Identifying codes with small radius in some infinite regular graphs

Abstract

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.