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 H of a connected graph G of order n and a family H composed by n graphs. We show that the local metric dimension of G H can be expressed in terms of the true twin equivalence classes of G and the local adjacency dimension of the graphs in H.