Metric dimension of lexicographic product of some known graphs

TL;DR

This paper studies the metric dimension, which measures how well a set of vertices can uniquely identify all other vertices by distances, specifically focusing on the lexicographic product of certain known graphs.

Contribution

It provides new insights into the metric dimension of the lexicographic product of graphs, extending known results to this specific graph operation.

Findings

Determined the metric dimension for specific classes of graph products.

Extended existing theories to the lexicographic product of graphs.

Provided formulas or bounds for the metric dimension in new graph classes.

Abstract

For an ordered set W = {w1,w2,...,wk} of vertices and a vertex v in a connected graph G, the ordered k-vector r(v|W) := (d(v,w1),d(v,w2),...,d(v,wk)) is called the (metric) representation of v with respect to W, where d(x,y) is the distance between the vertices x and y. The set W is called a resolving set for G if distinct vertices of G have distinct representations with respect to W. The minimum cardinality of a resolving set for G is its metric dimension. In this paper, we investigate the metric dimension of the lexicographic product of graphs G and H, G[H] for some known graphs.