International Journal of Nonlinear Analysis and Applications

The arrow edge domination in graphs

Document Type : Research Paper

Authors

College of Education for Pure Sciences, University of Thi-Qar, Thi-Qar, Iraq

Abstract

The idea of this paper is to study the arrow edge domination. The arrow edge dominating set $D_{e}$ of a graph $G$ is an arrow edge dominating set if every edge from $D$ dominates exactly one edge from $V-D$ and is adjacent to two or more edges from $D$. The arrow edge domination number $\gamma_{\text {are }}(G)$ is the minimum cardinality of all arrow edge dominating sets in $G$. Several properties and bounds are introduced here. Our results are applied in some graphs such that the path graph, cycle graph, complete graph, wheel graph, complete bipartite graph, Barbell graph, helm graph, big helm graph, complement path graph, complement cycle graph, the complement of complete graph and complement of complete bipartite graph. An important fact given here is if $G$ has no arrow vertex dominating set, then $G$ may have an arrow edge dominating set and an example is given.

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. http://dx.doi.org/10.5269/bspm.52252.
[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 its 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 its inverse for corona and join operations in graphs, Proc. Int. Math. Sci. 1 (2019), no. 2, 51–55.
[9] M.A. Abdlhusein and M.N. Al-Harere, Pitchfork domination and its inverse for complement graphs, Proc. Instit. Appl. Math. 9 (2020), no. 1, 13–17.
[10] M.A. Abdlhusein and M.N. Al-Harere, Some modified types of pitchfork domination and its inverse, Bol. Soc. Paran. Mat. 40 (2022), 1–9.
[11] M.A. Abdlhusein and Z.H. Abdulhasan, Modified types of triple effect domination, reprinted, 2022.
[12] M.A. Abdlhusein and Z.H. Abdulhasan, Stability and some results of triple effect domination, Int. J. Nonlinear Anal. Appl. Accepted to appear, (2022).
[13] Z.H. Abdulhasan and M.A. Abdlhusein, Triple effect domination in graphs, AIP Conf. Proc. 2022, 2386, 060013.
[14] 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.
[15] M.N. Al-Harere and M.A. Abdlhusein, Pitchfork domination in graphs, Discrete Math. Algor. Appl. 12 (2020), no. 2, 2050025.
[16] 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.
[17] 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.
[18] F. Harary, Graph theory, Addison-Wesley, Reading, MA, 1969.
[19] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of domination in graphs, Marcel Dekker Inc. New York, 1998.
[20] T.W. Haynes, M.A. Henning and P. Zhang, A survey of stratified domination in graphs, Discrete Math. 309 (2009), 5806–5819.
[21] M.K. Idan and M.A. Abdlhusein, Some properties of discrete topological graph, IOP Conf. Proc. (2022), accepted to appear.
[22] A.A. Jabor and A.A. Omran, Domination in discrete topological graph, AIP Conf. Proc. 2138 (2019), 030006.
[23] A.A. Jabor and A.A. Omran, Topological domination in graph theory, AIP Conf. Proc. 2334 (2021), 020010.
[24] Z.N. Jweir and M.A. Abdlhusein, Appling some dominating parameters on the topological graph, IOP Conf. Proc., accepted to appear. (2022).
[25] Z.N. Jweir and M.A. Abdlhusein, Constructing new topological graph with several properties, reprinted, 2022.
[26] S.S. Kahat, A.A. Omran and M.N. Al-Harere, Fuzzy equality co-neighborhood domination of graphs, Int. J. Nonlinear Anal. Appl. 12 (2021), no. 2, 537–545.
[27] 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.
[28] S.J. Radhi, M.A. Abdlhusein and A.E. Hashoosh, Some modified types of arrow domination, Int. J. Nonlinear Anal. Appl. 13 (2021), no. 1, 1451–1461.
International Journal of Nonlinear Analysis and Applications
Volume 13, Issue 2
July 2022
Pages 591-597
  • Receive Date: 02 January 2022
  • Revise Date: 18 February 2022
  • Accept Date: 25 March 2022