Relations between Metric Dimension and Domination Number of Graphs

A set $W\subseteq V(G)$ is called a resolving set, if for each two distinct vertices $u,v\in V(G)$ there exists $w\in W$ such that $d(u,w)\neq d(v,w)$, where $d(x,y)$ is the distance between the vertices $x$ and $y$. The minimum cardinality of a resolving set for $G$ is called the metric dimension of $G$, and denoted by $β(G)$. In this paper, we prove that in a connected graph $G$ of order $n$, $β(G)\leq n-γ(G)$, where $γ(G)$ is the domination number of $G$, and the equality holds if and only if $G$ is a complete graph or a complete bipartite graph $K_{s,t}$, $s,t\geq 2$. Then, we obtain new bounds for $β(G)$ in terms of minimum and maximum degree of $G$.

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