Fuzzy Off-SuperHyperGraphs: Extending Uncertainty Modeling Beyond Classical Boundaries

Graph theory provides a mathematical framework for representing relationships and connectivity through vertices and edges [1,2]. Hypergraphs extend this classical notion by introducing hyperedges that may connect more than two vertices at once [3]. Superhypergraphs further enrich the model by employing iterated powerset constructions, thereby capturing hierarchical and self-referential structures among hyperedges [4, 5]. A fuzzy 𝑛-SuperHyperGraph advances this approach by assigning membership degrees to both supervertices and superedges, offering a flexible tool for modeling uncertainty in complex systems. In this paper, we propose an extension of this framework, termed the Fuzzy Off-SuperHyperGraph, which integrates the offgraph paradigm into fuzzy 𝑛-SuperHyperGraphs. We establish its formal definition, investigate its structural properties, and discuss its potential applications in uncertainty modeling and hierarchical network analysis.