We provide a constructive proof of the Seymour Second Neighborhood Conjecture (SSNC) by reframing the problem as a set-packing optimization problem. The universal family of oriented graphs $\mathcal{O}$ is classified by their minimum out-degree $δ$. This shifts the objective to maximizing the number of non-Seymour vertices. A minimum counterexample (MCE) is a maximal packing of vertices that fail the SSNC. To prove such a packing is unsustainable, we introduce the Graph Level Order (GLOVER). This BFS-based coordinate system partitions $\mathcal{O}$ into rooted neighborhoods $R_i$ from a minimum out-degree node. Set-theoretic multiple parents resolve the double-counting that has plagued Seymour diamonds. This coordinate system also categorizes transitive triangles into eight distinct types and proves that seven are inconsistent in an MCE environment. Distinguishing it from BFS, the MCE environment forces cycles in the first neighborhood of every parent. These cause neighborhoods to become quadratically dense as they both decrease in size and need more arcs. The proof concludes with a supply-demand collision. Arc capacity is consumed when $i > \fracδ{3}$. This makes the packing of non-Seymour vertices unsustainable, forcing the appearance of a Seymour vertex in every graph of $\mathcal{O}$. The algorithm to identify these vertices is $O(|V|+|E|)$. This confirms that it can operate on large oriented networks that are dense and detectable in polynomial time.

翻译：我们通过将问题重新表述为集合打包优化问题，为 Seymour 第二邻域猜想（SSNC）提供了一个构造性证明。有向图的通用族 $\mathcal{O}$ 根据其最小出度 $δ$ 进行分类。这将目标转变为最大化非 Seymour 顶点的数量。最小反例（MCE）是不满足 SSNC 的顶点的最大打包。为了证明这种打包是不可持续的，我们引入了图层级序（GLOVER）。这种基于 BFS 的坐标系从最小出度节点出发，将 $\mathcal{O}$ 划分为有根的邻域 $R_i$。集合论的多重父节点解决了长期困扰 Seymour 菱形结构的重复计数问题。该坐标系还将传递三角形分类为八种不同的类型，并证明其中七种在 MCE 环境下是不一致的。与 BFS 不同，MCE 环境迫使每个父节点的第一邻域中出现环。这导致邻域在规模减小的同时需要更多弧，从而变得二次稠密。证明以供需冲突结束。当 $i > \fracδ{3}$ 时，弧容量被耗尽。这使得非 Seymour 顶点的打包无法持续，从而迫使 $\mathcal{O}$ 中每个图都出现 Seymour 顶点。识别这些顶点的算法复杂度为 $O(|V|+|E|)$。这证实了该算法可以在多项式时间内处理稠密且可检测的大型有向网络。