ABSTRACT

We introduced the lexicographic product in Section 4.4 and examined its metric properties in Section 5.4. We now address deeper questions such as prime factorization and cancellation. We prove that all prime factorizations of a given graph G have the same number of factors and that there exists a canonical way of transforming any prime factorization into any other one. We also show the existence of a large class of graphs with unique prime factor decomposition with respect to the lexicographic product.