Optimal lower bound for 2-identifying codes in the hexagonal grid

Abstract

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 non- empty 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 density smaller than 2/11. Recently, the lower bound has been improved to 1/5 by Martin and Stanton (2010). In this paper, we prove that the 2-identifying code with density 4/19 is optimal, i.e. that there does not exist a 2-identifying code in the hexagonal grid with smaller density.