Note for Line and Total SuperHyperGraphs: Connecting Vertices, Edges, Edges of Edges, Edges of Edges of Edges in Hierarchical Systems

Abstract

Hypergraphs extend classical graphs by allowing hyperedges to connect any nonempty subset of vertices, thereby capturing complex group-level relationships. Superhypergraphs advance this framework by introducing recursively nested powerset layers, enabling the representation of hierarchical and self-referential links among hyperedges. A line graph encodes the adjacencies between edges of an original graph by transforming each edge into a vertex and connecting two vertices if their corresponding edges share a common endpoint. A total graph incorporates both the vertices and edges of the original graph as its own vertices, with edges representing adjacency or incidence between these entities. An iterated line graph arises from the repeated application of the line graph construction, where each iteration takes the previous line graph as its input. Similarly, an iterated total graph is generated by iteratively applying the total graph transformation a specified number of times. This paper investigates the hypergraph and superhypergraph analogues of these constructions, providing a foundation for further theoretical development.