Isometric Hamming embeddings of weighted graphs
share
A mapping α : V(G) → V(H) 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 pseudofactorization, we show that every Hamming embedding of G may be partitioned into Hamming embeddings for each irreducible pseudofactor graph of G, which we call its canonical partition. This implies that G permits a Hamming embedding if and only if each of its irreducible pseudofactors 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 irreducible pseudofactor is a complete graph. When a graph G has nontrivial pseudofactors, determining whether G has a Hamming embedding can be simplified to checking embeddability of two or more smaller graphs.
READ FULL TEXT