A Note on Line and Total Directed SuperHyperGraphs, Line Bidirected Graphs, Line MultiDirected Graphs, and Related Structures

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. Various extensions of these graph concepts exist that incorporate directional information, such as Directed Graphs, Bidirected Graphs, and Multidirected Graphs. In this paper, we investigate the notions of line graphs and total graphs within the settings of Directed HyperGraphs, Directed SuperHyperGraphs, Bidirected Graphs, and Multidirected Graphs.