# Distinguishing number (index) and domination number of a graph

Document Type : Original Article

Authors

Department of Mathematics, Yazd University, 89195-741, Yazd, Iran

10.22034/kjm.2020.109817

Abstract

The distinguishing number (index)  of a graph $G$ is the least integer $d$
such that $G$ has a vertex labeling (edge labeling)  with $d$ labels  that is preserved only by the 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 $G$. In this paper, we obtain some upper bounds for the distinguishing number and the distinguishing index of a graph based on its domination number.

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