Evolving Dependencies: From Graphs to Hypergraphs and SuperHypergraphs

Graph theory studies the mathematical structure of vertices and edges to model relationships and connectivity [1, 2]. Hypergraphs extend this framework by allowing hyperedges to connect arbitrarily many vertices at once [3], and superhypergraphs further generalize hypergraphs via iterated powerset constructions to capture hierarchical linkages among edges [4, 5]. A Dependency graph is a finite directed acyclic graph whose vertices denote tasks and whose edges encode prerequisite relationships. In this paper, we extend this notion by introducing Dependency hypergraphs and Dependency superhypergraphs, and we investigate their fundamental properties. A Dependency hypergraph relates each nonempty set of prerequisite vertices to exactly one dependent vertex, while a Dependency superhypergraph is an 𝑛-level acyclic structure whose supervertices lie in iterated powersets and whose superedges encode multi-layer dependencies. We establish how these models generalize classical Dependency graphs and demonstrate their potential for representing higher-order Dependency systems.