Polyhexes that are ℓ1ℓ1 graphs

A connected graph is said to be ℓ1ℓ1 if its path distance isometrically embeds into the space ℓ1ℓ1. Following the work of Deza, Grishukhin, Shtogrin, and others on polyhedral ℓ1ℓ1 graphs, we determine all finite closed polyhexes (trivalent surface graphs with hexagonal faces) that are ℓ1ℓ1.