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Seymour第二邻域猜想（SSNC）重新表述为集合包装优化问题，给出了该猜想的一个构造性证明。有向图的全族$\mathcal{O}$根据其最小出度$δ$进行分类。这使目标转化为最大化非Seymour顶点的数量。最小反例（MCE）是违反SSNC的顶点的最大包装。为证明此类包装不可持续，我们引入了图层次序（GLOVER）。这一基于BFS的坐标系从最小出度节点出发将$\mathcal{O}$划分为有根邻域$R_i$。集合论中的多重父节点解决了长期困扰Seymondiamond的重数计算问题。该坐标系还将传递三角形划分为八种不同类型，并证明其中七种在MCE环境中矛盾。区别于BFS，MCE环境迫使每个父节点的第一邻域中出现环。这些环导致邻域在尺寸减小且需更多弧时呈二次密度增长。证明最终以供需矛盾收尾：当$i > \fracδ{3}$时，弧容量被消耗。这使得非Seymour顶点的包装不可持续，迫使$\mathcal{O}$中每个图均出现Seymour顶点。识别这些顶点的算法复杂度为$O(|V|+|E|)$，证实该算法可运行于稠密且可在多项式时间内检测的大规模有向网络。