Isometric Hamming embeddings of weighted graphs

TL;DR

This paper characterizes when weighted graphs can be isometrically embedded into unweighted Hamming graphs, introducing a decomposition approach that simplifies the embeddability problem by analyzing smaller irreducible components.

Contribution

It extends previous unweighted graph results to weighted graphs using pseudofactorization, providing a canonical partition approach for Hamming embeddings.

Findings

Hamming embedding of a graph depends on its irreducible pseudofactors.

Every Hamming embedding can be partitioned into embeddings of pseudofactors.

Determining embeddability reduces to checking smaller subgraphs.

Abstract

A mapping $\alpha : V(G) \to V(H)$ 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.