Fast Recognition of Partial Star Products and Quasi Cartesian Products

TL;DR

This paper introduces efficient algorithms for recognizing approximate and quasi Cartesian graph products, utilizing special data structures for constant-time computation of Partial Star Products and linear-time recognition for graphs with bounded degree.

Contribution

It presents novel methods for fast recognition of approximate and quasi Cartesian products, including linear and sublinear time algorithms, and introduces the concept of quasi Cartesian products.

Findings

Constant-time computation of Partial Star Products.

Linear-time recognition of quasi Cartesian products for bounded degree graphs.

Sublinear-time recognition possible with parallel algorithms.

Abstract

This paper is concerned with the fast computation of a relation $\R$ on the edge set of connected graphs that plays a decisive role in the recognition of approximate Cartesian products, the weak reconstruction of Cartesian products, and the recognition of Cartesian graph bundles with a triangle free basis. A special case of $\R$ is the relation $\delta^\ast$, whose convex closure yields the product relation $\sigma$ that induces the prime factor decomposition of connected graphs with respect to the Cartesian product. For the construction of $\R$ so-called Partial Star Products are of particular interest. Several special data structures are used that allow to compute Partial Star Products in constant time. These computations are tuned to the recognition of approximate graph products, but also lead to a linear time algorithm for the computation of $\delta^\ast$ for graphs with maximum bounded degree. Furthermore, we define \emph{quasi Cartesian products} as graphs with non-trivial $\delta^\ast$. We provide several examples, and show that quasi Cartesian products can be recognized in linear time for graphs with bounded maximum degree. Finally, we note that quasi products can be recognized in sublinear time with a parallelized algorithm.