Minimum density of identifying codes of king grids

A set C⊆V(G) is an identifying code in a graph G if for all v∈V(G), C[v]≠∅, and for all distinct u,v∈V(G), C[u]≠C[v], where C[v]=N[v]∩C and N[v] denotes the closed neighbourhood of v in G. The minimum density of an identifying code in G is denoted by d⁎(G). In this paper, we study the density of king grids which are strong product of two paths. We show that for every king grid G,d⁎(G)≥2/9. In addition, we show this bound is attained only for king grids which are strong products of two infinite paths. Given k≥3, we denote by Kk the (infinite) king strip with k rows. We prove that d⁎(K3)=1/3, d⁎(K4)=5/16, d⁎(K5)=4/15 and d⁎(K6)=5/18. We also prove that 29+881k≤d⁎(Kk)≤29+49k for every k≥7.