The cost number and the determining number of a graph

TL;DR

This paper explores the relationship between the cost of graph distinguishing labelings and the determining number, providing bounds and exact values for specific graph classes like friendship graphs and corona products.

Contribution

It introduces bounds for the cost of $d$-distinguishing in graphs based on the determining number and computes these parameters for specific graph families.

Findings

Established bounds for $ ho_d(G)$ using Det(G)

Computed cost and determining number for friendship graphs

Analyzed these parameters for corona product graphs

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 $G$, Det(G), is the minimum cardinality of determining sets of $G$. In this paper we obtain some general upper and lower bounds for $\rho_d(G)$ based on Det(G). Finally, we compute the cost and the determining number for the friendship graphs and corona product of two graphs.