Authors:
Iqbal M. Batiha,Nidal Anakira,Basma Mohamed,DOI NO:
https://doi.org/10.26782/jmcms.2024.09.00003Keywords:
Domination Number,Metric Dimension,Resolving Dominating Set,Abstract
A minimum resolving set is a resolving set with the lowest cardinality and its cardinality is a dimension of connected graph , represented by . A dominating set is a set of vertices such that each of is either in or has at least one neighbor in . The dominance number of is the lowest cardinality of such a set. The lowest cardinality of the dominant resolving set is called a dominant metric dimension of , represented by . This paper presents an algorithm for finding the domination resolving number of a graph.Refference:
I. A. A. Khalil. : ‘Determination and testing the domination numbers of Helm graph, web graph and Levi graph using MATLAB’. Journal of Education Science. Vol. 24, pp. 103-116, 2011. https://www.iasj.net/iasj/download/2b430f4e0c4f89fd
II. A. Sugumaran, E. Jayachandran. : ‘Domination number of some graphs’. International Journal of Scientific Development and Research. Vol. 3, pp. 386-391, 2018. https://api.semanticscholar.org/CorpusID:213194763
III. B. Mohamed. : ‘A comprehensive survey on the metric dimension problem of graphs and its types’. International Journal of Theoretical and Applied Mathematics. Vol. 9, pp. 1-5, 2023. 10.11648/j.ijtam.20230901.11
IV. B. Mohamed, L. Mohaisen, M. Amin. : ‘Binary equilibrium optimization algorithm for computing connected domination metric dimension problem’. Scientific Programming. Vol. 2022, pp. 1-15, 2022. 10.1155/2022/6076369
V. B. Mohamed, L. Mohaisen, M. Amin. : ‘Computing connected resolvability of graphs using binary enhanced Harris Hawks optimization’. Intelligent Automation & Soft Computing. Vol. 36, pp. 2349-2361, 2023. 10.32604/iasc.2023.032930
VI. B. Mohamed, M. Amin. : ‘A hybrid optimization algorithms for solving metric dimension problem’. Graph-HOC. Vol. 15, pp. 1-10, 2023. https://ssrn.com/abstract=4504670
VII. B. Mohamed, M. Amin. : ‘Domination number and secure resolving sets in cyclic networks’. Applied and Computational Mathematics. Vol. 12, pp. 42-45, 2023. 10.11648/j.acm.20231202.12
VIII. B. Mohamed, M. Amin. : ‘The metric dimension of subdivisions of Lilly graph, tadpole graph and special trees’. Applied and Computational Mathematics. Vol. 12, pp. 9-14, 2023. 10.11648/j.acm.20231201.12
IX. B. Mohamed. : ‘Metric dimension of graphs and its application to robotic navigation’. International Journal of Computer Applications. Vol. 184, pp. 1-3, 2022. 10.5120/ijca2022922090
X. B. N. Kavitha, I. Kelkar. : ‘Split and equitable domination in book graph and stacked book graph’. International Journal of Advanced Research in Computer Science. Vol. 8, pp. 108-112, 2017. 10.26483/ijarcs.v8i6.4475
XI. C. S. Nagabhushana, B. N. Kavitha, H. M. Chudamani. : ‘Split and equitable domination of some special graph’. International Journal of Science Technology & Engineering. Vol. 4, pp. 50-54, 2017.
