Generalized m-Polar Fuzzy Planar Graph and Its Application

Planarity of crisp graphs is a well-established field, whereas planarity within a fuzzy framework has seen recent development and extensive exploration. In an m -polar fuzzy graph ( m PFG), each node and edge is associated with m -components, connected through minimal relationships. However, if one desires to incorporate maximum, average, or other intermediate relationships between nodes and edges, the m PFG concept becomes inadequate as in the m -polar fuzzy model, only minimum relation is considered. To address this limitation, a generalized model of m PFG is introduced in this article, allowing for a broader range of relationships to be considered simultaneously. This paper also discusses the properties of generalized m -polar fuzzy environments and generalized m -polar fuzzy graphs ( Gm PFGs), highlighting their isomorphism. Several significant findings and insights are presented in this paper. The article delves into the properties and characteristics of generalized m -polar fuzzy planar graphs ( Gm PFPGs) and explores various intriguing aspects related to them. Additionally, a novel concept of a generalized m -polar fuzzy dual graph ( Gm PFDG) is introduced, derived from Gm PFPGs. The paper establishes a relationship between the dual of a Gm PFG and Gm PFG, examining their properties in the context of dual Gm PFPGs. Lastly, the article discusses an illustrative example of a social group network problem assessing the group’s activity based on attributes such as cooperation, team spirit, awareness, controlling power, good behaviour, and creativity.