JACIII Vol.27 No.5 pp. 837-847
doi: 10.20965/jaciii.2023.p0837

Research Paper:

Novel Concepts on Domination in Neutrosophic Incidence Graphs with Some Applications

Siti Nurul Fitriah Mohamad*,**, Roslan Hasni**,† ORCID Icon, and Florentin Smarandache*** ORCID Icon

*College of Computing, Informatics and Mathematics, Universiti Teknologi MARA
Machang, Kelantan 18500, Malaysia

**Special Interest Group on Modeling and Data Analytics, Faculty of Ocean Engineering Technology and Informatics, Universiti Malaysia Terengganu
Kuala Nerus, Terengganu 21030, Malaysia

Corresponding author

***Mathematics & Science Department, The University of New Mexico
705 Gurley Avenue, Gallup, New Mexico 87301, USA

October 21, 2022
May 8, 2023
September 20, 2023
dominating set, domination number, neutrosophic graph, single-valued neutrosophic incidence graph

In graph theory, the concept of domination is essential in a variety of domains. It has broad applications in diverse fields such as coding theory, computer network models, and school bus routing and facility location problems. If a fuzzy graph fails to obtain acceptable results, neutrosophic sets and neutrosophic graphs can be used to model uncertainty correlated with indeterminate and inconsistent information in arbitrary real-world scenario. In this study, we consider the concept of domination as it relates to single-valued neutrosophic incidence graphs (SVNIGs). Given the importance of domination and its utilization in numerous fields, we propose the application of dominating sets in SVNIG with valid edges. We present some relevant definitions such as those of valid edges, cardinality, and isolated vertices in SVNIG along with some examples. Furthermore, we also show a few significant sets connected to the dominating set in an SVNIG such as independent and irredundant sets. We also investigate a relationship between the concepts of dominating sets and domination numbers as well as irredundant and independence sets in SVNIGs. Finally, a real-life deployment of domination in SVNIGs is investigated in relation to COVID-19 vaccination locations as a practical application.

