Spatial HyperGraphs and Spatial SuperHyperGraphs

Graph theory studies the mathematical structures of vertices and edges to model relationships and connectivity [1, 2]. Hypergraphs extend this framework by allowing hyperedges to connect arbitrarily many vertices at once [3], and superhypergraphs further generalize hypergraphs via iterated powerset constructions to capture hierarchical linkages among edges [4, 5]. A spatial hypergraph is a hypergraph in which each vertex is assigned a fixed location in Euclidean space through an embedding. In this paper, we introduce the spatial 𝑛-SuperHyperGraph, an extension of spatial hypergraphs within the 𝑛-SuperHyperGraph framework. This generalization provides a clear and intuitive means of representing the hierarchical structures inherent in spatial graphs, yielding significant advantages for modeling and analysis.