Directed Fuzzy Edge Graphs Under q-ROF Environment: A Framework for Optimal Pathfinding

This paper introduces a novel framework of Directed Edge q-Rung Orthopair Fuzzy Graphs (DEq-ROFGs), where graph vertices are crisp, and edges are characterized by q-rung orthopair fuzzy numbers (q-ROFNs). This structure captures the uncertainty in edge relationships while retaining deterministic node identities, making it ideal for applications in uncertain environments such as social networks, supply chains, healthcare systems, and recommendation systems. The paper defines foundational properties of DEq-ROFGs including subgraphs, completeness, and various degree-based metrics, and it establishes a proposition regarding the balance between in-degrees and out-degrees. The core contribution is a novel path-finding algorithm based on Hamacher operators and an improved score function, which identifies optimal paths between nodes under uncertainty. Unlike classical algorithms, it considers the suitability of a path, not just its length. Applied to an emergency road network scenario, the algorithm successfully determines the optimal route for service vehicles, and the choice between these routes can be made based on the score of the resulting path length. Comparative simulations show their effectiveness over traditional methods. Further analysis shows that increasing the q-value reduces both path score and length, and that Einstein operators yield higher destination scores than Hamacher and Dombi, confirming the model’s adaptability and robustness.