Metric dimension of generalized wheels

Abstract

In a graph $G$, a vertex $w\in V\left(G\right)$ resolves a pair of vertices $u,v\in V\left(G\right)$ if $d(u,w)\ne d(v,w)$. A resolving set of $G$ is a set of vertices $S$ such that every pair of distinct vertices in $V\left(G\right)$ is resolved by some vertex in $S$. The minimum cardinality among all the resolving sets of $G$ is called the metric dimension of $G$, denoted by $\beta \left(G\right)$. The metric dimension of a wheel has been obtained in an earlier paper (Shanmukha et al., 2002). In this paper, the metric dimension of the family of generalized wheels is obtained. Further, few properties of the metric dimension of the corona product of graphs have been discussed and some relations between the metric dimension of a graph and its generalized corona product are established.