Odd and even repetition sequences of independent domination number

Leomarich F. Casinillo
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 26, 2020, Number 1, Pages 8—20
DOI: 10.7546/nntdm.2020.26.1.8-20
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Leomarich F. Casinillo
Department of Mathematics and Physics, Visayas State University
Visca, Baybay City, Leyte, Philippines

Abstract

Let {Pn}n=1 be a sequence of paths. The odd repetition sequence denoted by {𝜌𝑘𝑜:𝑘∈ℕ} is a sequence of natural numbers in which odd numbers are repeated once and defined by {𝜌𝑘𝑜}={1,1,2,3,3,4,5,5,…}={𝑖(𝑃𝑛)} where 𝑛=2𝑘−1. The even repetition sequence denoted by {𝜌𝑘𝑒:𝑘∈ℕ} is a sequence of natural numbers, in which even numbers are repeated once and defined by {𝜌𝑘𝑒}={1,2,2,3,4,4,5,6,6,…}={𝑖(𝑃𝑛)}, where 𝑛 = 2𝑘. In this paper, the explicit formula that shows the values of the element of two sequences {𝜌𝑘𝑜}} and {𝜌𝑘𝑒} that depends on the subscript 𝑘 were constructed. Also, the formula that relates the partial sum of the elements of the said sequences, which depends on the subscript 𝑘 and order of the sequence of paths, were established. Further, the independent domination number of the triangular grid graph 𝑇𝑚 = (𝑉(𝑇𝑚), 𝐸(𝑇𝑚)) will be determined using the said sequences and the two sequences will be evaluated in relation to the Fibonacci sequence {𝐹𝑛} along with the order of the path.

Keywords

  • Odd repetition sequence
  • Even repetition sequence
  • Independent domination number
  • Fibonacci numbers
  • Triangular grid graph

2010 Mathematics Subject Classification

  • 05C69

References

  1. Canoy, Jr. S. R., & Garces, I. J. L. (2002). Convex sets under some graph operations. Graphs and Combinatorics, 18, 787–793.
  2. Casinillo, L. F. (2018). A note on Fibonacci and Lucas number of domination in path. Electronic Journal of Graph Theory and Applications, 6(2), 317–325.
  3. Casinillo, L. F., Lagumbay, E. T. & Abad, H. R. F. (2017). A note on connected interior domination in join and corona of two graphs. IOSR Journal of Mathematics, 13(2), 66–69.
  4. Chartrand, G. & Zhang, P. (2012). A First Course in Graph Theory. Dover Publication Inc., New York.
  5. Cockayne, E. J., & Hedetniemi, S. T. (1977). Towards a theory of domination in graph. Networks Advanced Topics, 7, 247–261.
  6. Dorfling, M., & Henning, M. A. (2006). A note on power domination in grid graphs. Discrete Applied Mathematics, 154, 1023–1027.
  7. Haynes, T. W., Hedetniemi, S. T., & Slater, P. J. (1998). Fundamentals of Domination in Graphs. Marcel Dekker. New York.
  8. Koshy, T. (2001). Fibonacci and Lucas Numbers with Application. Wiley-Interscience, New York.
  9. Ore, O. (1962). Theory of Graphs. American Mathematical Society Providence, R. I.
  10. Perderson, A. S., & Vestergaard, P. D. (2005). The number of independent sets in unicyclic graphs. Discrete Applied Mathematics, 152, 246–256.
  11. Prodinger, H., & Tichy, R. (1982). Fibonacci numbers of graphs. Fibonacci Quarterly, 20(1), 16–21.
  12. Singh, B. Sisodiya, K., & Ahmad, F. (2014). On the products of k-Fibonacci numbers and k-Lucas numbers. International Journal of Mathematics and Mathematical Science, 21, 1–4.
  13. Tarr, J. M. (2010). Domination in Graphs. Graduate Theses and Dissertations. Retrieved from https://scholarcommons.usf.edu/etd/1786.
  14. Vajda, S. (2008). Fibonacci and Lucas Numbers and the Golden Section: Theory and Applications. Dover Publications Inc., New York.

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Cite this paper

Casinillo, L. F. (2020). Odd and even repetition sequences of independent domination number. Notes on Number Theory and Discrete Mathematics, 26(1), 8-20, doi: 10.7546/nntdm.2020.26.1.8-20.

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