A Comprehensive Survey of Soft Graphs and Rough Graphs

Graph theory provides a fundamental mathematical framework for representing and analyzing networks through vertices, edges, and their structural relationships. However, classical graph models are often insufficient for systems characterized by uncertainty, vagueness, incomplete information, or parameter-dependent relationships. This book presents a comprehensive and systematic survey of two major approaches for extending graph theory to such environments: Soft Graphs and Rough Graphs. Soft Graphs associate parameters with subgraphs of an ambient graph, allowing network structures to vary according to attributes, criteria, or contextual information, whereas Rough Graphs employ lower and upper approximations to distinguish structural components that are certainly present, possibly present, or situated within a boundary region. Beginning with the foundations of soft sets, rough sets, uncertain sets, graphs, HyperGraphs, and SuperHyperGraphs, the book develops an extensive taxonomy of Soft Graph and Rough Graph models. It examines numerous uncertainty-based, structural, and parameter-based extensions, including HyperSoft, SuperHyperSoft, fuzzy, intuitionistic fuzzy, neutrosophic, plithogenic, bipolar, multipolar, interval-valued, Pythagorean fuzzy, and q-rung orthopair formulations, as well as Soft HyperGraphs, Soft SuperHyperGraphs, Rough HyperGraphs, and Rough SuperHyperGraphs. The treatment also encompasses important special classes and structural properties, including soft and rough trees, regular and complete graphs, connectivity, Hamiltonian structures, paths, cycles, and covering-based models. By organizing these diverse frameworks within a unified graph-theoretic perspective, the book clarifies their mathematical foundations, structural relationships, and modeling capabilities, providing a reference for researchers working on uncertain networks, generalized graph structures, decision modeling, and related areas of applied mathematics.