Isometric Hamming embeddings of weighted graphs

A mapping α:V(G)→V(H) from the vertex set of one graph G to another H is an isometric embedding if shortest path distance between any two vertices in equals their images H. Here, we consider embeddings a weighted into unweighted Hamming graphs, called embeddings, when satisfies property that every edge its endpoints. Using Cartesian product decomposition canonical representation, show may be partitioned partition, whose parts provide for each factor representation G. This implies permits and only embeddable. result extends prior work on graphs showed complete graph. When has nontrivial determining whether can simplified checking embeddability or more smaller graphs.