Seymour's Second Neighbourhood Conjecture for Oriented Graphs of Order at Most Seven and Split-Twin Extensions

TL;DR

We study Seymour's Second Neighborhood Conjecture for oriented graphs by formalizing $N_1^+(v)$, $N_2^+(v)$, and the invariant $\Delta(D)=\max_v(|N_2^+(v)|-|N_1^+(v)|)$; the conjecture is equivalent to $\Delta(D)\ge 0$. We report exhaustive verification that $\Delta(D)\ge 0$ for all oriented graphs on at most seven vertices with minimum outdegree at least one, using precise bitmask computations over labeled graphs. We introduce the split--twin extension, a local operation that preserves the nonnegativity of $\Delta(D)$, and prove an inductive consequence showing that repeated extensions yield infinite families of graphs with the second-neighborhood property. Together, these results provide a complete small-graph validation and a scalable inductive mechanism toward understanding Seymour's conjecture in larger graphs.

Abstract

For an oriented graph $D$, let $N_1^+(v)$ denote the out-neighborhood of a vertex $v$, and let $N_2^+(v)$ be the set of vertices reachable from $v$ by a directed path of length two that are neither out-neighbors of $v$ nor equal to $v$. The Second Neighborhood Conjecture of Seymour asserts that every oriented graph contains a vertex $v$ with $|N_2^+(v)| \ge |N_1^+(v)|$. Equivalently, if one defines the second neighborhood invariant \[ Δ(D)=\max_{v\in V(D)}\bigl(|N_2^+(v)|-|N_1^+(v)|\bigr), \] the conjecture asserts that $Δ(D)\ge 0$ for all oriented graphs. We prove by exhaustive computation that $Δ(D)\ge 0$ for every oriented graph on at most seven vertices. We also introduce a local graph operation, called a split--twin extension, and prove that it preserves the inequality $Δ(D)\ge 0$. Consequently, $Δ(D)\ge 0$ holds for infinite inductively generated families of oriented graphs.

Theorems & Definitions (13)

• Definition 1
• Definition 2
• proof
• Theorem 4
• proof
• Remark 5
• Definition 6: Split--Twin Extension
• Lemma 7
• proof
• Theorem 8