On the density of (1,≤ℓ)-locating-dominating codes in the infinite square grid

Given a simple graph G , let B r ( v ) be the set of vertices at a distance of at most r from v . A vertex subset C is an ( r , ≤ ℓ ) -locating-dominating code of type A or ( r , ≤ ℓ ) -LDA code if the sets I r ( F ) = B r ( F ) ∩ C are distinct for all vertex subsets F ⊆ V ( G ) of size at most ℓ that share the same vertices in C . Similarly, C is referred to as an ( r , ≤ ℓ ) -locating-dominating code of type B or ( r , ≤ ℓ ) -LDB code if the identifying sets I r ( F ) are distinct for all vertex subsets F ⊆ V ( G ) ∖ C of size at most ℓ . In this article, we present optimal (with respect to density) ( 1 , ≤ ℓ ) -LDA and ( 1 , ≤ ℓ + 1 ) -LDB codes in the infinite square grid for all ℓ ≥ 2 . For ( 1 , ≤ 2 ) -LDB codes in infinite square grid, we show that the optimal code will have density between [ 2 5 , 5 11 ] .