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

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.