Abstract
A vertex coloring of a graph G is distinguishing if non-identity automorphisms do not preserve it. The distinguishing number, D(G), is the minimum number of colors required for such a coloring and the distinguishing threshold, \(\theta (G)\), is the minimum number of colors k such that any arbitrary k-coloring is distinguishing. Moreover, \(\Phi _k (G)\) is the number of distinguishing coloring of G using at most k colors. In this paper, for some graph operations, namely, vertex-sum, rooted product, corona product and lexicographic product, we find formulae of the distinguishing number and threshold using \(\Phi _k (G)\).
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Acknowledgements
The authors would like to express their gratitude to the referees for their careful reading and helpful comments. The research of the first author was in part supported by a grant from Yazd University research council as Post-doc research project.
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Shekarriz, M.H., Talebpour, S.A., Ahmadi, B. et al. Distinguishing Threshold for Some Graph Operations. Iran J Sci 47, 199–209 (2023). https://doi.org/10.1007/s40995-022-01379-2
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DOI: https://doi.org/10.1007/s40995-022-01379-2