TARU PUBLICATIONS
Journal of Discrete Mathematical Sciences and Cryptography cover
Open Access ·Peer-reviewed·ISSN (Online): 2169-0065·ISSN (Print): 0972-0529

Monthly Journal: Publishes theoretical and applied research in all areas of Discrete Mathematical Sciences, Cryptography, Combinatorics, Elliptic Curves and Information Security.

Issues up to 2022 co-published with and available at:Taylor & Francis Online
submissions@tarupublications.com
Open Access Research Article

Secure 2-domination in graphs

, * ,

* Corresponding author · click or hover a name for details

pp. 2723–2736Vol. 28Issue 7October 2025DOI: 10.47974/JDMSC-2067 Crossmark XML
Received:
10 Oct 2023
Published Online:
12 Jul 2025
Article type:
Research Article
Language:
EN
Article no.:
JDMSC-2067
Pages:
2723–2736

Abstract

Let G = (V, E) be a simple graph. A set D ⊆ V is a dominating set of G if every vertex in V\D is adjacent to at least one vertex in D. D is a 2-dominating set if every vertex in V\D has at least two neighbors in D. A secure dominating set S of a graph G is a dominating set with the property that every vertex u in V\S is adjacent to a vertex v∈S such that (S\{v}) ∪‍ {u} is a dominating set. In this paper, we define and study a new invariant of domination in graphs which is the secure 2-dominating set. A 2-dominating set S is secure 2-dominating if for every vertex u in V\S, ∃v∈(S ∩‍ N(u)) such that (S\{v}) ∪‍ {u} is a 2-dominating set. The secure 2-domination number γ2s(G) is equal to the minimum cardinality of a secure 2-dominating set. First, we establish lower and upper bounds on γ2s (G) of a graph and a triangle-free graph. Then we determine the secure 2-domination number γ2s (G) of paths and cycles. After, we study the complexity of secure 2-domination problem by proving that the determining of the parameters value is NP-complete, also for split and bipartite graphs. Finally, we provide an upper bound of parameter in terms of clique number and independent number for perfect graphs. 

Keywords

Subject Classifications

05C6968Q25

References

[1] S. Alikhani and M. Safazadeh, “Fair dominating sets of paths,” Journal of Information and Optimization Sciences, vol. 44, no. 5, pp. 855–864 (2023).
[2] C. Berge, Theory of Graphs and Its Applications, Dunod, Paris (1958).
[3] M. Blidia, M. Chellali, and O. Favaron, “Independence and 2-domination in trees,” Australasian Journal of Combinatorics, vol. 33, pp. 317–327 (2005).
[4] M. Blidia, M. Chellali, and L. Volkmann, “Bounds on the 2-domination number of graphs,” Utilitas Mathematica, vol. 71, pp. 209–216 (2006).
[5] I. Boufelgha and M. Ahmia, “Global dominating broadcast in graphs,” Discrete Mathematics, Algorithms and Applications, vol. 15, no. 2 (2023).
[6] E. J. Cockayne, R. M. Dawes, and S. T. Hedetniemi, “Total domination in graphs,” Networks, vol. 10, pp. 211–219 (1980).
[7] E. J. Cockayne, O. Favaron, and C. M. Mynhardt, “Secure domination, weak Roman domination and forbidden subgraphs,” Bulletin of the Institute of Combinatorics and its Applications, vol. 39, pp. 87–100 (2003).
[8] E. J. Cockayne, P. J. P. Grobler, W. R. Grundlingh, J. Munganga, and J. H. van Vuuren, “Protection of a graph,” Utilitas Mathematica, vol. 67, pp. 19–32 (2005).
[9] E. J. Cockayne, “Irredundance, secure domination and maximum degree in trees,” Discrete Mathematics, vol. 307, pp. 12–17 (2007).
[10] L. Devendra, P. V. Subba Reddy, and J. P. Kumar, “Complexity issues of variants of secure domination in graphs,” Electronic Notes in Discrete Mathematics, vol. 63, pp. 77–84 (2017).
[11] J. F. Fink and M. S. Jacobson, “n-Domination in graphs,” Graph Theory with Applications to Algorithms and Computer Science, John Wiley and Sons, New York, pp. 283–300 (1985).
[12] P. J. P. Grobler and C. M. Mynhardt, “Secure domination critical graphs,” Discrete Mathematics, vol. 309, pp. 5820–5827 (2009).
[13] M. Guettiche and H. Kheddouci, “Critical links detection in stochastic networks: application to the transport networks,” International Journal of Intelligent Computing and Cybernetics (2018).
[14] C. L. Liu, Introduction to Combinatorial Mathematics, McGraw-Hill, New York (1968).
[15] P. J. Looges and S. Olariu, “Optimal greedy algorithms for indifference graphs,” Computers and Mathematics with Applications, vol. 25, no. 7, pp. 15–25 (1993).
[16] H. B. Merouane and M. Chellali, “On secure domination in graphs,” Information Processing Letters, vol. 115, pp. 786–790 (2015).
[17] C. M. Mynhardt, H. C. Swart, and L. Ungerer, “Excellent trees and secure domination,” Utilitas Mathematica, vol. 67, pp. 255–267 (2005).
[18] A. P. De Villiers, Edge Criticality in Secure Graph Domination, Ph.D. dissertation, Stellenbosch University, Stellenbosch, South Africa (2014).

Views: 237Downloads: 129Citations: 0