A note on the partition dimension of Cartesian product graphs

Abstract

Let $G=(V,E)$ be a connected graph. The distance between two vertices $u,v\in V$, denoted by $d(u, v)$, is the length of a shortest $u-v$ path in $G$. The distance between a vertex $v\in V$ and a subset $P\subset V$ is defined as $min\{d(v, x): x \in P\}$, and it is denoted by $d(v, P)$. An ordered partition $\{P_1,P_2, ...,P_t\}$ of vertices of a graph $G$, is a \emph{resolving partition}of $G$, if all the distance vectors $(d(v,P_1),d(v,P_2),...,d(v,P_t))$ are different. The \emph{partition dimension} of $G$, denoted by $pd(G)$, is the minimum number of sets in any resolving partition of $G$. In this article we study the partition dimension of Cartesian product graphs. More precisely, we show that for all pairs of connected graphs $G, H$, $pd(G\times H)\le pd(G)+pd(H)$ and $pd(G\times H)\le pd(G)+dim(H).$ Consequently, we show that $pd(G\times H)\le dim(G)+dim(H)+1.$