Incidence graph

Incidence graph

Cite this article as:
S. Mohamad, R. Hasni, and F. Smarandache, “Novel Concepts on Domination in Neutrosophic Incidence Graphs with Some Applications,” J. Adv. Comput. Intell. Intell. Inform., Vol.27 No.5, pp. 837-847, 2023.
Data files:
  1. [1] O. Ore, “Theory of Graphs,” American Mathematical Society, 1962.
  2. [2] C. Berge, “The Theory of Graphs and its Applications,” Wiley, 1962.
  3. [3] T. Dinesh, “A Study on Graph Structures, Incidence Algebras and Their Fuzzy Analogues,” Ph.D. thesis, Kannur University, 2012.
  4. [4] L. A. Zadeh, “Fuzzy Sets,” Inf. Control, Vol.8, No.3, pp. 338-353, 1965.
  5. [5] A. Rosenfeld, “Fuzzy Graphs,” L. A. Zadeh et al. (Eds.), “Fuzzy Sets and Their Applications to Cognitive and Dexision Processes,” pp. 77-95, Academic Press, 1975.
  6. [6] A. Kaufmann, “Introduction to the Theory of Fuzzy Subsets,” Academic Press, 1975.
  7. [7] F. Smarandache, “Neutrosophy: Neutrosophic Probability, Set, and Logic: Analytic Synthesis & Synthetic Analysis,” American Research Press, 1998.
  8. [8] H. Wang et al., “Single Valued Neutrosophic Sets,” F. Smarandache (Ed.), “Multispace & Multistructure: Neutrosophic Transdisciplinarity (100 Collected Papers of Sciences), Vol.4,” pp. 410-413, North-European Scientific Publishers, 2010.
  9. [9] M. Akram and S. Shahzadi, “Neutrosophic Soft Graphs with Application,” J. Intell. Fuzzy Syst., Vol.32, No.1, pp. 841-858, 2017.
  10. [10] S. Broumi et al., “Interval Valued Pentapartitioned Neutrosophic Graphs with an Application to MCDM,” Oper. Res. Eng. Sci.: Theory Appl., Vol.5, No.3, pp. 68-91, 2022.
  11. [11] S. Broumi et al., “Interval-Valued Fermatean Neutrosophic Graphs,” Decision Making: Application in Management and Engineering, Vol.5, No.2, pp. 176-200, 2022.
  12. [12] S. Broumi et al., “Complex Fermatean Neutrosophic Graph and Application to Decision Making,” Decis. Mak.: Appl. Manag. Eng., Vol.6, No.1, pp. 474-501, 2023.
  13. [13] S. S. Hussain et al., “Domination Number in Neutrosophic Soft Graphs,” Neutrosophic Sets Syst., Vol.28, pp. 228-244, 2019.
  14. [14] R. N. Devi, “Minimal Domination via Neutrosophic Over Graphs,” AIP Conf. Proc., Vol.2277, Article No.100019, 2020.
  15. [15] M. Mullai and S. Broumi, “Dominating Energy in Neutrosophic Graphs,” Int. J. Neutrosophic Sci., Vol.5, No.1, pp. 38-58, 2020.
  16. [16] S. U. Khan et al., “Graphical Analysis of Covering and Paired Domination in the Environment of Neutrosophic Information,” Math. Probl. Eng., Vol.2021, Article No.5518295, 2021.
  17. [17] S. Afsharmanesh and R. A. Borzooei, “Domination in Fuzzy Incidence Graphs Based on Valid Edges,” J. Appl. Math. Comput., Vol.68, No.1, pp. 101-124, 2022.
  18. [18] Y. Rao et al., “A Study on Domination in Vague Incidence Graph and its Application in Medical Sciences,” Symmetry, Vol.12, No.11, Article No.1885, 2020.
  19. [19] R. A. Brualdi and J. J. Q. Massey, “Incidence and Strong Edge Colorings of Graphs,” Discrete Math., Vol.122, Nos.1-3, pp. 51-58, 1993.
  20. [20] T. Dinesh, “Fuzzy Incidence Graph – An Introduction,” Adv. Fuzzy Sets Syst., Vol.21, No.1, pp. 33-48, 2016.
  21. [21] S. Mathew and J. N. Mordeson, “Connectivity Concepts in Fuzzy Incidence Graphs,” Inf. Sci., Vols.382-383, pp. 326-333, 2017.
  22. [22] D. S. Malik, S. Mathew, and J. N. Mordeson, “Fuzzy Incidence Graphs: Applications to Human Trafficking,” Inf. Sci., Vol.447, pp. 244-255, 2018.
  23. [23] S. Mathew, J. Mordeson, and H.-L. Yang, “Incidence Cuts and Connectivity in Fuzzy Incidence Graphs,” Iran. J. Fuzzy Syst., Vol.16, No.2, pp. 31-43, 2019.
  24. [24] S. Mathew and J. N. Mordeson, “Fuzzy Incidence Blocks and their Applications in Illegal Migration Problems,” New Math. Nat. Comput., Vol.13, No.3, pp. 245-260, 2017.
  25. [25] M. Akram, S. Sayed, and F. Smarandache, “Neutrosophic Incidence Graphs with Application,” Axioms, Vol.7, No.3, Article No.47, 2018.
