The Arc-Weighted Version of the Second Neighborhood Conjecture

TL;DR

This paper extends Seymour's second neighborhood conjecture to arc-weighted digraphs, proves it for arc-weighted tournaments, and provides an alternative proof of Fisher's theorem.

Contribution

It introduces a new arc-weighted version of the conjecture and proves it for tournaments, offering an independent proof of Fisher's theorem.

Findings

Proved the conjecture for arc-weighted tournaments.

Provided an alternative proof of Fisher's theorem.

Extended the conjecture to arc-weighted digraphs.

Abstract

Seymour conjectured that every oriented simple graph contains a vertex whose second neighborhood is at least as large as its first. Seymour's conjecture has been verified in several special cases, most notably for tournaments by Fisher. One extension of the conjecture that has been used by several researchers is to consider vertex-weighted digraphs. In this paper we introduce a version of the conjecture for arc-weighted digraphs. We prove the conjecture in the special case of arc-weighted tournaments, strengthening Fisher's theorem. Our proof does not rely on Fisher's result, and thus can be seen as an alternate proof of said theorem.