Optimal $(1,\le \mathrm{\ell })$-locating-dominating codes in infinite triangular grid

We study two variants of location-domination in infinite triangular grids which allow locating up to $\mathrm{\ell }$ vertices simultaneously. A locating-dominating code $S$ is a dominating vertex set such that every vertex outside of it has a unique neighborhood within the code $S$. As this neighborhood is unique, we may use it as an identifier for locating the considered vertex outside of the vertex set. With the variants we study, we may also locate sets of vertices instead of only single vertices. In particular, we obtain the exact minimum densities of these types of sets in the infinite triangular grid for every value of $\mathrm{\ell }$.