XII. F. Muhammad, L. Susilowati. : ‘Algorithm and computer program to determine metric dimension of graph’. Journal of Physics. Vol. 1494, 012018, 2020. 10.1088/1742-6596/1494/1/012018
XIII. H. Al-Zoubi, H. Alzaareer, A. Zraiqat, T. Hamadneh, W. Al-Mashaleh. : ‘On ruled surfaces of coordinate finite type’. WSEAS Transactions on Mathematics. Vol. 21, pp. 765–769, 2022. 10.37394/23206.2022.21.87
XIV. H. Iswadi, E. T. Baskoro, A. N. M. Salman, R. Simanjuntak. : ‘The resolving graph of amalgamation of cycles’. Utilitas Mathematica. Vol. 83, pp. 121-132, 2010. https://api.semanticscholar.org/CorpusID:55139163
XV. I. M. Batiha, B. Mohamed. : ‘Binary rat swarm optimizer algorithm for computing independent domination metric dimension problem’. Mathematical Models in Engineering. Vol. 10, pp. 6-13, 2024. 10.21595/mme.2024.24037
XVI. I. M. Batiha, B. Mohamed, I. H. Jebril. : ‘Secure metric dimension of new classes of graphs’. Mathematical Models in Engineering. Vol. 10, pp. 1-6, 2024. 10.21595/mme.2024.24168
XVII. I. M. Batiha, J. Oudetallah, A. Ouannas, A. A. Al-Nana, I. H. Jebril. : ‘Tuning the fractional-order PID-Controller for blood glucose level of diabetic patients’. International Journal of Advances in Soft Computing and its Applications. Vol. 13, pp. 1–10, 2021. https://www.i-csrs.org/Volumes/ijasca/2021.2.1.pdf
XVIII. I. M. Batiha, M. Amin, B. Mohamed, H. I. Jebril. : ‘Connected metric dimension of the class of ladder graphs’. Mathematical Models in Engineering. Vol. 10, pp. 65–74, 2024. 10.21595/mme.2024.23934
XIX. I. M. Batiha, N. Anakira, A. Hashim, B. Mohamed. : ‘A special graph for the connected metric dimension of graphs’. Mathematical Models in Engineering. Vol. 10, pp. 1-8, 2024. 10.21595/mme.2024.24176
XX. I. M. Batiha, S. A. Njadat, R. M. Batyha, A. Zraiqat, A. Dababneh, S. Momani. : ‘Design fractional-order PID controllers for single-joint robot ARM model’. International Journal of Advances in Soft Computing and its Applications. Vol. 14, pp. 97–114, 2022. 10.15849/IJASCA.220720.07
XXI. K. B. Murthy. : ‘The end equitable domination of dragon and some related graphs’. Journal of Computer and Mathematical sciences. Vol. 7, pp. 160-167, 2016.
XXII. L. Susilowati, I. Sa’adah, R. Z. Fauziyyah, A. Erfanian. : ‘The dominant metric dimension of graphs’. Heliyon. Vol. 6, 03633, 2020. 10.1016/j.heliyon.2020.e03633
XXIII. P. Sumathi, A. Rathi, A. Mahalakshmi. : ‘Quotient labeling of corona of ladder graphs’. International Journal of Innovative Research in Applied Sciences and Engineering. Vol. 1, pp. 1-12, 2017. 10.29027/IJIRASE.v1.i3.2017.80-85
XXIV. R. Alfarisi, Dafik, A. Kristiana. : ‘Resolving domination number of graphs’. Discrete Mathematics, Algorithms and Applications. Vol. 11, 1950071, 2019. 10.1142/S179383091950071X
XXV. R. C. Brigham, G. Chartrand, R. D. Dutton, P. Zhang. : ‘Resolving domination in graphs’. Mathematica Bohemica. Vol. 128, pp. 25-36, 2003. 10.21136/MB.2003.133935
XXVI. S. Kurniawati, D. A. R. Wardani, E. R. Albirri. : ‘On resolving domination number of friendship graph and its operation’. Journal of Physics. Vol. 1465, 012019, 2020. 10.1088/1742-6596/1465/1/012019
XXVII. R. P. Adirasari, H. Suprajitno, L. Susilowati. : ‘The dominant metric dimension of corona product graphs’. Baghdad Science Journal. Vol. 18, 0349, 2021. https://bsj.uobaghdad.edu.iq/index.php/BSJ/article/view/5039
XXVIII. T. Mazidah, Dafik, Slamin, I. H. Agustin, R. Nisviasari. : ‘Resolving independent domination number of some special graphs’. Journal of Physics. Vol. 1832, 012022, 2021. 10.1088/1742-6596/1832/1/012022