‎Application of superhypergraphs-based domination number in real world

Document Type : Research Paper


Department of Mathematics, Payame Noor University, Tehran, Iran


    The concept of (quasi) superhypergraphs as a generalization of graphs makes a relation between some sets of elements in detail and in general (in the form of parts to parts, parts to whole, and whole to whole elements of sets) and is very useful in the real world. This paper considers the novel concept of (quasi) superhypergraphs and introduces the notation of dominating set and domination number of (quasi) superhypergraphs. Especially, we have analyzed the domination number of uniform (quasi) superhypergraphs and computed their domination number on different cases. The flows (from right to left, from left to right, and two-sided) as maps play a main role in (quasi) superhypergraphs and it is proved that domination numbers of (quasi) superhypergraphs are dependent on the flows. We define the valued-star (quasi) superhypergraphs for the design of hypernetworks and compute their domination numbers. We have shown that the domination numbers of valued-star (quasi) superhypergraphs are distinct in  different flow states. In final, we introduce some applications of dominating sets of (quasi) superhypergraphs in hypernetwork as computer networks and treatment networks with the optimal application.


Main Subjects

[1] M. Akram, and A. Luqman, Fuzzy Hypergraphs and Related Extensions, Studies in Fuzziness and Soft Computing DOI: 10.1007/978-981-15-2403-5 vol. 390 (2020).
[2] M. Akram, N. Waseem, and B. Davvaz, Certain Types of Domination in m-polar Fuzzy Graphs, Journal of Multiple-Valued Logic and Soft Computing vol. 29, no. 5 (2017) 619{646.
[3] M. Akram, N. Waseem, Novel decision making method based on domination in m-polar fuzzy graphs, Communications of the Korean Mathematical Society vol. 32, no. 4 (2017) 1077-1097.
[4] B. D. Acharya, Domination in Hypergraphs, AKCE International Journal of Graphs and Combinatorics vol. 4, no. 2 (2007) 117{126.
[5] J. S. Alameda, F. Kenter, K. Meagher, M. Young, An upper bound for the k-power domination number in r-uniform hypergraphs, Discrete Mathematics vol. 345, no. 11 (2022)113038 1-9.
[6] C. Bujtas, M. A. Henning, Z. Tuz, Transversals and domination in uniform hypergraphs, European Journal of Combinatorics vol. 33, no. 1 (2012) 62-71.
[7] B. Bjorkman, Infectious power domination of hypergraphs, Discrete Mathematics vol. 343, no.3 (2020) 111724 1-12.
[8] C. Berge, Graphs and Hypergraphs, North Holland 1979.
[9] K. Cardoso and V. Trevisan, The signless Laplacian matrix of hypergraphs, Special Matrices vol. 10 (2022) 327-342.
[10] Y. Dong, E. Shana, L. Kanga, S. Li, Domination in intersecting hypergraphs, Discrete Applied Mathematics vol. 251, no. 31 (2018) 155{159.
[11] M. W. Y. Dong, A Critical Analysis on Complex Urban Systems and Complex Systems Theory, Journal of Computing and Information Science in Engineering vol. 3, no. 1 (2023) 24{34.
[12] M. Fazil, I. Javaid, M. Salman & U. Ali, Locating-dominating sets in hypergraphs, Periodica Mathematica Hungarica vol. 72 (2016) 224{234.
[13] M. Goberna and M. Verdu, Cautionary notes on the use of co-occurrence networks in soil ecology, Soil Biology & Biochemistry vol. 166 (2022) 108534 1-10.
[14] T. Gaudelet, N. M. Dognin and N. Przulj, Higher-order molecular organization as a source of biological function, Journal of Bioinformatics and Computational Biology vol. 34 (2018) i944{i953.
[15] M. Hamidi, F. Smarandache, and E. Davneshvar, Spectrum of Superhypergraphs via Flows, Journal of Mathematics vol. 2022 (2022) 1-12.
[16] M. Hamidi and A. Borumand Saeid, Creating and Computing Graphs from Hypergraphs, Kragujevac Journal of Mathematics vol. 43, no. 1 (2019) 139{164.
