Generalized <italic>m</italic>-Polar Fuzzy Planar Graph and Its Application

Abstract

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 ( $\text{G}m$ 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 ( $\text{G}m$ PFPGs) and explores various intriguing aspects related to them. Additionally, a novel concept of a generalized $m$ -polar fuzzy dual graph ( $\text{G}m$ PFDG) is introduced, derived from $\text{G}m$ PFPGs. The paper establishes a relationship between the dual of a $\text{G}m$ PFG and $\text{G}m$ PFG, examining their properties in the context of dual $\text{G}m$ 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.

Keywords

• Fuzzy graph
• m-polar fuzzy graph
• planarity of generalized m-polar fuzzy graph
• dual of generalized m-polar fuzzy planar graph
• fuzzy face