Polyhexes that are graphs

A connected graph is said to be ℓ 1 if its path distance isometrically embeds into the space ℓ 1 . Following the work of Deza, Grishukhin, Shtogrin, and others on polyhedral ℓ 1 graphs, we determine all finite closed polyhexes (trivalent surface graphs with hexagonal faces) that are ℓ 1 .