About the second neighborhood conjecture for tournaments missing two stars or disjoint paths

TL;DR

This paper proves Seymour's Second Neighborhood Conjecture for tournaments missing two stars and explores its validity for tournaments missing disjoint paths, especially length-2 paths, identifying vertices with the SNP.

Contribution

It establishes SSNC for a new class of tournaments missing two stars and analyzes its validity for tournaments missing disjoint paths, including length-2 paths.

Findings

Proved SSNC for tournaments missing two stars

Identified vertices with SNP in certain missing path cases

Extended SSNC analysis to tournaments missing disjoint paths

Abstract

Seymour's Second Neighborhood Conjecture (SSNC) asserts that every oriented finite simple graph (without digons) has a vertex whose second out-neighborhood is at least as large as its first out-neighborhood. Such a vertex is said to have the second neighborhood property (SNP). In this paper, we prove SSNC for tournaments missing two stars. We also study SSNC for tournaments missing disjoint paths and, particularly, in the case of missing paths of length 2. In some cases, we exhibit at least two vertices with the SNP.