Recognizing Cartesian products in linear time

Abstract

We present an algorithm that determines the prime factors of connected graphs with respect to the Cartesian product in linear time and space. This improves a result of Aurenhammer et al. [Cartesian graph factorization at logarithmic cost per edge, Comput. Complexity 2 (1992) 331-349], who compute the prime factors in O(m log n) time, where m denotes the number of vertices of G and n the number of edges. Our algorithm is conceptually simpler. It gains its efficiency by the introduction of edge-labellings.

Cartesian product of hypergraphs: properties and algorithms

“…A straightforward consequence of the Square Lemma given in [IP07] is that every triangle of G is necessarily contained in the same layer. From these facts we get easily the following result.…”

“…In the sequel, we use some of the results from Imrich and Peterin in [IP07]. In order to find prime factorizations of conformal hypergraphs, we extend the algorithm given in [IP07].…”

“…In order to find prime factorizations of conformal hypergraphs, we extend the algorithm given in [IP07]. This algorithm is based on a coloring of the edges of the graph G to be factorized in such a way that the factors are proper subgraphs of G said layers.…”

“…We show that the algorithm of Imrich and Peterin [IP07] can be used to find the prime factorization of connected conformal hypergraphs by considering the 2-section and the labelled 2-section of the hypergraph to be factorized.…”

“…These relations are of particular interest as they allow us to break down problems by transferring algorithmic complexity from the product to the factors. In 2006, Imrich and Peterin [IP07] gave an algorithm able to compute the prime factorization of connected graphs in linear time and space, making the use of Cartesian product decomposition particularly attractive.…”