Research Article

Fuzzy Off-Hypergraphs and Directed Offgraphs:Real-World Case Studies in Uncertainty Modeling

Abstract

Handling uncertainty in mathematical models has led to a variety of frameworks—fuzzy sets [1], rough sets [2], intuitionistic fuzzy sets [3], neutrosophic sets [4], hyperneutrosophic sets [5–7], and plithogenic sets [8]—each addressing different aspects of imprecision. In parallel, graph-theoretic generalizations—hypergraphs [9], directed and bidirected graphs [10], multidirected graphs [11], and superhypergraphs [12, 13]—have expanded our ability to model complex relationships. More recently, the notions of “offset” and “offgraph” have emerged to allow membership and relation degrees to overflow the classical [0,1] interval. In this paper, we introduce two new constructs—the Fuzzy Off-Hypergraph and the Fuzzy Directed Offgraph—and present a concise study of their fundamental mathematical properties. By permitting vertex and edge memberships to exceed the unit range, these models capture uncertainty with greater precision. We anticipate that they will inspire further advances in fuzzy graph theory and its applications.