A Note on Triple Repetition Sequence of Domination Number in Graphs

Leomarich F. Casinillo, Emily L. Casinillo, Lanndon A. Ocampo


A set D subset of V(G) is a dominating set of a graph G if for all x ϵ V(G)\D, for some y ϵ D such that xy ϵ E(G). A dominating set D subset of V(G) is called a connected dominating set of a graph G if the subgraph <D> induced by D is connected. A connected domination number of G, denoted by γ_c(G), is the minimum cardinality of a connected dominating set D. The triple repetition sequence denoted by {S_n:n ϵ Z+} is a sequence of positive integers which is repeated thrice, i.e., {S_n}={1,1,1,2,2,2,3,3,3, ...}. In this paper, we construct a combinatorial explicit formula for the triple repetition sequence of connected domination numbers of a triangular grid graph.

Keywords: connected domination number; triangular grid graph; triple repetition sequence.



Suatu himpunan D subhimpunan dari V(G) adalah himpunan pendominasi graf G apabila untuk semua x ϵ V(G)\D, untuk suatu y ϵ D sehingga xy ϵ E(G). Suatu himpunan pendominasi D subhimpunan dari V(G) dikatakan himpunan pendominasi terhubung dari graf G apabila subgraf <D> yang diinduksi oleh D terhubung. Suatu bilangan pendominasi dari G, dinotasikan dengan γ_c(G), adalah kardinalitas minimum dari himpunan pendominasi terhubung D. Barisan pengulangan rangkap tiga yang dinotasikan dengan {S_n:n ϵ Z+} adalah suatu barisan bilangan bulat positif yang setiap sukunya berulang tiga kali, yakni, {S_n}={1,1,1,2,2,2,3,3,3, ...}. Dalam paper ini dikonstruksi suatu rumus eksplisit kombinatorial untuk barisan pengulangan rangkap tiga dari bilangan pendominasi terhubung graf grid triangular.

Kata Kunci: bilangan pendominasi terhubung; graf grid triangular; barisan pengulangan rangkap tiga.


2020MSC: 05C69


connected domination number; triangular grid graph; triple repetition sequence.


J. L. Gross and J. Yellen, Handbook of graph theory, CRC press, 2003.

R. J. Wilson, History of graph theory, Handbook of Graph Theory, 29, 2003.

G. Chartrand and P. Zhang, A First Course in Graph Theory, New York: Dover Publication Inc., 2012.

E. J. Cockayne and S. T. Hedetniemi, "Towards a theory of domination in graph," Networks Advanced Topics, vol. 7, pp. 247-261, 1977.

L. F. Casinillo, E. T. Lagumbay and H. R. F. Abad, "A note on connected interior domination in join and corona of two graphs," IOSR Journal of Mathematics, vol. 13, no. 2, pp. 66-69, 2017.

E. L. Casinillo , L. F. Casinillo and J. Valenzona, "On Triangular Secure Domination Number," InPrime: Indonesian Journal of Pure and Applied Mathematics, vol. 2, no. 2, pp. 105-110, 2020.

M. T. K. Varkey and M. S. Thomas, "Some tree and tree related cubic graceful graphs," International Journal of Advanced Scientific and Technical Research, vol. 7, no. 4, pp. 239-244, 2017.

A. Gagarin, A. Poghosyan and V. Zverovich, "Randomized algorithms and upper bounds for multiple domination in graphs and networks," Discrete Applied Mathematics, vol. 161, no. 4-5, pp. 604-611, 2013.

E. C. Castillano, R. L. Ugbinada and S. R. Canoy Jr., "Secure domination in the joins of graphs," Applied Mathematical Sciences, vol. 8, no. 105, pp. 5203-5211, 2014.

J. Quadras, A. S. M. Mahizl , I. Rajasingh and R. S. Rajan, "Domination in certain chemical graphs," Journal of Mathematical Chemistry, vol. 53, no. 1, pp. 207-219, 2015.

E. A. Gnanasoundari, M. Raji and G. Jayalalitha, "Equitable Domination in Chemical Structural Graph," European Journal of Molecular & Clinical Medicine, vol. 7, no. 2, pp. 5246-5249, 2020.

R. A. Borzooei and H. Rashmanlou, "Domination in vague graphs and its applications," Journal of Intelligent & Fuzzy Systems, vol. 29, no. 5, pp. 1933-1940, 2015.

H. Rashmanlou, A. Lakdashti and S. S. Sabet, "A Study on Domination Sets in Vague Graphs with Novel Applications," Annals of Pure and Applied Mathematics, vol. 16, no. 1, pp. 223-234, 2018.

L. F. Casinillo, "Odd and even repetition sequences of independent domination number," Notes on Number Theory and Discrete Mathematics, vol. 26, no. 1, pp. 8-20, 2020.

L. F. Casinillo, "Some new notes on Mersenne primes and perfect numbers," Indonesian Journal of Mathematics Education, vol. 3, no. 1, pp. 15-22, 2020.

L. F. Casinillo, "A note on Fibonacci and Lucas number of domination in path," Electronic Journal of Graph Theory and Applications, vol. 6, no. 2, pp. 317-325, 2018.

L. F. Casinillo, "New counting formula for dominating sets in path and cycle graphs," Journal of Fundamental Mathematics and Applications (JFMA), vol. 3, no. 2, pp. 170-177, 2020.

E. L. Casinillo and L. F. Casinillo, "A Note on Full and Complete Binary Planar Graphs," Indonesian Journal of Mathematics Education, vol. 3, no. 2, pp. 70-75, 2020.

J. von Postel, T. Schweser and M. Stiebitz, "Point partition numbers: perfect graphs," Graphs and Combinatorics, vol. 38, no. 2, pp. 1-11, 2011.

S. Ali, M. K. Mahmood and K. P. Shum, "Novel classes of integers and their applications in graph labeling," Hacettepe Journal of Mathematics and Statistics, vol. 50, no. 4, pp. 1-17, 2021.

Full Text: PDF

DOI: 10.15408/inprime.v4i2.24573


  • There are currently no refbacks.