Threshold Hypergraphs, Hyper-PolyGraphs, and Their Extensions

Modern graph theory investigates the interplay of vertices and edges to represent complex relationships and connectivity patterns [1, 2]. Hypergraphs extend this paradigm by allowing hyperedges to join any number of vertices in a single relation [3], and superhypergraphs further enrich these models via iterated powerset constructions that capture multi-layered, hierarchical connections among edges [4, 5]. Threshold Graphs form a classical family built by iteratively adding either isolated vertices or vertices adjacent to all existing ones according to a numeric threshold rule. PolyGraphs have likewise been studied extensively in combinatorial and applied settings. In this work, we introduce the concepts of Threshold SuperHypergraphs and SuperHyperPolyGraphs, thereby generalizing Threshold Graphs and PolyGraphs within the superhypergraph framework and providing a unified, hierarchical approach to threshold-based and polygraphic structures.