A constructive, algorithmic approach to the Seymour Second Neighborhood Conjecture built on the Graph Level Order (GLOVER) data structure. GLOVER orders the vertices of an oriented graph by shortest-path (BFS) distance from a minimum-out-degree node and imposes a well-ordering on rooted neighborhoods. From this ordering one builds decreasing sequences of vertex subsets and partitions transitive triangles into distinct sets, which the author uses to argue no counterexample can exist and to construct an explicit path to a Seymour vertex (one whose second out-neighborhood is at least as large as its first). The framework also surfaces dense subgraphs within rooted neighborhoods and is proposed for network-optimization and load-balancing applications.
Presents an algorithmic, constructive attack on the Seymour Second Neighborhood Conjecture (SSNC), which asserts that every oriented graph has a vertex whose second out-neighborhood N++(v) is at leas…