[17] M. Hamidi and F. Smarandache, Single-Valued Neutrosophic Directed (Hyper)Graphs And Applications in Networks, Journal of Intelligent & Fuzzy Systems vol. 37, no. 2 (2019) 2869{2885.
[18] M. Hamidi and A. Borumand Saeid, Achievable Single-Valued Neutrosophic Graphs in Wireless Sensor Networks, New Mathematics and Natural Computation vol. 14, no. 2 (2018) 157{185.
[19] M. Hamidi, A. Borumand Saeid, On Derivable Tree, Transactions on Combinatorics, vol. 8, no. 2 (2019) 21{43.
[20] M. Hamidi, A. Borumand saied, Accessible single-valued neutrosophic graphs, Journal of Applied Mathematics and Computing vol. 57 (2018) 121{146.
[21] C. F. De. Jaenisch, Applications de Lanaluse Mathematique an Jen des Echecs, Petrograd, 1862.
[22] A. Jing, The technology and digital  nancial risk management model using intelligent data processing, Journal of Optics vol. 273 (2023)170410 1-13.
[23] J. Kim, M. Kim, Domination numbers and noncover complexes of hypergraphs, Journal of Combinatorial Theory, Series A vol. 180 (2021)105408 1-27.
[24] A. Luqman, M. Akram, and A. N. A. Koam, Granulation of Hypernetwork Models under the q-Rung Picture Fuzzy Environment, Journal of Mathematics vol. 7, no. 6 (2019)496 1{25.
[25] C. Y. Lee, H. Y. Chong, P. C. Liao X. Wang, Critical Review of Social Network Analysis Applications in Complex Project Management, Journal of Management in Engineering vol. 34, no. 2 04017061-2{ 04017061-15.
[26] A. S. d. Mata, Complex Networks: a Mini-review, Brazilian Journal of Physics vol. 50 (2020) 658{672.
[27] A. Nagoorgani, Mu. Akram and S. Anupriya, Double domination on intuitionistic fuzzy graphs , Journal of Computational and Applied Mathematics vol. 52, no. 1-2 (2016) 515-528.
[28] J. Ribeiro, K. Davids, D. Araujo, P. Silva, J. Ramos, R.Lopes and J. Garganta, The Role of Hypernetworks as a Multilevel Methodology for Modelling and Understanding Dynamics of Team Sports Performance, The American Journal of Sports Medicine vol. 49 (2019) 1337{1344.
[29] S . S. Sahak, S. K. Panda, On the Laplacian spectrum of k-uniform hypergraphs, Linear Algebra and its Applications vol. 655, no. 15 (2022) 1-27.
[30] F. Smarandache, Extension of Hypergraph to n-Super hypergraph and to Plithogenic n-Super hypergraph, and Extension of Hyperalgebra to n-ary (Classical-/Neutro-/Anti-)Hyperalgebra, Neutrosophic Sets and Systems vol. 33 (2020) 290-296.
[31] C. Sergiou, M. Lestas, P. Antonious, C. Liaskos and A. Pitsillides, Complex Systems: A Communication Networks Perspective Towards 6G, IEEE Access vol. 8 (2020) 89007-89030.
[32] M. Saleh, Y. Esa and A. Mohamed, Applications of Complex Network Analysis in Electric Power Systems, Energy vol. 11, no. 6 1381 (2018) 1{16.
[33] E. Sampathkumar and Hanumappa B Walikar, The connected domination number of a graph, Journal of Mathematical Physics vol. 13, no. 6 (1979) 607-613.
[34] L. Trajkovic, Analysis of Internet topologies, IEEE Circuits and Systems Magazine vol. 10, no. 3 (2010) 48{54.
[35] J. Zhu, X. Ma, L. Martnez and J. Zhan, A probabilistic linguistic three-way decision method with regret theory via fuzzy c-means clustering algorithm, IEEE Transactions on Fuzzy Systems DOI: 10.1109/Tfuzz.2023.3236386 (2023).
[36] J. Zhan, J. Wang, W.Ding, and Y. Yao, Three-way behavioral decision making with hesitant fuzzy information systems: survey and challenges, IEEE/CAA Journal of Automatica Sinica vol. 2 (2023) 330-350.
[37] J. Zhan, J. Deng, Z. Xu, L. Martinez, A three-way decision methodology with regret theory via triangular fuzzy numbers in incomplete multi-scale decision information systems, IEEE Transactions on Fuzzy Systems DOI: 10.1109/Tfuzz.2023.3237646 (2023).