Generalized <i>m</i>-Polar Fuzzy Planar Graph and Its Application
Uttam Mondal, Tanmoy Mahapatra, Qin Xin, Madhumangal Pal
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 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> -polar fuzzy graph ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> PFG), each node and edge is associated with <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> -components, connected through minimal relationships. However, if one desires to incorporate maximum, average, or other intermediate relationships between nodes and edges, the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> PFG concept becomes inadequate as in the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> -polar fuzzy model, only minimum relation is considered. To address this limitation, a generalized model of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> 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 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> -polar fuzzy environments and generalized <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> -polar fuzzy graphs (G <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> PFGs), highlighting their isomorphism. Several significant findings and insights are presented in this paper. The article delves into the properties and characteristics of generalized <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> -polar fuzzy planar graphs (G <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> PFPGs) and explores various intriguing aspects related to them. Additionally, a novel concept of a generalized <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> -polar fuzzy dual graph (G <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> PFDG) is introduced, derived from G <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> PFPGs. The paper establishes a relationship between the dual of a G <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> PFG and G <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> PFG, examining their properties in the context of dual G <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> 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.