Abstract

Let be a simple graph of order . The certified domination polynomial of is the polynomial, , where is the minimum cardinality of a certified dominating set of and   is the number of certified dominating sets of  of size . In this article, we study the certified dominating sets and the certified domination polynomial of the book graph and obtain some properties of this polynomial.

Keywords

Certified dominating set, Certified domination number, Certified domination polynomial.

References

[1] T. Anitha Baby, and Y.A. Shiny, “Connected 2-Domination Polynomials of Some Graph Operations,” Malaya Journal of Matematik. vol. 8, no. 4, pp. 2329- 2332, 2020.
[CrossRef] [Google Scholar] [Publisher Link]

[2] A.M. Anto, and P. Paul Hawkins, “Perfect Dominating Sets and Perfect Domination Polynomial of Some Standard Graphs,” Turkish Journal of Computer and Mathematics Education, vol. 11, no. 3, pp. 1178-1182, 2020.
[CrossRef] [Google Scholar] [Publisher Link]

[3] Saeid Alikhani, and Somayeh Jahari, “Domination Polynomial of Generalized Book Graphs,” Journal of Mathematical Extension, vol. 10, no.3, pp. 35-45, 2016.
[Google Scholar] [Publisher Link]

[4] Jyoti Rani, and Seema Mehra, “Total Domination Polynomial of Some Graphs,” Advances and Applications in Discrete Mathamatics, vol. 25, no. 2, pp.161-171, 2020.
[CrossRef] [Google Scholar] [Publisher Link]

[5] Magda Dettlaf et al., “Certified Domination,” AKCE International Journal of Graphs and Combinatorics, vol. 17, no. 1, pp 86-97, 2020.
[CrossRef] [Google Scholar] [Publisher Link]

[6] Rafia Yoosuf, and Preethi Kuttipulackal, “The Connected Domination Polynomial of Some Graph Constructions,” Journal of Physics: Conference Series, vol. 1850, pp. 1-9, 2021.
[CrossRef] [Google Scholar] [Publisher Link]

[7] Nechrivan B. Ibrahim, and Assad A. Jund, “Edge Connected Domination Polynomial of a Graph,” Palestine Journal of Mathamatics, vol. 7, no. 2, pp. 458-467, 2018.
[Google Scholar] [Publisher Link]

[8] P.C. Priyanka Nair, and T. Anitha Baby, “2-Dominating Sets and 2-Domination Polynomials of Cycles,” Turkish Online Journal of Qualitative Inquiry, vol.12, no. 10, pp. 4325-4332, 2021.
[CrossRef] [Google Scholar] [Publisher Link]

[9] V.S. Radhika, and A. Vijayan, “Connected Vertex-Edge Domination Polynomials of Some Graphs,” Advances and Applications in Discrete Mathematics, vol. 27, no. 2, pp. 183-192, 2021.
[CrossRef] [Google Scholar] [Publisher Link]

[10] Saeid Alikhani, and Yee-Hock Peng, “Introduction to Domination Polynomial of a Graph,” arXiv:0905.2251, pp. 1-10, 2009.
[CrossRef] [Google Scholar] [Publisher Link]

[11] Y.A. Shiny, and T. Anitha Baby, “Connected 2-Dominating Sets and Connected 2-Domination Polynomials of the Complete Bipartite Graph ,” Ratio Mathematica, vol. 44, pp. 51-55, 2022.
[CrossRef] [Google Scholar] [Publisher Link]

[12] A. Vijayan, and M. Felix Nes Mabel, “Connected Domination Polynomial of Some Graphs,” IOSR Journal of Mathematics, vol. 12, no. 4, pp. 13-16, 2016.

       [Crossref] [Google Scholar] [Publisher Link]