The Metric Dimension of Lexicographic Product of Graphs

TL;DR

This paper investigates the metric dimension of the lexicographic product of graphs, introducing a new parameter called adjacency metric dimension, and derives formulas relating it to the product's metric dimension.

Contribution

It introduces the adjacency metric dimension and provides a formula for the metric dimension of the lexicographic product of graphs in terms of this new parameter.

Findings

Derived the metric dimension of G[H] using adjacency metric dimension of H.

Introduced the adjacency metric dimension as a new graph parameter.

Connected the metric dimension of graph products to existing graph parameters.

Abstract

For an ordered set $W=\{w_1,w_2,...,w_k\}$ of vertices and a vertex $v$ in a connected graph $G$, the ordered $k$-vector $r(v|W):=(d(v,w_1),d(v,w_2),...,d(v,w_k))$ 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 study the metric dimension of the lexicographic product of graphs $G$ and $H$, $G[H]$. First, we introduce a new parameter which is called adjacency metric dimension of a graph. Then, we obtain the metric dimension of $G[H]$ in terms of the order of $G$ and the adjacency metric dimension of $H$.