Abstract
The metric dimension is quite a well-studied graph parameter. Recently, the local metric dimension and the adjacency 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 of n graphs. We show that the local metric dimension of $$G \circ \mathcal {H}$$ can be expressed in terms of the numbers of vertices in the true twin equivalence classes of G, and the local adjacency dimension of the graphs in $$\mathcal {H}$$ .
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