Optimal (r,≤3)-locating-dominating codes in the infinite king grid

Assume that G=(V,E) is an undirected graph with vertex set V and edge set E. The ball Br(v) denotes the vertices within graphical distance r from v. A subset C⊆V is called an (r,≤l)-locating-dominating code of type B if the sets Ir(F)=⋃v∈F(Br(v)∩C) are distinct for all subsets F⊆V∖C with at most l vertices. We give examples of optimal (r,≤3)-locating-dominating codes of type B in the infinite king grid for all r∈N+ and prove optimality. The infinite king grid is the graph with vertex set Z2 and edge set {{(x1,y1),(x2,y2)}∣|x1−x2|≤1,|y1−y2|≤1}.