On the Concept of Outerplanarity in Pythagorean Fuzzy Graph Structures

Abstract

This study introduces the concept of Pythagorean fuzzy outerplanar graphs (PFOGs), integrating the theory of Pythagorean fuzzy sets with outerplanar graph structures. In PFOGs, uncertainty is modeled by assigning to each vertex and edge a degree of membership and a degree of non-membership satisfying the Pythagorean condition, where the sum of the squares of these degrees does not exceed one. This representation provides a more flexible and expressive mechanism for handling ambiguity and partial information compared to classical fuzzy outerplanar graphs. Outerplanar graphs, as a significant subclass of planar graphs, possess structural properties that are valuable in simplifying complex network systems. In this work, we generalize crisp outerplanar graphs within the Pythagorean fuzzy framework and examine their structural behavior. We introduce vertex deleted (VD) and edge deleted (ED) Pythagorean fuzzy outerplanar subgraphs, along with the notions of maximum and maximal Pythagorean fuzzy outerplanar subgraphs under vertex and edge deletions. Furthermore, a novel concept of the dual of Pythagorean fuzzy outerplanar graphs is formulated and investigated. Theoretical developments are established through formal definitions, illustrative examples, and rigorous theorems.