On Partition Dimension of Some Cycle-Related Graphs

Abstract

Let $G$ be a simple connected graph. Suppose $\mathrm{\Delta }=\left\{{\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2,\dots ,{\mathrm{\Delta }}_l\right\}$ an $l$ -partition of $V\left(G\right)$ . A partition representation of a vertex $\alpha \text{\hspace{0.17em}w}.\text{r}.\text{t\hspace{0.17em}}\mathrm{\Delta }$ is the $l-$ vector $\left(d\left(\alpha ,{\mathrm{\Delta }}_1\right),d\left(\alpha ,{\mathrm{\Delta }}_2\right),\dots ,d\left(\alpha ,{\mathrm{\Delta }}_l\right)\right)$ , denoted by $r\left(\alpha |\mathrm{\Delta }\right)$ . Any partition $\mathrm{\Delta }$ is referred as resolving partition if $\forall {\alpha }_i\ne {\alpha }_j\in V\left(G\right)$ such that $r\left({\alpha }_i|\mathrm{\Delta }\right)\ne r\left({\alpha }_j|\mathrm{\Delta }\right)$ . The smallest integer $l$ is referred as the partition dimension $\text{pd}\left(G\right)$ of $G$ if the $l$ -partition $\mathrm{\Delta }$ is a resolving partition. In this article, we discuss the partition dimension of kayak paddle graph, cycle graph with chord, and a graph generated by chain of cycles. It has been shown that the partition dimension of the said families of graphs is constant.