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

The third volume of “Advancing Uncertain Combinatorics through Graphization, Hyperization, and Uncertainization” delves into groundbreaking developments in uncertain combinatorics and set theory. It highlights methodologies such as graphization, hyperization, and uncertainization, seamlessly integrating fuzzy, neutrosophic, soft, and rough set theories to address uncertainty in complex systems. These innovations bridge the gap between combinatorics and graph theory, resulting in novel structures like hypergraphs and superhypergraphs. The first chapter introduces Upside-Down Logic, a framework that reverses truths and falsehoods based on contextual shifts. The second chapter focuses on Local-Neutrosophic Logic, which incorporates the concept of locality to better model uncertainty. In the third chapter, advanced set-theoretic offsets are explored, while the fourth chapter extends Plithogenic Graphs to innovative structures such as Plithogenic OffGraphs. The fifth chapter expands matroid theory with the introduction of Neutrosophic Closure Matroids, which integrate uncertainty and indeterminacy. Finally, the sixth chapter examines graph parameters, including Superhypertree-width and Neutrosophic Tree-width, to deepen the understanding of graph characteristics in uncertain contexts. This volume establishes a comprehensive framework for addressing mathematical and real-world uncertainties, laying a strong foundation for future innovations in combinatorics, set theory, and graph theory.