Introduction for Superhyperfunction and Superhypergraph with some algorithms

Abstract

A finite hypergraph generalizes the classical graph model by allowing hyperedges that can connect any nonempty subset of vertices. Building on this foundation, a finite SuperHyperGraph is obtained through iterative application of the powerset construction, thereby creating nested families of vertex and edge sets that capture multi-layered relationships.
In this paper, we introduce the concept of an h, k-ary m, n-superHyperGraph, which extends the already wellstudied SuperHyperGraph by employing the framework of h, k-ary m, n-SuperHyperFunctions. An h, k-ary m, n-SuperHyperFunction is a structured mapping that takes h input sets at level m and produces k output sets at level n, enabling the representation of complex multi-input, multi-output relationships. An h, k-ary m, n-SuperHyperGraph is then defined as a higher-order hypergraph whose vertices are such superhyperfunctions, while its hyperedges group these functions to represent contextual and hierarchical dependencies among the mappings. In addition, recognition algorithms and construction algorithms of h, k-ary m, n-SuperHyperGraph are examined with respect to both their validity and their computational complexity.