International Journal of Nonlinear Analysis and Applications
Some dominating results of the join and corona operations between discrete topological graphs
Document Type : Research Paper
Authors
Department of Mathematics, College of Education for Pure Sciences, University of Thi-Qar, Thi-Qar, Iraq
Abstract
The dominating sets play an important role in applications of graph theory. In this field, some recent studies discussed the properties of the minimum dominating set ($\gamma $-set). The other type of study produces a topological space from the set of vertices or the set of edges of a graph. The previous paper introduces a new method to construct a graph $G_{\tau }$ from the topological space. In this paper, the dominating set and the domination number of $G_{\tau }$ are proved with their inverse. The domination number and inverse domination number of corona operation and join operation between two graphs are discussed.
Keywords
[1] M.A. Abdlhusein, New approach in graph domination, Ph. D. Thesis, University of Baghdad, Iraq, 2020.
[2] M.A. Abdlhusein, Doubly connected bi-domination in graphs, Discrete Math. Algor. Appl. 13 (2021), no. 2, 2150009.
[3] M.A. Abdlhusein, Stability of inverse pitchfork domination, Int. J. Nonlinear Anal. Appl. 12 (2021), no. 1, 1009– 1016.
[4] M.A. Abdlhusein, Applying the (1, 2)-pitchfork domination and its inverse on some special graphs, Bol. Soc. Paran. Mat. Accepted to appear, (2022).
[5] M.A. Abdlhusein and M.N. Al-Harere, Total pitchfork domination and its inverse in graphs, Discrete Math. Algor. Appl. 13 (2021), no. 4, 2150038.
[6] M.A. Abdlhusein and M.N. Al-Harere, New parameter of inverse domination in graphs, Indian J. Pure Appl. Math. 52 (2021), no. 1, 281–288.
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[11] M.A. Abdlhusein and S.J. Radhi, The arrow edge domination in graphs, Int. J. Nonlinear Anal. Appl. 13 (2022), no. 2, 591–597.
[12] Z.H. Abdulhasan and M.A. Abdlhusein, Triple effect domination in graphs, AIP Conf. Proc. 2386 (2022), 060013.
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[3] M.A. Abdlhusein, Stability of inverse pitchfork domination, Int. J. Nonlinear Anal. Appl. 12 (2021), no. 1, 1009– 1016.
[4] M.A. Abdlhusein, Applying the (1, 2)-pitchfork domination and its inverse on some special graphs, Bol. Soc. Paran. Mat. Accepted to appear, (2022).
[5] M.A. Abdlhusein and M.N. Al-Harere, Total pitchfork domination and its inverse in graphs, Discrete Math. Algor. Appl. 13 (2021), no. 4, 2150038.
[6] M.A. Abdlhusein and M.N. Al-Harere, New parameter of inverse domination in graphs, Indian J. Pure Appl. Math. 52 (2021), no. 1, 281–288.
[7] M.A. Abdlhusein and M.N. Al-Harere, Doubly connected pitchfork domination and it’s inverse in graphs, TWMS J. App. Eng. Math. 12 (2022), no. 1, 82-–91.
[8] M.A. Abdlhusein and M.N. Al-Harere, Pitchfork domination and it’s inverse for corona and join operations in graphs, Proc. Int. Math. Sci. 1 (2019), no. 2, 51–55.
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[12] Z.H. Abdulhasan and M.A. Abdlhusein, Triple effect domination in graphs, AIP Conf. Proc. 2386 (2022), 060013.
[13] Z.H. Abdulhasan and M.A. Abdlhusein, An inverse triple effect domination in graphs, Int. J. Nonlinear Anal. Appl. 12 (2021), no. 2, 913–919.
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[17] M.N. Al-Harere and A.T. Breesam, Further results on bi-domination in graph, AIP Conf. Proc. 2096 (2019), no. 1, 020013-020013-9.
[14] Z.H. Abdulhasan and M.A. Abdlhusein, Stability and some results of triple effect domination, Int. J. Nonlinear Anal. Appl. In Press, Doi: 10.22075/IJNAA.2022.6473
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[16] M.N. Al-Harere and M.A. Abdlhusein, Pitchfork domination in graphs, Discrete Math. Algor. Appl. 12 (2020), no. 2, 2050025.
[17] M.N. Al-Harere and A.T. Breesam, Further results on bi-domination in graph, AIP Conf. Proc. 2096 (2019), no. 1, 020013-020013-9.
