Secure domination number of generalized thorn graphs

Document Type : Research Paper

Authors

1 Research Scholar, Dept. of Mathematics, St. Paul's College, Kalamassery, India.

2 Dept. of Mathematics, St. Paul's College, Kalamassery, India.

Abstract

A secure dominating set S ⊆ V is a dominating set of G satisfying the condition that for each u ∈ V \ S, there exists a vertex v ∈ N(u) ∩ S such that (S \ {v}) S {u} is a dominating set of G. The minimum cardinality of a secure dominating set of G is called the secure domination number of G, γs(G). In this paper, we obtain the secure domination number of generalized thorn paths, thorn graphs, and some special graph classes like thorn rod, thorn star and Kragujevac trees, where the generalized thorn paths are important in the study of chemical compounds.

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References

[1] R. Arasu and N. Parvathi, Secure domination parameters of Halin graph with perfect k-ary tree, J. Appl. Math. Inf., 41(4), (2023), 839–848.
[2] M. Azari, On the Gutman index of thorn graphs, Kragujevac J. Sci., 40, (2018), 33–48.
[3] M. Azari and A. Iranmanesh, Dendrimer graphs as thorn graphs and their topological edge properties, Natl. Acad. Sci. Lett., 39, (2016), 455–460.
[4] R. Balakrishnan and K. Ranganathan, A textbook of graph theory, Springer, 2012.
[5] D. Bonchev and D. Klein, On the wiener number of thorn trees, stars, rings, and rods, Croat. Chem. Acta, 75(2), (2002), 613–620.
[6] R. Burdett, M. Haythorpe, and A. Newcombe, Variants of the domination number for flower snarks, Ars Math.  Contemp., 24(3), (2024), P3.04, 26 pp.
[7] A. P. Burger, A. P. de Villiers, and J. H. van Vuuren, A linear algorithm for secure domination in trees, Discrete Appl. Math., 171, (2014), 15–27.
[8] A. P. Burger, M. A. Henning, and J. H. Van Vuuren, Vertex covers and secure domination in graphs, Quaest. Math., 31, (2008), 163–171.
[9] A. Cayley, On the mathematical theory of isomers, Phil. Magazine, 47, (1874), 444–447.
[10] E. J. Cockayne, Irredundance, Secure domination and maximum degree in trees, Discrete Math., 307, (2007), 12–17.
[11] E. J. Cockayne, P. J. P. Grobler, W. R. Gr ¨undlingh, J. Munganga, and J. H. Van Vuuren, Protection of a graph, Util. Math., 67, (2005), 19–32.
[12] Gisha Saraswathy and Manju K. Menon, Secure domination parameters in Sierpi´nski graphs, IAENG Int. J. Appl. Math., 53(2), (2023), 573–577.
[13] P. J. P. Grobler and C. M. Mynhardt, Secure domination critical graphs, Discrete Math., 309, (2009), 5820–5827.
[14] I. Gutman, Distance of thorny graphs, Publications De L’institut Mathematique, 63(77), (1998), 31–36.
[15] M. Haythorpe and A. Newcombe, The secure domination number of Cartesian products of small graphs with paths and cycles, Discrete Appl. Math., 309, (2022), 32–45.
[16] S. A. Hosseini, M. B. Ahmadi, and I. Gutman, Kragujevac Trees with minimal Atom-Bond Connectivity Index, MATCH Commun. Math. Comput. Chem., 71, (2014), 5–20.
[17] C. M. Mynhardt, H. C. Swart, and E. Ungerer, Excellent trees and secure domination, Util. Math., 67 (2005), 255—267.
[18] P. G. Nayana and I. R. Rajamani, On secure domination number of generalized Mycielskian of some graphs, J. Intell. Fuzzy Syst., 44(3), (2023), 4831–4841.
[19] G. Polya, Kombinatorische Anzahlbestimmungen fur Gruppen, Graphen und chemische Verbindungen, Acta Math., 68, (1937), 145–254.
[20] S. Shanmugavelan and C. Natarajan, On hop domination number of some generalized graph structures, Ural Math. J., 7, (2021), 121–135.
[21] D. Yun-Ping, W. Haichao, and Z. Yancai, The complexity of secure domination problem in graphs, Discuss. Math. Graph Theory, 38, (2018), 385–396.
Journal of Hyperstructures
Volume 14, Issue 1
June 2025
Pages 93-105

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