Metric dimension of lexicographic product of some known‎ ‎graphs

Abstract

‎For an ordered set $W=\{w_1,w_2,\ldots,w_k\}$ of vertices and a vertex $v$ in a connected graph $G$, the ordered $k$-vector $r(v|W):=(d(v,w_1),d(v,w_2),\ldots,d(v,w_k))$ 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 investigate the metric dimension of the lexicographic product  of graphs $G$ and $H$, $G[H]$, for some known graphs