Hyperstar Decomposition of r-partite complete, Knodel and Fibonacci Hypergraphs

Abstract

A graph is a representation of a relation between a pair of objects in a set called vertex set. A relation is a set of two-element subsets of the vertex set, called edge set. If the restriction of the two-element subset is relaxed to any number, the corresponding representation is a hypergraph. An r-uniform hypergraph is a Steiner system, in which every pair of nodes is contained in exactly a single edge. Knodel graphs, Fibonacci graphs, and r-partite complete graphs are structures with optimal communication properties. Knodel hypergraph and Fibonacci hypergraph are introduced. Partitioning of the edge set leads to the decomposition of a given graph into sub-structures. If all such sub-structures are isomorphic, the decomposition is an isomorphic decomposition. The isomorphic decomposition of an r-partite hypergraph, Knodel hypergraph, and Fibonacci hypergraph into hyperstars are presented.