Research PaperNo Access

Captive domination in graphs

Manal N. Al-Harere

http://orcid.org/0000-0003-1125-4439

Department of Applied Sciences, University of Technology, Baghdad, Iraq

E-mail Address: 100035@uotechnology.edu.iq

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Ahmed A. Omran

Department of Mathematics, College of Education for Pure Science, University of Babylon, Babylon, Iraq

E-mail Address: a.a.mamoory@gmail.com

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, and

Athraa T. Breesam

Department of Applied Sciences, University of Technology, Baghdad, Iraq

E-mail Address: A.T.Breesam@gmail.com

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https://doi.org/10.1142/S1793830920500767Cited by:22 (Source: Crossref)

Abstract

In this paper, a new definition of graph domination called “Captive Domination” is introduced. The proper subset of the vertices of a graph $G$ is a captive dominating set if it is a total dominating set and each vertex in this set dominates at least one vertex which doesn’t belong to the dominating set. The inverse captive domination is also introduced. The lower and upper bounds for the number of edges of the graph are presented by using the captive domination number. Moreover, the lower and upper bounds for the captive domination number are found by using the number of vertices. The condition when the total domination and captive domination number are equal to two is discussed and obtained results. The captive domination in complement graphs is discussed. Finally, the captive dominating set of paths and cycles are determined.

Keywords:

AMSC: 05C69

References

1. M. A. Abdlhusein and M. N. Al-Harere, Pitchfork Domination and It’s Inverse for Corona and Join Operations in Graphs, Proceedings of International Mathematical Sciences I(2) (2019) 51–55. Google Scholar
2. M. N. Al-Harere and M. A. Abdlhusein, Pitchfork domination in graphs, Discrete Mathematics, Algorithms and Applications 12(2) (2020) 2050025. Link, Google Scholar
3. M. N. Al-Harere and A. T. Breesam, Further results on bi-domination in graphs, AIP Conf. Proc. 2096 (2019) 020013. Crossref, Google Scholar
4. M. N. Al-Harere and P. A. Khuda Bakhash, Tadpole domination in graphs, Baghdad Sci. J. 15 (2018) 466–471. Crossref, Google Scholar
5. E. J. Cockayne, R. M. Dawes, S. T. Hedetniemi, Total domination in graphs, Networks: Int. J. 10(3) (1980) 211–219. Crossref, Google Scholar
6. M. Chellali, T. W. Haynes, S. T. Hedetniemi and A. M. Rae, [1,2]-Set in graphs, Discrete Appl. Math. 161(18) (2013) 2885–2893. Crossref, Google Scholar
7. F. Harary, Graph Theory (Addison-Wesley, Reading Mass, 1969). Crossref, Google Scholar
8. T. W. Haynes, S. T. Hedetniemi and P. J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, Inc., New York, 1998). Google Scholar
9. T. W. Haynes, S. T. Hedetniemi and P. J. Slater, Domination in Graphs-Advanced Topics (Marcel Dekker Inc., 1998). Google Scholar
10. A. A. Jabour and A. A. Omran, Domination in discrete topology graph, AIP Conf. Proc. 2183 (2019) 030006. Crossref, Google Scholar
11. A. Khodkar, B. Samadi and H. R. Golmohammadi, ( $k, k^{'}, k^{''}$ )-domination in graphs, J. Combin. Math. Combin.Comput. 98 (2016) 343–349. Google Scholar
12. C. Natarajan, S. K. Ayyaswamy and G. Sathiamoorthy, A note on hop domination number of some special families of graphs, Int. J. Pure Appl. Math. 119(12) (2018) 14165–14171. Google Scholar
13. A. A. Omran and H. Hamed Oda, Hn domination in graphs, Baghdad Sci. J. 16(1) (2019) 242–247. Crossref, Google Scholar
14. A. A. Omran and Y. Rajihy, Some properties of frame domination in graphs, J. Eng. Appl. Sci. 12 (2017) 8882–8885. Google Scholar
15. A. A. Omran and H. J. Al Hwaeer, Modern roman domination in graphs, Basrah Journal of Science 36(1) (2018) 45–54. Google Scholar
16. O. Ore, Theory of Graphs (American Mathematical Society, Providence, R.I., 1962). Crossref, Google Scholar
17. N. Saradha and V. Swaminathan, Connected equitable co-independent domination of a graph, Int. J. Pure Appl. Math. 101(5) (2015) 721–726. Google Scholar
18. X. Yang and B. Wu, [1, 2]-Domination in graphs, Discrete Appl. Math. 175 (2014) 79–86. Crossref, Google Scholar
19. X. Zhang, Z. Shao and H. Yang, The [ $a, b$ ]-domination and [ $a, b$ ]-total domination of graphs, J. Math. Res. 9(3) (2017) 38–45. Crossref, Google Scholar