The local metric dimension of the lexicographic product of graphs

Abstract

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 $\mathcal{H}$.