Domination in Inverse Fuzzy Mixed Graphs with Application
Rahul Mondal
1
(
Department of Mathematics, Vivekananda Satavarshiki Mahavidyalaya, Manikpara, Jhargram -721513, West Bengal, India.
)
Ganesh Ghorai
2
(
Department of Applied Mathematics, Vidyasagar University, Midnapore - 721 102, West Bengal, India.
)
Keywords: Inverse fuzzy mixed dominating function, Inverse fuzzy mixed domination number, Upper inverse fuzzy mixed domination number, (g, h)- inverse fuzzy mixed dominating set.,
Abstract :
In the first part of our work, the concepts of inverse fuzzy mixed dominating set (IFMDS) and inverse fuzzy mixed domination number (IFMDN) have been explored for inverse fuzzy mixed graphs (IFMGs). Based on these ideas, the inequality $\mathcal{R}^{ir} \leq \mathcal{R}^{\gamma} \leq \mathcal{R}^{i} \leq \mathcal{R}^{\beta_{0}} \leq \mathcal{R}^{\Gamma} \leq \mathcal{R}^{IR}$ has been established for an IFMG $\mathcal{R}$. In the second part of our work, we have explored the concept of $(g, h)$-IFMDS, which is defined as a fuzzy subset of the membership function of nodes. Most importantly, the relation $ (g,h)$-IN$(\mathcal{R}$) $\leq (g,h)$-DN$(\mathcal{R}$)$\leq (g,h)$-UDN$(\mathcal{R}$)$\leq (g,h)$-UIN$(\mathcal{R})$ has been established for an IFMG $\mathcal{R}$. At the end, a real-life application of the concept of $(g, h)$-IFMDS is given.
[1] Rosenfeld A. Fuzzy graphs. In: Zadeh LA, Fu K-S, Tanaka K, Shimura M. (eds) Fuzzy sets and their applications. New York: Academic Press; 1975. p.77-95. DOI: https://doi.org/10.1016/B978-0-12-775260- 0.50008-6
[2] Akram M, Sattar A, Saeid AB. Competition graphs with complex intuitionistic fuzzy information. Granular Computing. 2022; 7: 25-47. DOI: https://doi.org/10.1007/s41066-020-00250-2
[3] Akram M, Siddique S, Alcantud JCR. Connectivity indices of m-polar fuzzy network model, with an application to a product manufacturing problem. Artificial Intelligence Review. 2023; 56: 7795-7838. DOI: https://doi.org/10.1007/s10462-022-10360-9
[4] Bhutani KR, Rosenfeld A. Geodesics in fuzzy graphs. Electronic Notes in Discrete Mathematics. 2003; 15: 49-52. DOI: https://doi.org/10.1016/S1571-0653(04)00526-8
[5] Brigham RC, McMorris FR, Vitray RP. Tolerance competition graphs. Linear Algebra and its Application. 1995; 217: 41-52. DOI: https://doi.org/10.1016/0024-3795(94)00059-M
[6] Cockayne EJ, Mynhardt CM. The sequence of upper and lower domination, independence and irredundance numbers of a graph. Discrete Mathematics. 1993; 122(1-3): 89-102. DOI: https://doi.org/10.1016/0012-365X(93)90288-5
[7] Haynes TW, Hedetniemi ST, Slater PJ. Fundamentals of Domination in Graphs. Cleveland: CRC Press; 1998.
[8] Mathew S, Sunitha, MS. Types of arcs in a fuzzy graph. Information Sciences. 2009; 179(11): 1760-1768. DOI: https://doi.org/10.1016/j.ins.2009.01.003
[9] Mordeson JN, Nair PS. Fuzzy Graphs and Fuzzy Hypergraphs. New York: Physica-Verlag; 2000. https://doi.org/10.1007/978-3-7908-1854-3
[10] Borzooei RA, Almallah R, Jun YB, Ghaznavi H. Inverse fuzzy graphs with applications. New Mathematics and Natural Computation. 2020; 16(02): 397-418. DOI: https://doi.org/10.1142/S1793005720500246
[11] Almallah R, Borzooei RA, Jun YB. Domination numbers of inverse fuzzy graphs with application in decision-making problems. New Mathematics and Natural Computation. 2022; 18(01): 19-42. DOI: https://doi.org/10.1142/S179300572250003X
[12] Poulik S, Ghorai G. New concepts of inverse fuzzy mixed graphs and its application. Granular computing. 2021; 7(3): 549-559. DOI: https://doi.org/10.1007/s41066-021-00284-0
[13] Das K, Naseem U, Samanta S, Khan SK, De K. Fuzzy mixed graphs and its application to identification of COVID19 affected central regions in India. Journal of Intelligent & Fuzzy Systems. 2020; 40(1):1051-1064. DOI: https://doi.org/10.3233/JIFS-201249
[14] Mondal R, Ghorai G. On the m-step inverse fuzzy mixed competition graphs with application. International Journal of Mathematics for Industry. 2024; 16(01): 2450006. DOI:
https://doi.org/10.1142/S2661335224500060
[15] Mondal R, Ghorai G. Inverse fuzzy mixed planar graphs with application. International Journal of Applied and Computational Mathematics. 2024; 10(131). DOI: https://doi.org/10.1007/s40819-024-01764-y
[16] Mondal R, Ghorai G. Wiener index of inverse fuzzy mixed graphs with application in education system. Journal of Applied Mathematics and Computing. 2025; 71: 725-742. DOI:
https://doi.org/10.1007/s12190-024-02263-5
[17] Wiener H. Structural determination of paraffin boiling points. Journal of the American Chemical Society. 1947; 69(1): 17-20. DOI: https://doi.org/10.1021/ja01193a005
[18] Selvakumar K, Gangaeswari P, Arunkumar G. The wiener index of the zero-divisor graph of a finite commutative ring with unity. Discrete Applied Mathematics. 2022; 311: 72-84. DOI: https://doi.org/10.1016/j.dam.2022.01.012
[19] Somasundaram A, Somasundaram S. Domination in fuzzy graphs-I. Pattern Recognition Letters. 1998; 19(9): 787-791. DOI: https://doi.org/10.1016/S0167-8655(98)00064-6
[20] Samanta S, Akram M, Pal M. m-Step fuzzy competition graphs. Journal of Applied Mathematics and Computing. 2015; 47: 461-472. DOI: https://doi.org/10.1007/s12190-014-0785-2