Soft Directed n-SuperHyperGraphs with Some Real-World Applications

Abstract

Graph theory offers a powerful language for encoding relationships via vertices and edges [1]. Hypergraphs extend this by introducing hyperedges that may join any number of vertices at once [2], and superhypergraphs go further still by iteratively applying the powerset construction to capture multi-level, hierarchical connections [3]. Meanwhile, a variety of frameworks for managing uncertainty—fuzzy sets [4], soft sets [5], intuitionistic fuzzy sets [6], rough sets[7], neutrosophic sets[8], and plithogenic sets[9]—have been proposed to handle imprecision and indeterminacy. In directed settings, one encounters directed graphs[10], directed hypergraphs[11], and directed superhypergraphs[12], as well as soft directed graphs[13]. In this work, we unify these ideas by introducing the Directed Soft SuperHyperGraph, marrying the notions of directionality, hierarchical hyperstructure, and soft-set parameterization. We anticipate that this new model will be especially well suited to representing complex, multi-layered, directed networks—such as those arising in urban infrastructure, transportation planning, and information flow—where both hierarchy and uncertainty must be managed simultaneously.