An r-identifying code in a graph G = (V,E) is a subset C ⊆ V such that for each u ∈ V the intersection of C and the ball of radius r centered at u is nonempty and unique. Previously, r-identifying codes have been studied in various grids. In particular, it has been shown that there exists a 2-identifying code in the hexagonal grid with density 4/19 and that there are no 2-identifying codes with...

An identifying code of a graph is a dominating set which uniquely determines all the vertices by their neighborhood within the code. Whereas graphs with large minimum degree have small domination number, this is not the case for the identifying code number (the size of a smallest identifying code), which indeed is not even a monotone parameter with respect to graph inclusion. We show that every...

In an undirected graph G = (V; E) a subset C V is called an identifying code, if the sets B1 (v) \ C consisting of all elements of C within distance one from the vertex v are nonempty and diierent. We take G to be the innnite hexagonal grid, and show that the density of any identifying code is at least 16=39 and that there is an identifying code of density 3=7.

LetG = (V,E) be a simple undirected graph. We call any subset C ⊆ V an identifying code if the sets I(v) = {c ∈ C | {v, c} ∈ E or v = c} are distinct and non-empty for all vertices v ∈ V . A graph is called twin-free if there is an identifying code in the graph. The identifying code with minimum size in a twin-free graph G is called the optimal identifying code and the size of such a code is de...