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.

翻译：映射 $ ALpha : V( G)\ a (H) 映射 $ : V( G)\ a( G) 至 V( H) $ 。 从一个图形的顶端集, V( G) 美元到另一个图形的顶点, V( G) 美元 $( G) 美元 $( H) 。 如果以美元计算的任何两个顶点之间的最短路径距离, 以美元计算的任何两个顶点之间的最短路径距离等于以美元表示图像之间的距离。 这里, 我们考虑将一个加权图形( G) $( $) 嵌入未加权的 Hammum 地图, 称为 Hamming 嵌入 : $( Hamm), $( g) 满足了每个边缘是其端点之间的最短路径。 使用卡通的 $( g) $( $) 的卡通产品分解调, 称为它的假相形图, 我们显示, $( G) $ ( ) $ ( ) ) 每一个不加权 或 硬形图显示一个不精化的底图是否 显示一个不精化的硬化的硬化图, 。