Split-Twin Extensions Preserving Seymour Vertices

TL;DR

This paper introduces split-twin extensions, a local graph operation that preserves Seymour vertices in oriented graphs, and demonstrates how this leads to constructing infinite classes of graphs satisfying Seymour's Second Neighborhood Conjecture.

Contribution

The paper presents a novel split-twin extension operation that preserves Seymour vertices and enables the creation of infinite graph classes satisfying the conjecture.

Findings

Split-twin extension preserves Seymour vertices under certain conditions.

Infinite classes of graphs satisfying Seymour's conjecture can be constructed.

The operation facilitates understanding of the conjecture's validity in larger graphs.

Abstract

The Second Neighborhood Conjecture of Seymour asserts that every oriented graph contains a vertex $v$ satisfying \[ |N_2^+(v)| \ge |N_1^+(v)|. \] Vertices with this property are called Seymour vertices. In this paper we introduce a local graph operation, called a \emph{split--twin extension}, which replaces a vertex by two new vertices that inherit all of its incoming edges while partitioning its outgoing edges. We prove that this operation preserves the existence of a Seymour vertex under a natural separation condition. More precisely, if $v$ is a Seymour vertex in an oriented graph $D$, and $x\notin N_1^+(v)$, then every split--twin extension of $D$ at $x$ again contains a Seymour vertex. As a consequence, any oriented graph containing a Seymour vertex gives rise to arbitrarily large graphs whenever vertices outside the first out-neighborhood of the preserved Seymour vertex exist. In particular, combining our result with the theorem of Kaneko and Locke implies the existence of infinite classes of oriented graphs satisfying Seymour's conjecture.