Seymour's second neighborhood conjecture for tournaments missing a generalized star

TL;DR

This paper proves a weighted version of Seymour's Second Neighborhood Conjecture for a class of tournaments missing a generalized star, extending the conjecture's validity to related structures.

Contribution

It introduces a proof of the weighted conjecture for tournaments missing a generalized star, broadening the scope of the original conjecture.

Findings

Weighted version holds for tournaments missing a generalized star

Consequence: conjecture holds for tournaments missing a sun, star, or complete graph

Extends the validity of Seymour's conjecture to new classes of digraphs

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. We prove its weighted version for tournaments missing a generalized star. As a consequence the weighted version holds for tournaments missing a sun, star, or a complete graph.