Directed n-Superhypergraphs Incorporating Bipolar Fuzzy Information: A Multi-Tier Framework for Modeling Bipolar Uncertainty in Complex Networks

Graph theory studies the mathematical structures of vertices and edges to model relationships and connectivity. Hypergraphs extend this framework by allowing hyperedges to connect arbitrarily many vertices at once [1], and Super-HyperGraphs further generalize hypergraphs via iterated powerset constructions to capture hierarchical linkages among edges [2, 3]. Bipolar fuzzy directed graphs assign positive and negative membership degrees to directed edges and vertices, and bipolar fuzzy directed hypergraphs extend this assignment to multi-vertex hyperedges. In this paper, we extend directed Super-HyperGraphs by incorporating bipolar fuzzy membership and introduce the Bipolar Fuzzy Directed n-Super-HyperGraph, whose structural properties we investigate.