The local metric dimension of the lexicographic product of graphs

TL;DR

This paper derives a general formula for the local metric dimension of the lexicographic product of graphs, linking it to twin classes of the base graph and the local adjacency dimensions of component graphs.

Contribution

It introduces a formula connecting the local metric dimension of the lexicographic product to twin classes and local adjacency dimensions, expanding understanding of graph product parameters.

Findings

Formula for local metric dimension of lexicographic product

Relation to true twin equivalence classes

Expression in terms of local adjacency dimensions

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 $\mathcal{H}$.