Hyperfuzzy OffGraphs: A Unified Graph-Based Theoretical Decision Framework for Hierarchical Under Off-Uncertainty

Abstract

Uncertainty-handling frameworks, such as fuzzy sets, rough sets, intuitionistic fuzzy sets, hesitant fuzzy sets, spherical fuzzy sets, neutrosophic sets, and plithogenic sets, have been extensively investigated. These models have further been generalized into HyperUncertain and SuperHyperUncertain sets to represent more complex forms of uncertainty. Related concepts have also been examined in graph theory, where fuzzy offsets with freely varying interval values have been explored. While research on fuzzy graphs continues to be significant, extensions of fuzzy offgraphs remain relatively underdeveloped. This paper introduces the notion of the hyperfuzzy offgraph, a novel structure that extends fuzzy offgraphs by incorporating hierarchical, over-uncertainty, and under-uncertainty behaviors on both vertices and edges. We provide rigorous definitions, establish key properties, and present illustrative examples to demonstrate expressive real-world applications.