Density of identifying codes of hexagonal grids with finite number of rows

Abstract

In a graph G, a set C ⊆ V (G) is an identifying code if, for all vertices v in G, the sets N[v] ∩ C are all nonempty and pairwise distinct, where N[v] denotes the closed neighbourhood of v. We focus on the minimum density of identifying codes of infinite hexagonal grids Hk with k rows, denoted by d*(Hk), and present optimal solutions for k ≤ 5. Using the discharging method, we also prove a lower bound in terms of maximum degree for the minimum-density identifying codes of well-behaved infinite graphs. We prove that d*(H2) = 9/20, d*(H3) = 6/13 ≈ 0.4615, d*(H4) = 7/16 = 0.4375 and d*(H5) = 11/25 = 0.44. We also prove that H2 has a unique periodic identifying code with minimum density.