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

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.