The local metric dimension of the lexicographic product of graphs
  The metric dimension is quite a well-studied graph parameter. Recently, the adjacency dimension and the local metric dimension have been introduced and studied. In this paper, we give a general formula for the local metric dimension of the lexicographic product $G \circ \mathcal{H}$ of a connected graph $G$ of order $n$ and a family $\mathcal{H}$ composed by $n$ graphs. We show that the local metric dimension of $G \circ \mathcal{H}$ can be expressed in terms of the true twin equivalence classes of $G$ and the local adjacency dimension of the graphs in $\mathcal{H}$. 
G. A. Barrag\'an-Ram\'irez, A. Estrada-Moreno, Y. Ram\'irez-Cruz and   J. A. Rodr\'iguez-Vel\'azquez
math.CO