Isometric Hamming embeddings of weighted graphs

Abstract

A mapping  from the vertex set of one graph  to another graph  is an isometric embedding if the shortest path distance between any two vertices in  equals the distance between their images in . Here, we consider isometric embeddings of a weighted graph  into unweighted Hamming graphs, called Hamming embeddings, when  satisfies the property that every edge is a shortest path between its endpoints. Using a Cartesian product decomposition of  called its canonical isometric representation, we show that every Hamming embedding of  may be partitioned into a canonical partition, whose parts provide Hamming embeddings for each factor of the canonical isometric representation of . This implies that  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  has nontrivial isometric representation, determining whether  has a Hamming embedding can be simplified to checking embeddability of two or more smaller graphs.