Utilizing m -Polar Fuzzy Saturation Graphs for Optimized Allocation Problem Solutions

Abdulaziz M. Alanazi, Ghulam Muhiuddin (), Bashair M. Alenazi, Tanmoy Mahapatra and Madhumangal Pal
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Abdulaziz M. Alanazi: Department of Mathematics, Faculty of Science, University of Tabuk, P.O. Box 741, Tabuk 71491, Saudi Arabia
Ghulam Muhiuddin: Department of Mathematics, Faculty of Science, University of Tabuk, P.O. Box 741, Tabuk 71491, Saudi Arabia
Bashair M. Alenazi: Department of Mathematics, Faculty of Science, University of Tabuk, P.O. Box 741, Tabuk 71491, Saudi Arabia
Tanmoy Mahapatra: Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore 721 102, India
Madhumangal Pal: Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore 721 102, India

Mathematics, 2023, vol. 11, issue 19, 1-18

Abstract: It is well known that crisp graph theory is saturated. However, saturation in a fuzzy environment has only lately been created and extensively researched. It is necessary to consider m components for each node and edge in an m -polar fuzzy graph. Since there is only one component for this idea, we are unable to manage this kind of circumstance using the fuzzy model since we take into account m components for each node as well as edges. Again, since each edge or node only has two components, we are unable to apply a bipolar or intuitionistic fuzzy graph model. In contrast to other fuzzy models, m PFG models produce outcomes of fuzziness that are more effective. Additionally, we develop and analyze these kinds of m PFGs using examples and related theorems. Considering all those things together, we define saturation for a m -polar fuzzy graph ( m PFG) with multiple membership values for both vertices and edges; thus, a novel approach is required. In this context, we present a novel method for defining saturation in m PFG involving m saturations for each element in the membership value array of a vertex. This explains α -saturation and β -saturation. We investigate intriguing properties such as α -vertex count and β -vertex count and establish upper bounds for particular instances of m PFGs. Using the concept of α -saturation and α -saturation, block and bridge of m PFG are characterized. To identify the α -saturation and β -saturation m PFGs, two algorithms are designed and, using these algorithms, the saturated m PFG is determined. The time complexity of these algorithms is O ( | V | 3 ) , where | V | is the number of vertices of the given graph. In addition, we demonstrate a practical application where the concept of saturation in m PFG is applicable. In this application, an appropriate location is determined for the allocation of a facility point.

Keywords: m -polar fuzzy graph; saturated fuzzy graph; ? -saturation; ? -saturation; saturation in m -polar fuzzy graph (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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