Topological Generalizations of Graphs: Integrating Hypergraph and Superhypergraph Perspectives

Year 2025, Volume: 17 Issue: 2, 322 - 337, 30.12.2025

Takaaki Fujita , Florentin Smarandache

https://doi.org/10.47000/tjmcs.1729279

Abstract

A hypergraph is a generalization of a graph in which edges—called hyperedges—can connect any number of vertices, not just two. A superhypergraph further extends this framework by introducing recursive powerset constructions, enabling the representation of hierarchical relationships among hyperedges themselves. A topological graph is a graph embedded in the plane, where each vertex is represented by a distinct point and each edge is drawn as a continuous curve connecting its endpoints. A topological hypergraph extends this idea by representing hyperedges as closed curves that enclose sets of vertices on the plane [13].
In this paper, we introduce the concept of a Topological n-SuperHypergraph, which unifies the structural hierarchy of n-superhypergraphs with the geometric intuition of topological hypergraphs. We anticipate that this new formulation will contribute to future developments in graph theory, topology, and network theory.

Keywords

SuperHyperGraph , HyperGraph , Topological Hypergraph , Topology , Topological Graph , topological super- HyperGraph

Ethical Statement

The authors declare that there are no competing interests related to this manuscript.

Supporting Institution

This research received no financial support from any external agencies or organizations.

Thanks

The authors gratefully acknowledge the support, insights, and encouragement provided by colleagues, mentors, and reviewers. We appreciate the interest of our readers and the foundational work of scholars whose publications informed this study. We also thank the institutions and individuals who provided essential resources and infrastructure. Finally, we are indebted to everyone who offered assistance in any form.

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