Factoring cartesian-product graphs

In a fundamental paper, G. Sabidussi “Graph Multiplication,” Mathematische Zeitschrift, Vol. 72 (1960), pp. 446–457] used a tower of equivalence relations on the edge set E(G) of a connected graph G to decompose G into a Cartesian product of prime graphs. Later, a method by R.L. Graham and P.M. Winkler “On Isometric Embeddings of Graphs,” Transactions of the American Mathematics Society, Vol. 288 (1985), pp. 527–533] of embedding a connected graph isometrically into Cartesian products opened another approach to this problem. In both approaches an equivalence relation σ that determines the prime factorization is constructed. The methods differ by the starting relations used. We show that σ can be obtained as the convex hull of the starting relation used by Sabidussi. Our result also holds for the relation determining the prime decomposition of infinite connected graphs with respect to the weak Cartesian product. Moreover, we show that this relation is the transitive closure of the union of the starting relations of Sabidussi and Winkler “Factoring a Graph in Polynomial Time,” European Journal of Combinatorics, Vol. 8 (1987), pp. 209–212], thereby generalizing the result of T. Feder “Product Graph Representations,” Journal of Graph Theory, Vol 16 (1993), pp. 467–488] from finite to infinite graphs.