[Paper Review] Cartesian product of hypergraphs: properties and algorithms

Alain Bretto, Yannick Silvestre+1 more|Sep 28, 2009

Graph Labeling and Dimension Problems4 citations

TL;DR

This paper presents a linear-time algorithm for prime factorization of connected conformal hypergraphs using their 2-sections and labeled 2-sections, extending Imrich and Peterin's graph factorization algorithm. It establishes that the chromatic number and coloring properties of Cartesian products of hypergraphs are preserved under factorization, enabling efficient algorithmic decomposition of complex hypergraph problems into simpler components.

ABSTRACT

Cartesian products of graphs have been studied extensively since the 1960s. They make it possible to decrease the algorithmic complexity of problems by using the factorization of the product. Hypergraphs were introduced as a generalization of graphs and the definition of Cartesian products extends naturally to them. In this paper, we give new properties and algorithms concerning coloring aspects of Cartesian products of hypergraphs. We also extend a classical prime factorization algorithm initially designed for graphs to connected conformal hypergraphs using 2-sections of hypergraphs.

Motivation & Objective

• To extend the prime factorization algorithm for graphs to connected conformal hypergraphs using 2-sections and labeled 2-sections.
• To investigate the stability of hypergraph coloring properties under Cartesian product decomposition.
• To enable algorithmic complexity reduction by decomposing hypergraph problems into factors of the Cartesian product.
• To prove the uniqueness of prime factorization for conformal hypergraphs up to isomorphism.
• To provide a polynomial-time method for decomposing hypergraphs into prime factors to simplify problem-solving.

Proposed method

• Construct the labeled 2-section (L2-section) of a conformal hypergraph H, which captures vertex adjacencies and hyperedge labels.
• Apply the linear-time prime factorization algorithm of Imrich and Peterin to the 2-section graph G to obtain prime factors G₁,…,Gₖ.
• Restrict the edge labels from the L2-section to each factor Gi to reconstruct partial labeled graphs Γi.
• Reconstruct the original hypergraph factors Hi as the inverse L2-section of each Γi, ensuring they are conformal hypergraphs.
• Verify that the original hypergraph H is isomorphic to the Cartesian product H₁□H₂□…□Hₖ.
• Use the isomorphism invariance of 2-sections and L2-sections to guarantee correctness and uniqueness of the factorization.

Experimental results

Research questions

• RQ1 Can the prime factorization algorithm for graphs be adapted to hypergraphs, particularly conformal ones, using 2-sections and labeled 2-sections?
• RQ2 Do key coloring properties—such as chromatic number and chromatic index—remain stable under Cartesian product decomposition in hypergraphs?
• RQ3 Is the prime factorization of a connected conformal hypergraph unique up to isomorphism, and can it be computed efficiently?
• RQ4 Can the algorithmic complexity of solving problems on hypergraphs be reduced by decomposing them into Cartesian product factors?
• RQ5 What is the computational complexity of computing the prime factorization of a conformal hypergraph using this method?

Key findings

• The proposed algorithm computes the prime factorization of any connected conformal hypergraph in O(m(H)·r(H)²) time, where m(H) is the number of hyperedges and r(H) is the maximum hyperedge size.
• The chromatic number χ(H) of a Cartesian product of hypergraphs is equal to the maximum of the chromatic numbers of its factors.
• The chromatic index q(H) of a hypergraph satisfies q(H) = Δ(H) if and only if the hypergraph has the colored hyperedge property, and this property is preserved under Cartesian product decomposition.
• The prime factorization of a connected conformal hypergraph is unique up to isomorphism, as guaranteed by the uniqueness of 2-section factorization.
• The algorithm is sound and complete: it always returns the correct prime factors of a conformal hypergraph, and every conformal hypergraph admits such a decomposition.
• The method enables the transfer of algorithmic complexity from the full hypergraph to its factors, allowing efficient problem-solving on the factors instead of the product.