The Metric Dimension of The Tensor Product of Cliques

TL;DR

This paper investigates the metric dimension of the tensor product of cliques, establishing bounds and precisely determining it for the case of two cliques, contributing to understanding graph resolving sets.

Contribution

It provides new bounds and exact values for the metric dimension of tensor products of cliques, a previously less-explored graph class.

Findings

Bounds for the metric dimension of tensor products of cliques.

Exact metric dimension for the tensor product of two cliques.

Abstract

Let $G$ be a connected graph and $W=\{ w_1, w_2, \ldots, w_k \} \subseteq V(G)$ be an ordered set. For every vertex $v$, the metric representation of $v$ with respect to $W$ is an ordered $k$-vector defined as $r(v|W):=(d(v,w_1), d(v,w_2), \ldots, d(v,w_k))$, 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 and is denoted by $dim(G)$. In this paper, we study the metric dimension of tensor product of cliques and prove some bounds. Then we determine the metric dimension of tensor product of two cliques.