The distinguishing number (index) and the domination number of a graph

Abstract

The distinguishing number (index) $D(G)$ ($D'(G)$) of a graph $G$ is the least integer $d$ such that $G$ has an vertex labeling (edge labeling) with $d$ labels that is preserved only by a trivial automorphism. A set $S$ of vertices in $G$ is a dominating set of $G$ if every vertex of $V(G)\setminus S$ is adjacent to some vertex in $S$. The minimum cardinality of a dominating set of $G$ is the domination number of $G$ and denoted by $\gamma (G)$. In this paper, we obtain some upper bounds for the distinguishing number and the distinguishing index of a graph based on its domination number.