A Connection between Metric Dimension and Distinguishing Number of Graphs

Abstract

In this paper, we introduce a connection between two classical concepts of graph theory: \; metric dimension and distinguishing number. For a given graph $G$, let ${\rm dim}(G)$ and $D(G)$ represent its metric dimension and distinguishing number, respectively. We show that in connected graphs, any resolving set breaks the symmetry in the graphs. Precisely, if $G$ is a connected graph with a resolving set $S=\{v_1, v_2, \ldots, v_n \}$, then $\{\{v_1\}, \{v_2\}, \ldots, \{v_n\}, V(G)\setminus S \}$ is a partition of $V(G)$ into a distinguishing coloring, and as a consequence $D(G)\leq {\rm dim}(G)+1$. Furthermore, we construct graphs $G$ such that $D(G)=n$ and ${\rm dim}(G)=m$ for all values of $n$ and $m$, where $1\leq n< m$. Using this connection, we have characterized all graphs $G$ of order $n$ with $D(G) \in \{n-1, n-2\}$. For any graph $G$, let $G_c = G$ if $G$ is connected, and $G_c = \overline{G}$ if $G$ is disconnected. Let $G^{\ast}$ denote the twin graph obtained from $G$ by contracting any maximal set of vertices with the same open or close neighborhood into a vertex. Let {\rsfs F} be the set of all graphs except graphs $G$ with the property that ${\rm dim}(G_c)=|V(G)|-4$, ${\rm diam}(G_c) \in \{2, 3\}$ and $5\leq |V(G_{c}^{\ast})| \leq 9$. We characterize all graphs $G \in$ {\rsfs F} of order $n$ with the property that $D(G)= n-3$.