The Cartesian product of graphs with loops

TL;DR

This paper extends the Cartesian product concept to graphs with loops, proves the Sabidussi-Vizing factorization theorem still applies, and provides an efficient O(m) algorithm for factorization.

Contribution

It generalizes the Cartesian product to graphs with loops and establishes an efficient factorization algorithm.

Findings

Sabidussi-Vizing theorem holds for graphs with loops with at least one unlooped vertex

Factorization can be computed in linear time O(m)

Extends graph product theory to a broader class of graphs

Abstract

We extend the definition of the Cartesian product to graphs with loops and show that the Sabidussi-Vizing unique factorization theorem for connected finite simple graphs still holds in this context for all connected finite graphs with at least one unlooped vertex. We also prove that this factorization can be computed in O(m) time, where m is the number of edges of the given graph.