Strong Products of Hypergraphs: Unique Prime Factorization Theorems and Algorithms

TL;DR

This paper extends the concept of prime factorization from graphs to hypergraphs, introducing algorithms for unique decomposition based on the Cartesian skeleton, applicable to hypergraphs with bounded degree and rank.

Contribution

It establishes the existence and uniqueness of prime factorization for connected thin hypergraphs and provides efficient algorithms for computing the Cartesian skeleton and PFD.

Findings

Unique prime factorization for hypergraphs established.

Algorithms for Cartesian skeleton and PFD run in polynomial time.

Applicable to hypergraphs with bounded degree and rank.

Abstract

It is well-known that all finite connected graphs have a unique prime factor decomposition (PFD) with respect to the strong graph product which can be computed in polynomial time. Essential for the PFD computation is the construction of the so-called Cartesian skeleton of the graphs under investigation. In this contribution, we show that every connected thin hypergraph H has a unique prime factorization with respect to the normal and strong (hypergraph) product. Both products coincide with the usual strong graph product whenever H is a graph. We introduce the notion of the Cartesian skeleton of hypergraphs as a natural generalization of the Cartesian skeleton of graphs and prove that it is uniquely defined for thin hypergraphs. Moreover, we show that the Cartesian skeleton of hypergraphs can be determined in O(|E|^2) time and that the PFD can be computed in O(|V|^2|E|) time, for hypergraphs H = (V,E) with bounded degree and bounded rank.