Isometric Embeddings of Graphs into Cartesian Products

Abstract

We have characterized in the previous chapter the graphs that can be isometrically embedded into a hypercube. The hypercube is the simplest example of a Cartesian product of graphs; indeed, the m-hypercube is nothing but (K ₂)^m. We consider here isometric embeddings of graphs into arbitrary Cartesian products. It turns out that every graph can be isometrically embedded in a canonical way into a Cartesian product whose factors are “irreducible”, i.e., cannot be further embedded into Cartesian products. We present two applications of this result, for finding the prime factorization of a graph, and for showing that the path metric of every bipartite graph can decomposed in a unique way as a nonnegative combination of primitive semimetrics.