On (r,≤ 2)-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.Let Iᵣ(F)=v∈ F(Bᵣ(v) ∩ C) be a set of codewords in the neighbourhoods of vertices v∈ F.A subset C⊆ V is called an (r,≤ l)-locating-dominating code of type A if sets Iᵣ(F₁) and Iᵣ(F₂)are distinct for all subsets F₁,F₂⊆ V where F₁≠ F₂, F₁∩ C= F₂ ∩ C and |F₁|,|F₂| ≤ l.A subset C⊆ V is an (r,≤ l)-locating-dominating code of type B if the sets Iᵣ(F) are distinct for all subsetsF⊆ V C with at most l vertices. We study (r,≤ l)-locating-dominating codes in the infinite king gridwhen r≥ 1 and $l=2$. The infinite king grid is the graph with vertex set Z² and edge set\\(x₁,y₁),(x₂,y₂)\||x₁-x₂|≤ 1, |y₁-y₂|≤ 1, (x₁,y₁)≠(x₂,y₂)\.