes of digraphs whose missing graph is a comb, a complete graph minus 2 independent edges, or a complete graph minus the edges of a cycle of length 5.