l1-embeddability under the edge-gluing operation on graphs

A graph is an l1-graph if its vertices can be labeled by binary vectors in such a way that the Hamming distance between the binary addresses is, up to scale, the distance in the graph between the corresponding vertices. In this note, we study when the result of gluing two l1-graphs along an edge is an l1-graph. If at least one of the constituent graphs is bipartite, then the resulting graph is also an l1-graph. For two nonbipartite l1-graphs, this is not always the case.