Strong Fuzzy Planar Graphs: Theoretical Foundations and Applications in Traffic Network Planning Under Uncertainty

Abstract

This paper introduces strong fuzzy planar graphs (SFPLGs), extending fuzzy graph theory with a quantitative planarity measure $\vartheta(\Omega)=\frac{1}{1+\sum_{i=1}^{n} \Lambda\left(\theta_{i}\right)}$ that classifies networks as strong or weak based on controlled edge crossings. Formal definitions establish fuzzy strong-weak arcs, face memberships, dual graph constructions, and key theorems, including the 0.67 threshold that prohibits strong-strong intersections and maintains planarity values through isomorphism. Theoretical results reconcile classical Kuratowski's graphs with fuzzy gradations. The framework proves effective in the planning of the traffic network, modelling a 10 urban core intersections with vertex memberships of $0.70-0.90$ and edge strengths revealing connectivity bottlenecks $C O N N$ limited by weak segments $5-10,9-10$ at 0.70. Strong edges form reliable backbones, while weak links identify upgrade priorities, balancing costs with necessary intersections in environments with uncertain capacities. SFPLGs provide transportation engineers with interpretable tools for durable infrastructure design, with zero-crossing embeddings verifying planarity and edge analysis guiding investments. Future work will investigate dynamic traffic data, multi-layer networks, and intuitionistic variants.