[18] L.K. Alzaki, M.A. Abdlhusein and A.K. Yousif, Stability of (1, 2)-total pitchfork domination, Int. J. Nonlinear Anal. Appl. 12 (2021), no. 2, 265–274.
[19] F. Harary, Graph theory, Addison-Wesley, Reading, MA, 1969.
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[21] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Domination in graphs-advanced topics, Marcel Dekker, Inc., 1998.
[22] T.W. Haynes, M.A. Henning and P. Zhang, A survey of stratified domination in graphs, Discrete Math. 309 (2009), 5806–5819.
[23] M.K. Idan and M.A. Abdlhusein, Some properties of discrete topological graph, J. Phys.: Conf. Ser. to appear, 2022.
[24] M.K. Idan and M.A. Abdlhusein, Different types of dominating sets of the discrete topological graph, Int. J. Nonlinear Anal. Appl. In Press, 10.22075/IJNAA.2022.6452.
[25] A.A. Jabor and A.A. Omran, Domination in discrete topological graph, AIP Conf. Proc. 2138 (2019), 030006.
[26] A.A. Jabor and A.A. Omran, Topological domination in graph theory, AIP Conf. Proc. 2334 (2021), 020010.
[27] Z.N. Jweir and M.A. Abdlhusein, Appling some dominating parameters on the topological graph, IOP Conf. Proc. to appear, 2022.
[28] Z.N. Jwair and M.A. Abdlhusein, Some dominating results of the topological graph, Int. J. Nonlinear Anal. Appl. In Press, Doi:10.22075/IJNAA.2022.6404.
[29] Z.N. Jwair and M.A. Abdlhusein, Constructing new topological graph with several properties, Iraqi J. Sci. to appear, 2022.
[30] S.J. Radhi, M.A. Abdlhusein and A.E. Hashoosh, The arrow domination in graphs, Int. J. Nonlinear Anal. Appl. 12 (2021), no. 1, 473–480.
[31] S.J. Radhi, M.A. Abdlhusein and A.E. Hashoosh, Some modified types of arrow domination, Int. J. Nonlinear Anal. Appl. 13 (2022), no. 1, 1451–1461.
[19] F. Harary, Graph theory, Addison-Wesley, Reading, MA, 1969.
[20] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of domination in graphs, Marcel Dekker, Inc. New York, 1998.
[21] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Domination in graphs-advanced topics, Marcel Dekker, Inc., 1998.
[22] T.W. Haynes, M.A. Henning and P. Zhang, A survey of stratified domination in graphs, Discrete Math. 309 (2009), 5806–5819.
[23] M.K. Idan and M.A. Abdlhusein, Some properties of discrete topological graph, J. Phys.: Conf. Ser. to appear, 2022.
[24] M.K. Idan and M.A. Abdlhusein, Different types of dominating sets of the discrete topological graph, Int. J. Nonlinear Anal. Appl. In Press, 10.22075/IJNAA.2022.6452.
[25] A.A. Jabor and A.A. Omran, Domination in discrete topological graph, AIP Conf. Proc. 2138 (2019), 030006.
[26] A.A. Jabor and A.A. Omran, Topological domination in graph theory, AIP Conf. Proc. 2334 (2021), 020010.
[27] Z.N. Jweir and M.A. Abdlhusein, Appling some dominating parameters on the topological graph, IOP Conf. Proc. to appear, 2022.
[28] Z.N. Jwair and M.A. Abdlhusein, Some dominating results of the topological graph, Int. J. Nonlinear Anal. Appl. In Press, Doi:10.22075/IJNAA.2022.6404.
[29] Z.N. Jwair and M.A. Abdlhusein, Constructing new topological graph with several properties, Iraqi J. Sci. to appear, 2022.
[30] S.J. Radhi, M.A. Abdlhusein and A.E. Hashoosh, The arrow domination in graphs, Int. J. Nonlinear Anal. Appl. 12 (2021), no. 1, 473–480.
[31] S.J. Radhi, M.A. Abdlhusein and A.E. Hashoosh, Some modified types of arrow domination, Int. J. Nonlinear Anal. Appl. 13 (2022), no. 1, 1451–1461.
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Volume 14, Issue 5
May 2023Pages 235-242
- Receive Date: 05 January 2022
- Revise Date: 20 February 2022
- Accept Date: 26 March 2022