Research Article

Expanding the Horizons of Graph Theory: Rough Tree-Width, Hyperrough Structures, and Superhyperrough Generalizations

Abstract: Rough set theory provides a formal framework for managing imprecise information by approximating subsets of a universe through lower and upper bounds derived from equivalence relations. This foundational idea has been extended to more expressive structures such as the Hyperrough Set and the Superhyperrough Set. Building upon rough set theory, rough graphs are introduced to represent uncertain relationships among elements, where the existence of edges is determined by the rough membership values of their endpoints. To analyze structural properties of graphs, various parameters are employed—among them, graph width parameters play a central role by measuring the structural complexity of graphs through bounded decompositions. One widely studied example is tree-width, which has recently been generalized to the context of rough graphs as rough tree-width. In this paper, We extend the framework by proposing several new generalizations: the Hyperrough Tree-width, the Hyperrough Graph, and the Superhyperrough Graph. For each of these constructs, we provide formal definitions and initiate a preliminary exploration of their mathematical properties, laying the groundwork for future study and application in uncertainty-aware graph analysis.