Basic Theory of Hierarchical SuperHyperGraphs: Cross-LevelIncidence, Direction, and Bidirection

Resumen

Hypergraphs extend ordinary graphs by permitting a single hyperedge to connect any nonempty
set of vertices, thereby representing intrinsically multiway interactions. To capture hierarchical or multi-layer
organization, SuperHyperGraphs further iterate the powerset construction, so that aggregates formed at one
level can themselves act as vertices or edge endpoints at higher levels. While this modeling perspective is
increasingly relevant, a systematic treatment of hierarchical SuperHyperGraphs remains at an early stage and
many foundational notions are not yet widely standardized. Addressing this gap, we introduce and investigate
three hierarchical extensions: Hierarchical SuperHyperGraphs, Hierarchical Directed SuperHyperGraphs, and
Hierarchical Bidirected SuperHyperGraphs.