A family of optimal identifying codes in [formula omitted

Abstract

Assume that G = ( V , E ) is a simple undirected graph, and C is a nonempty subset of V. For every v ∈ V , we denote I r ( v ) = { u ∈ C | d G ( u , v ) ⩽ r } , where d G ( u , v ) denotes the number of edges on any shortest path between u and v. If the sets I r ( v ) for v ∈ V are pairwise different, and none of them is the empty set, we say that C is an r-identifying code in G. If C is r-identifying in every graph G ′ that can be obtained by adding and deleting edges in such a way that the number of additions and deletions together is at most t, the code C is called t-edge-robust. Let K be the graph with vertex set Z 2 in which two different vertices are adjacent if their Euclidean distance is at most 2 . We show that the smallest possible density of a 3-edge-robust code in K is r + 1 2 r + 1 for all r > 2 .