  26. [26] M. Akram et al., “Application of Bipolar Neutrosophic Sets to Incidence Graphs,” Neutrosophic Sets Syst., Vol.27, pp. 180-200, 2019.
  27. [27] S. S. Hussain, R. J. Hussain, and M. V. Babu, “Neutrosophic Vague Incidence Graph,” Int. J. Neutrosophic Sci., Vol.12, No.1, pp. 29-38, 2021.
  28. [28] S. N. F. Mohamad et al., “Novel Concept of Interval-Valued Neutrosophic Incidence Graphs with Application,” Neutrosophic Sets Syst., Vol.43, pp. 61-81, 2021.
  29. [29] E. J. Cockayne and S. T. Hedetniemi, “Towards a Theory of Domination in Graphs,” Networks, Vol.7, No.3, pp. 247-261, 1977.
  30. [30] A. Somasundaram and S. Somasundaram, “Domination in Fuzzy Graphs – I,” Pattern Recognit. Lett., Vol.19, No.9, pp. 787-791, 1998.
  31. [31] A. Somasundaram, “Domination in Products of Fuzzy Graphs.” Int. J. Uncertain. Fuzziness Knowl.-Based Syst., Vol.13, No.2, pp. 195-204, 2005.
  32. [32] A. N. Gani and M. B. Ahamed, “Strong and Weak Domination in Fuzzy Graphs,” East Asian Math. J., Vol.23, No.1, pp. 1-8, 2007.
  33. [33] C. Natarajan and S. K. Ayyaswamy, “On Strong (Weak) Domination in Fuzzy Graphs,” Int. J. Math Comput. Sci., Vol.4, No.7, pp. 1035-1037, 2010.
  34. [34] S. Vimala and J. S. Sathya, “Some Results on Point Set Domination of Fuzzy Graphs,” Cybern. Inf. Technol., Vol.13, No.2, pp. 58-62, 2013.
  35. [35] O. T. Manjusha and M. S. Sunitha, “Total Domination in Fuzzy Graphs Using Strong Arcs,” Ann. Pure Appl. Math., Vol.9, No.1, pp. 23-33, 2014.
  36. [36] O. T. Manjusha and M. S. Sunitha, “Notes on Domination in Fuzzy Graphs,” J. Intell. Fuzzy Syst., Vol.27, No.6, pp. 3205-3212, 2014.
  37. [37] O. T. Manjusha and M. S. Sunitha, “Strong Domination in Fuzzy Graphs,” Fuzzy Inf. Eng., Vol.7, No.3, pp. 369-377, 2015.
  38. [38] O. T. Manjusha and M. S. Sunitha, “Connected Domination in Fuzzy Graphs Using Strong Arcs,” Ann. Fuzzy Math. Inform., Vol.10, No.6, pp. 979-994, 2015.
  39. [39] O. T. Manjusha and M. S. Sunitha, “The Strong Domination Alteration Sets in Fuzzy Graphs,” Int. J. Math. Appl., Vol.4, No.2-D, pp. 109-123, 2016.
  40. [40] O. T. Manjusha and M. S. Sunitha, “Coverings, Matchings and Paired Domination in Fuzzy Graphs Using Strong Arcs,” Iran. J. Fuzzy Syst., Vol.16, No.1, pp. 145-157, 2019.
  41. [41] C. Y. Ponnappan, S. B. Ahamed, and P. Surulinathan, “Edge Domination in Fuzzy Graphs – New Approach,” Int. J. IT Eng. Appl. Sci. Res., Vol.4, No.1, pp. 14-17, 2015.
  42. [42] K. M. Dharmalingam and M. Rani, “Equitable Domination in Fuzzy Graphs,” Int. J. Pure Appl. Math., Vol.94, No.5, pp. 661-667, 2014.
  43. [43] K. M. Dharmalingam and P. Nithya, “Excellent Domination in Fuzzy Graphs,” Bull. Int. Math. Virtual Inst., Vol.7, No.2, pp. 257-266, 2017.
  44. [44] P. Nithya and K. M. Dharmalingam, “Very Excellent Domination in Fuzzy Graphs,” Int. J. Comput. Appl. Math., Vol.12, No.1, pp. 313-326, 2017.
  45. [45] A. Bozhenyuk, S. Belyakov, M. Knyazeva, and I. Rozenberg, “On Computing Domination Set in Intuitionistic Fuzzy Graph,” Int. J. Comput. Intell. Syst., Vol.14, No.1, pp. 617-624, 2021.
  46. [46] I. Nazeer et al., “Domination of Fuzzy Incidence Graphs with the Algorithm and Application for the Selection of a Medical Lab,” Math. Probl. Eng., Vol.2021, Article No.6682502, 2021.
  47. [47] I. Nazeer et al., “Domination in Join of Fuzzy Incidence Graphs Using Strong Pairs with Application in Trading System of Different Countries,” Symmetry, Vol.13, No.7, Article No.1279, 2021.
  48. [48] P. Bhattacharya, “Some Remarks on Fuzzy Graphs,” Pattern Recognit. Lett., Vol.6, No.5, pp. 297-302, 1987.
  49. [49] S. Broumi et al., “Single Valued Neutrosophic Graphs,” J. New Theory, Vol.10, pp. 86-101, 2016.
  50. [50] M. Akram, “Single-Valued Neutrosophic Graphs,” Springer, 2018.

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Last updated on Sep. 29, 2023