Twain Secure Perfect Dominating Sets and Twain Secure Perfect Domination Polynomials of Cycles

Document Type : Research Paper

Authors

1 Department of Mathematics, Scott Christian College (Autonomous), Nagercoil-629 003, Manonmaniam Sunderanar University, India.

2 Department of Mathematics,Scott Christian College (Autonomous), Nagercoil-629003, Manonmaniam Sunderanar University, India.

Abstract

Let G = (V,E) be a simple graph. A set S ⊆ V is a dominating set of G, if for every vertex in V\S is adjacent to at least one vertex in S. A subset S of V is called a twain secure perfect dominating set of G (TSPD-set) if for every vertex v ∈ V \S is adjacent to exactly one vertex u ∈ S and (S\{u})∪{v} is a dominating set of G. The minimum cardinality of a twain secure perfect dominating set of G is called the twain secure perfect domination number of G and is denoted by γtsp(G).
Let Dtsp(Cn, i) denote the family of all twain secure perfect dominating sets of Cn with cardinality i, for γtsp(Cn)≤ i≤ n. Let dtsp(Cn, i) = |Dtsp(Cn, i)|. In this article, we derive a recursive formula for dtsp(Cn, i) and construct Dtsp(Cn, i). We
consider the polynomial Dtsp(Cn, x) = Σn i=γtsp(Cn) dtsp(Cn, i)xi, which we refer to as the twain secure perfect domination polynomial of cycles using this recursive formula. In this research, we use a recursive technique to generate all twain secure perfect dominating sets of cycles and twain secure perfect domination polynomials of cycles.

Keywords

Main Subjects

References

[1] S. Alikhani and Y. H. Peng, Dominating sets and domination polynomials of paths, International Journal of Mathematics and Mathematical Sciences, (2009), Article ID. 542040, 1-10.
[2] F. Buckley and F. Harary Distance in Graphs, Addison-Wesley Publishing Company, Redwood City, California, 1990.
[3] E. J. Cockayne, P. J. Grobler, P. Grundlingh, J. Munganga and J. H. V. Vuuren , Protection of a graph, Utilitas Mathematica, 67(2005), 19-32.
[4] K. Lal Gipson and S. Arun Williams, 2-Edge dominating sets and 2-edge domination polynomials of paths, Malaya Journal of Matematik, 9(1)(2021), 844-849.
[5] K. Lal Gipson and T. Subha, Secure domination polynomials of paths, International Journal of scientific & Technology Research, 9(2)(2020), 6517-6521.
[6] K. Lal Gipson and T. Subha, Secure dominating sets of wheels, IOSR Journal of Mathematics, 18(1)(2022), 21-26.
[7] A. Vijayan and K. Lal Gipson, Dominating sets and domination polynomials of square of paths, Open Journal of Discrete Mathematics,3(1)(2013), 60-69.
[8] A. Vijayan and K. Lal Gipson, Dominating sets and domination polynomials of square of cycles, IOSR JOurnal of Mathematics, 3(4)(2012), 04-14.
[9] C. Vinisha and K. Lal Gipson, Twain Secure Perfect Dominating Sets and Twain Secure Perfect Domination Polynomials of Stars, Journal of Computational Analysis and Applications, 33(6)(2024), 416-419.
[10] P. M. Weichsel, Dominating Sets in n-cubes, Journal of Graph Theory, 18(5)(1994), 479-488.
Journal of Hyperstructures
Volume 14, Issue 2
December 2025
Pages 268-275

Files

Share

How to cite

Statistics