On Vertex-Robust Identifying Codes of Level Three

Assume that G = (V, E) is an undirected and connected graph, and consider C subset of V. For every v is an element of V, let I(r)(v) = {u is an element of C : d(u, v) = 2t+1 for any two different vertices u and v. Vertex-robust identifying codes of different levels are examined, in particular, of level 3. We give bounds (sometimes exact values) on the density or cardinality of the codes in binary hypercubes and in some infinite grids