The Cartesian product of graphs with loops
  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. 
Tetiana Boiko, Johannes Cuno, Wilfried Imrich, Florian Lehner,   Christiaan E. van de Woestijne
math.CO