The Metric Dimension of The Tensor Product of Cliques

Abstract

Let $G$ be a connected graph and $W=\{w_1,w_2,\dots ,w_k\}\subseteq V(G)$ be an ordered set. For every vertex $v$, the metric representation of $v$ with respect to $W$ is an ordered $k$-vector defined as $r(v|W):=(d(v,w_1),d(v,w_2),\dots ,d(v,w_k))$, 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.

arXiv comment9 pages, no figure

References in corpus (2)

• On the Metric Dimension of Cartesian Products of Graphs
• The Metric Dimension of Regular Bipartite Graphs