On Partition Dimension and Domination of Abid-Waheed 〖(AW)〗_r^4 Graph
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Abstract
A graph denoted by H, which has a simple link between its vertices, possesses the set of vertices V(H) . Given a graph, a set that is dominant, is a subset of vertex set such that any vertex outside of is close to at least one vertex inside of . The smallest size of for the dominating set is known as the graph’s domination number. When a linked graph H has a vertex x and a subset of the vertex set, the separation between x and S is given by. Pertaining to an ordered k-partition of , the illustration of in relation to Π is to be the k-vectorAbid-Waheed graph is a simply connected graph which contains vertices and edges for all and In this paper, we studied some results on the domination number, independent and restrained domination number denoted by respectively in the Abid-Waheed graphs and the relation between domination number, independent and restrained domination number. Also, the objective of this paper is to generate a partition dimension of.
Received 16/01/2023
Revised 19/05/2023
Accepted 21/05/2023
Published Online First 20/10/2023
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References
Laskar RC, Walikar HB. On Domination Related Concepts in Graph Theory. J Comb Math Comb Comput. 1981; 855(1): 308-320. https://dx.doi.org/10.1007/BFb0092276
Galleros DH., Enriquez EL. Fair Restrained Dominating Set in the Corona of Graphs. Int J Eng. 2020; 10(3): 110-4. https://dx.doi.org/10.31033/ijemr.10.3.17
Al-Harere MN, Bakhash KPA. Tadpole Domination in Graphs. Baghdad Sci J. 2018 Dec 9; 15(4): 466-471. https://dx.doi.org/10.21123/bsj.2018.15.4.0466
Omran AA., Oda HH. Hn Domination in Graphs. Baghdad Sci J. 2019; 16(1(Suppl.)): 242-247. https://dx.doi.org/10.21123/bsj.2019.16.1(Suppl.).0242
Gupta P. Domination in Graph with Application. Indian J Sci Res. 2013; 2(3): 115-117.
Omran AA., Swadi T. Observer Domination Number in Graphs. J Dyn Control Syst. 2019; 11(01-Special Issue).
Goddard W, Henning MA. Independent Domination in Graphs: A Survey and Recent Results. Discret Appl. Math. 2013 Apr 6; 313(7): 839-54. https://dx.doi.org/10.1016/j.disc.2012.11.031
Wen T, Cheong KH. The Fractal Dimension of Complex Networks: A Review. Inf Fusion. 2021; 73: 87-102. DOI: https://doi.org/10.1016/j.inffus.2021.02.001
Haynes TW, Hedetniemi ST, Henning MA, editors. Topics in Domination in Graphs. J Comb Math Comb Comput. 2020 Oct 19; 64. https://dx.doi.org/10.1007/978-3-030-51117-3
Rehman SU, Imran M, Javaid I. On the Metric Dimension of Arithmetic Graph of a Composite Number. Symmetry. 2020; 12(4); 607. https://dx.doi.org/10.3390/sym12040607
Ikhlaq HM, Hayat S, Siddiqui HM. Unique Identification and Domination of Edges in a Graph: The Vertex-Edge Dominant Edge Metric Dimension. ArXiv org. 2022 Nov 17; 2211(09327). https://dx.doi.org/10.48550/arXiv.2211.09327
Hwang SF, Chang G. The Edge Domination Problem Discuss. Math Graph Theory. 1995; 15(1): 51-57.
Lu CL, Ko MT, Tang CY. Perfect Edge Domination and Efficient Edge Domination in Graphs. Discret Appl Math. 2002; 119(3): 227-25. https://dx.doi.org/10.1016/S0166-218X(01)00198-6
Panda BS, Chaudhary J. Acyclic Matching in Some Subclasses of Graphs. Theor Comput Sci. 2020; 12126: 409–421. https://dx.doi.org/10.1016/j.tcs.2022.12.008
Żyliński P. Vertex-Edge Domination in Graphs. Aequ Math. 2019 Aug; 93(4): 735-42.
Dong F, Ge J, Yang Y. Upper Bounds on the Signed Edge Domination Number of a Graph. Discret Appl Math. 2021 Feb 1; 344(2): 112201. https://dx.doi.org/10.1016/j.disc.2020.112201
Grinstead DL, Slater PJ, Sherwani NA, Holmes ND. Efficient Edge Domination Problems in Graphs. Inf Process Lett. 1993; 48(5): 221-228. https://dx.doi.org/10.1016/0020-0190(93)90084-M
Vaidya SK., Ajani PD. Restrained Edge Domination Number of Some Path Related Graphs. J Sci Res. 2021; 13(1): 145-151. https://dx.doi.org/10.3329/jsr.v13i1.48520
Al-Harere MN, Metlif RJ, Sadiq FA. Variant Domination Types for a Complete h-ary Tree. Baghdad Sci J. 2021; 18(1): 797-802. https://dx.doi.org/10.21123/bsj.2021.18.1(Suppl.).0797
Raj RN, Raj FS. On the Partition Dimension of Honey Comb, Hexagonal Cage Networks and Quartz Network Ann. Romanian Soc. Cell Biol. 2021 Apr 11; 2811-7.
Tomescu I, Javaid I, Slamin I. On the Partition Dimension and Connected Partition Dimension of Wheels. Ars Comb. 2007; 84: 311-318.
Singh P, Sharma S, Sharma SK, Bhat VK. Metric Dimension and Edge Metric Dimension of Windmill Graphs. AIMS Mathematics. 2021 Jan 1; 6(9): 9138-53. https://dx.doi.org/10.3934/math.2021531
Wei C, Nadeem MF, Siddiqui HMF, Azeem M, Liu JB, Khalil A. On Partition Dimension of Some Cycle-Related Graphs. Math Probl Eng. 2021; 2021: 1-8. https://dx.doi.org/10.1155/2021/4046909
Chu YM, Nadeem MF, Azeem M, Siddiqui MK. On Sharp Bounds on Partition Dimension of Convex Polytopes. IEEE Access. 2020 Dec 14; 8: 224781-90. https://dx.doi.org/10.1109/ACCESS.2020.3044498
Chartrand G, Salehi E, Zhang P. On the Partition Dimension of a Graph. Congr Numer. 1998; 130: 157168.
Goddard W., Henning MA. Independent Domination in Outerplanar Graphs. Discret Appl Math. 2023 Jan 30; 325: 52-7. https://dx.doi.org/10.1016/j.dam.2022.10.003
Idrees M, Ma H, Wu M, Nizami AR, Munir M, Ali S. Metric Dimension of Generalized Möbius Ladder and its Application to WSN Localization. J Adv Comput Intell Intell Inform. 2020 Jan 20; 24(1): 3-11. https://dx.doi.org/10.20965/jaciii.2020.p0003
Mahboob A, Rasheed MW. Hosaya Polynomial and Weiner Index of Abid-Waheed Graph (AW)_p^6. Appl Comput Math. 2022; 10(3): 89-94. https://dx.doi.org/10.11648/j.ijebo.20221003.13