The Metric Dimension of Graph with Pendant Edges

Abstract

For an ordered set $W=\{w_1,w_2,\dots ,w_k\}$ of vertices and a vertex $v$ in a connected graph $G$, the representation of $v$ with respect to $W$ is the ordered $k$-tuple $r(v|W)=(d(v,w_1),d(v,w_2),\dots ,d(v,w_k))$ where $d(x,y)$ represents the distance between the vertices $x$ and $y$. The set $W$ is called a resolving set for $G$ if every two vertices of $G$ have distinct representations. A resolving set containing a minimum number of vertices is called a basis for $G$. The dimension of $G$, denoted by $\text{dim}(G)$, is the number of vertices in a basis of $G$. In this paper, we determine the dimensions of some corona graphs $G\odot K_1$, $G\odot {\overline{K}}_m$ for any graph $G$ and $m\ge 2$, and a graph with pendant edges more general than corona graphs $G\odot {\overline{K}}_m$.