New Optimal Results on Codes for Location in Graphs

In this paper, we broaden the understanding of the recently introduced concepts of solid-locating-dominating and self-locating-dominating codes in various graphs. In particular, we present the optimal, i.e., smallest possible, codes in the infinite triangular and king grids. Furthermore, we give optimal locating-dominating, self-locating-dominating and solid-locating-dominating codes in the direct product $Kn\times Km$ of complete graphs. We also present optimal solid-locating-dominating codes for the Hamming graphs $Kq\square Kq\square K_q$ with $q\geq2$.