Linear Diophantine Fuzzy SuperHyperGraphs and Fuzzy Planar SuperHyperGraphs

Abstract

Graph theory offers a rigorous framework for modeling relationships and connectivity via vertices and edges [1, 2]. Hypergraphs generalize this framework by allowing hyperedges that join more than two vertices [3]. Superhypergraphs further enrich the model through iterated powerset constructions, capturing hierarchical and self-referential structures among hyperedges [4]. A Linear Diophantine Fuzzy Graph assigns linear Diophantine fuzzy numbers to vertices and edges, with admissibility enforced by algebraic membership constraints. A planar graph admits a crossing-free drawing in the plane, preserving adjacency without edge intersections, and a fuzzy planar graph is one in which sufficiently strong edges form a planar subgraph, combining fuzziness with classical planar embeddings. In this paper, we extend Linear Diophantine Fuzzy Graphs and fuzzy planar graphs to the settings of HyperGraphs and SuperHyperGraphs by introducing new classes, including Linear Diophantine Fuzzy SuperHyperGraphs and Fuzzy Planar SuperHyperGraphs, and we investigate their fundamental properties.