Identifying Codes with Small Radius in Some Infinite Regular Graphs

Let $G=(V,E)$ be a connected undirected graph and S a subset of vertices. If for all vertices v ∈ V, the sets Bᵣ(v) ∩ S are all nonempty and different, where Bᵣ(v) denotes the set of all points within distance r from v, then we call S an r-identifying code. We give constructive upper bounds on the best possible density of r-identifying codes in four infinite regular graphs, for small values of r.