Document Type : Original Manuscript

Authors

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

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

Abstract

The distinguishing number $D(G)$ of a graph $G$ is the least integer $d$ such that $G$ has an vertex labeling with $d$ labels that is preserved only by a trivial automorphism. The minimum size of a label class in such a labeling of $G$ with $D(G) = d$ is called the cost of $d$-distinguishing $G$ and is denoted by $\rho_d(G)$. A set of vertices $S\subseteq V(G)$ is a determining set for $G$ if every automorphism of $G$ is uniquely determined by its action on $S$. The determining number of $G$, ${\rm Det}(G)$, is the minimum cardinality of determining sets of $G$. In this paper we compute the cost and the determining number for the friendship graphs and corona product of two graphs.

Keywords 20.1001.1.23455128.2021.8.2.6.1
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