Directed Acyclic SuperHypergraphs (DASH): A General Framework for Hierarchical Dependency Modeling

Abstract. Graph theory models pairwise relationships through vertices and edges, while hypergraphs generalize this framework by allowing hyperedges to connect multiple vertices simultaneously. SuperHyperGraphs, introduced by Smarandache [26, 27], extend hypergraphs via iterated powerset constructions. Directed Acyclic Graphs (DAGs) are cycle-free directed graphs widely used for dependency modeling, and Directed Acyclic Hypergraphs (DAHs) further generalize DAGs by capturing multi-way dependencies [18, 23]. In this paper, we introduce Directed Acyclic SuperHypergraphs (DASH), which unify and extend both DAGs and DAHs within the SuperHyperGraph framework. We present a formal definition of DASH, characterize acyclicity through directed superhyperedges, and establish fundamental properties such as the existence of source supervertices and topological orderings. Our work provides a rigorous theoretical foundation for hierarchical dependency modeling in complex systems and paves the way for future advances in both the theory and applications of Hypergraph and SuperHyperGraph Theory. Keywords: Superhypergraph, Hypergraph, Directed graph, Directed Hypergraph, Directed Superhypergraph, Directed Acyclic graph, Directed Acyclic Hypergraph