Some Properties of Dominant Local Metric Dimension

Abstract

Let $G$ be a connected graph with vertex set $V$. Let $W_l$ be an ordered subset defined by $W_l=\{w_1,w_2,\dots,w_n\}\subseteq V(G)$. Then $W_l$ is said to be a dominant local resolving set of $G$ if $W_l$ is a local resolving set as well as a dominating set of $G$. A dominant local resolving set of $G$ with minimum cardinality is called the dominant local basis of $G$. The cardinality of the dominant local basis of $G$ is called the dominant local metric dimension of $G$ and is denoted by $Ddim_l(G)$. We characterize the dominant local metric dimension for any graph $G$ and for some commonly known graphs in terms of their domination number to get some properties of dominant local metric dimension.