Advancing Uncertain Combinatorics through Graphization, Hyperization, and Uncertainization: Fuzzy, Neutrosophic, Soft, Rough, and Beyond (Second Volume)

The second volume of "Advancing Uncertain Combinatorics through Graphization, Hyperization, and Uncertainization" presents significant advancements in uncertain combinatorics through methodologies such as graphization and hyperization. It seamlessly integrates fuzzy, neutrosophic, soft, and rough set theories with combinatorics and graph theory to effectively model and manage uncertainties. This integration facilitates the exploration of novel structures, including hypergraphs and superhypergraphs, breaking new ground in both theoretical and practical applications across mathematical and real-world systems. The chapters span a diverse range of topics, beginning with an in-depth analysis of relationships among graph types such as fuzzy, neutrosophic, and plithogenic graphs, with a focus on their applications. The work also investigates advanced graph structures, including directed and bidirected superhypergraphs as well as soft and rough superhypergraphs, emphasizing their role in managing uncertainty. Computational challenges, such as partitioning and optimization in superhypergraphs, are addressed, with classical problems like the Traveling Salesman Problem adapted to this new context. Algorithmic approaches specific to Plithogenic Graphs are introduced, addressing complexity and validity. The volume further examines "bunch graphs," where nodes represent groups rather than individual entities, offering a robust model for collaboration and competition. Concepts like even-hole-free and planar graphs are extended to uncertain frameworks, exploring their relevance to uncertainty modeling. Additionally, new graph classes such as Turiyam Neutrosophic Soft Graphs and General Plithogenic Soft Graphs are introduced, tailored for handling uncertainty in diverse applications. The volume concludes with a comprehensive survey of trees, forests, and paths in fuzzy and neutrosophic contexts, providing valuable classifications and insights into graph structures under uncertainty. Overall, this work serves as a foundational framework for advancing uncertain combinatorics, setting the stage for future research and practical applications.