An Algorithmic Approach to Finding Degree-Doubling Nodes in Oriented Graphs

Seymour's Second Neighborhood Conjecture claims that there will always exist a node whose out-degree doubles in the square of an oriented graph. In this paper, we first present a novel data structure, GLOVER (Graph Level Order), which partitions nodes into a total ordering of containers. This data structure establishes a well-ordering on oriented graphs and allows for the construction of a decreasing sequence of subsets of nodes. This sequence proves the non-existence of counterexamples to the SSNC and precisely identifies the required node. Further, our approach finds the occurrence of dense regular graphs inside containers. This finding extends the SSNC to the discovery of multiple nodes satisfying the degree doubling property. Beyond theoretical implications, the algorithm and data structure have practical applications in data science, network optimization and algorithm design.