Isometric Hamming embeddings of weighted graphs

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