About 2- Isolate Inclusive Sets In Graphs

D.K. Thakkar; N.J. Savaliya

doi:https://doi.org/10.14445/22315373/IJMTT-V63P512

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 63 | Number 2 | Year 2018 | Article Id. IJMTT-V63P512 | DOI : https://doi.org/10.14445/22315373/IJMTT-V63P512

About 2- Isolate Inclusive Sets In Graphs

D.K. Thakkar, N.J. Savaliya

Abstract

In this paper we introduce in new concept call 2-isolate inclusive sets in graphs. Every 2-isolate inclusive set is an isolate inclusive set of 𝐺 . We characterize maximal 2-isolate inclusive set of a graph. We deduce that every maximal 2-isolate inclusive sets of 𝐺 is a distance-2 dominating set of 𝐺. We also define 2- isolate inclusive number of a graph and we observe that it is less then or equal to isolate inclusive number of the graph. We also prove that if the < 𝑆 > has the maximum number of 2-isolated vertices among all the 2-isolate inclusive sets then 𝑆 is a maximum 2-packing of 𝐺. We also prove several other related results.

Keywords

2-isolated vertex, 2-packing, 2-isolate inclusive set, maximum 2-isolate inclusive set, maximal 2- isolate inclusive set, distance-𝑘 dominating set, distance-2 open neighbourhood, 2-degree of vertex,2-isolate distance-2 dominating set, distance-2 private neighbourhood.

References

[1] D. K. Thakkar and J. C. Bosamiya Graph critical with respect to independent domination, Journal of Discrete Mathematical Sciences and Cryptography Vol.16(2013),179-186
[2] D. K. Thakkar and N. J. Savaliya , About Isolate Domination in Graphs , International Journal of Mathematical Archive Vol.8 (11) (2017), 171-178
[3] D. K. Thakkar and N. J. Savaliya , On Isolate Inclusive Sets in Graphs , International Journal of Innovation in Science and Mathematics Vol.5(3) (2017), 74-76
[4] I. Sahul Hamid and S. Balamurugan, Isolate Domination Number and Maximum Degree, bulletin of the international mathematical virtual institute Vol.3(2013),127-133
[5] I. Sahul Hamid and S. Balamurugan, Isolate Domination in Graphs, Arab Journal of Mathematical Sciences Math Sci 22(2016),232-241
[6] Michael A. Henning., Anders Yeo Total Domination in Graphs, Springer, New York, (2013)
[7] T. W. Haynes, S. T. Hedetniemi, P. J. Slater, Fundamental of Domination In graphs, Marcel Dekker, New York, (1998)
[8] T. W. Haynes, S. T. Hedetniemi, P. J. Slater, Domination In Graphs Advanced Topics , Marcel Dekker, New York, (1998)

Citation :

D.K. Thakkar, N.J. Savaliya, "About 2- Isolate Inclusive Sets In Graphs," International Journal of Mathematics Trends and Technology (IJMTT), vol. 63, no. 2, pp. 89-102, 2018. Crossref, https://doi.org/10.14445/22315373/IJMTT-V63P512