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

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, mPFG models produce outcomes of fuzziness that are more effective. Additionally, we develop and analyze these kinds of mPFGs using examples and related theorems. Considering all those things together, we define saturation for a m-polar fuzzy graph (mPFG) 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 mPFG 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 mPFGs. Using the concept of α-saturation and α-saturation, block and bridge of mPFG are characterized. To identify the α-saturation and β-saturation mPFGs, two algorithms are designed and, using these algorithms, the saturated mPFG 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 mPFG is applicable. In this application, an appropriate location is determined for the allocation of a facility point.