Also available at http://amc.imfm.si ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 2 (2009) 191–205 On Cartesian skeletons of graphs Richard H. Hammack Department of Mathematics and Applied Mathematics Virginia Commonwealth University, Richmond, VA, USA Wilfried Imrich Chair of Applied Mathematics, Montanuniversit¨ at Leoben A-8700 Leoben, Austria Received 27 July 2009, accepted 9 September 2009, published online 30 September 2009 Abstract Under suitable conditions of connectivity or non-bipartiteness, each of the three stan- dard graph products (the Cartesian product, the direct product and the strong product) satisfies the unique prime factorization property, and there are polynomial algorithms to determine the prime factors. This is most easily proved for the Cartesian product. For the other products, current proofs involve a notion of a Cartesian skeleton which transfers their multiplication properties to the Cartesian product. The present article introduces simplified definitions of Cartesian skeletons for the direct and strong products, and provides new, fast and transparent algorithms for their construc- tion. Since the complexity of the prime factorization of the direct and the strong product is determined by the complexity of the construction of the Cartesian skeleton, the new al- gorithms also improve the complexity of the prime factorizations of graphs with respect to the direct and the strong product. We indicate how these simplifications fit into the existing literature. Keywords: Graph product, Cartesian skeleton, prime factorization of graphs, graph algorithms. Math. Subj. Class.: 05C85, 05C99 1 Introduction We consider finite graphs G =(V (G),E(G)) which may have loops but not multiple edges. We let Γ 0 denote the class of all such graphs, while Γ ⊂ Γ 0 is the class of graphs without loops. An edge joining g to g ′ is denoted gg ′ . E-mail addresses: rhammack@vcu.edu (Richard H. Hammack), imrich@unileoben.ac.at (Wilfried Imrich) Copyright c  2009 DMFA Slovenije