Khan Waheed Ahmad, Niaz Arsh E Mah, Nguyen Trung Tuan, Pham Minh Hoan, Tong Thi Minh Ngoc, Pham Hai Van
Division of Science and Technology, Department of Mathematics, University of Education Lahore, Attock Campus, Attock, Punjab, 43600, Pakistan.
National Economics University, 207 Giai Phong Street, Hai Ba Trung, District, Hanoi, 11616, Vietnam.
Sci Rep. 2025 Jul 1;15(1):21809. doi: 10.1038/s41598-025-02880-5.
Fuzzy sets and soft sets are two distinct mathematical tools used for modeling real-world problems involving uncertainty. In this study, we combine these models to address vagueness and uncertainty within the framework of planar graphs. We initiate the notion of fuzzy soft planar graphs and explore their applications in image processing. Initially, several key terms related to these graphs such as fuzzy soft multi-graphs (FSMGs) and intersecting values of FSMGs are established. Based on these, we introduce the concepts of fuzzy soft planar graphs (FSPGs) and explore their various characterizations. The concept of dual FSPGs is also initiated and examined. Furthermore, FSPGs are studied through various types of edges including effective, considerable and non-considerable edges. Several types of faces such as fuzzy soft face, strong fuzzy soft face and weak fuzzy soft face are also analyzed. A detailed critical analysis is conducted between Kuratowski's theorem and FSPGs to highlight their respective similarities and differences. Additionally, we demonstrate the applicability of FSPGs in image segmentation and representation. In this context, we propose a model supported by an algorithm for converting a traditional crisp image of the Asakusa Tokyo Pyramid into a fuzzy soft image pyramid. Finally, a comparative analysis between our proposed model and the traditional fuzzy planar graph (FPG) model is conducted for segmenting the selected image. The results provide evidence that our FSPG-based model outperforms the traditional FPG approach.
