Abstract
Molodtsov’s theory of soft sets is free from the parameterizations insufficiency of fuzzy set theory. Type-2 soft set as an extension of a soft set has an essential mathematical structure to deal with parametrizations and their primary relationship. Fuzzy type-2 soft models play a key role to study the partial membership and uncertainty of objects along with underlying and primary set of parameters. In this research article, we introduce the concept of fuzzy type-2 soft set by integrating fuzzy set theory and type-2 soft set theory. We also introduce the notions of fuzzy type-2 soft graphs, regular fuzzy type-2 soft graphs, irregular fuzzy type-2 soft graphs, fuzzy type-2 soft trees, and fuzzy type-2 soft cycles. We construct some operations such as union, intersection, AND, and OR on fuzzy type-2 soft graphs and discuss these concepts with numerical examples. The fuzzy type-2 soft graph is an efficient model for dealing with uncertainty occurring in vertex-neighbors structure and is applicable in computational analysis, applied intelligence, and decision-making problems. We study the importance of fuzzy type-2 soft graphs in chemical digestion and national engineering services.
Highlights
IntroductionFuzzy set theory has its remarkable origin to the work of Zadeh [1] in 1965 to interact with vagueness and imprecision between absolute true and absolute false. e range of the values in a fuzzy set lies in [0, 1]. is remarkable discovery of fuzzy set theory paved a different way for dealing with uncertainties in various domains of science and technology
Fuzzy set theory has its remarkable origin to the work of Zadeh [1] in 1965 to interact with vagueness and imprecision between absolute true and absolute false. e range of the values in a fuzzy set lies in [0, 1]. is remarkable discovery of fuzzy set theory paved a different way for dealing with uncertainties in various domains of science and technology.Graph theory is moving quickly into the mainstream of mathematics, primarily due to its applications in engineering, communication networks, computer science, and artificial intelligence
E main contribution of this study is as follows: (1) e present study introduces the mathematical approaches of vertex-neighbors-based type-2 soft set and vertex-neighbors-based type-2 soft graphs under fuzzy environment. e notions of fuzzy type-2 soft graphs, regular fuzzy type-2 soft graphs, irregular fuzzy type-2 soft graphs, fuzzy type-2 soft trees, and fuzzy type-2 soft cycles are discussed with certain operations and numerical examples
Summary
Fuzzy set theory has its remarkable origin to the work of Zadeh [1] in 1965 to interact with vagueness and imprecision between absolute true and absolute false. e range of the values in a fuzzy set lies in [0, 1]. is remarkable discovery of fuzzy set theory paved a different way for dealing with uncertainties in various domains of science and technology. Akram and Zafar [22] introduced various hybrid models based on fuzzy sets, soft sets, and rough sets. Researchers are actively working on interval type-2 fuzzy arc lengths [27], trapezoidal interval type-2 fuzzy soft sets [28], total uniformity of graph under fuzzy soft information [29], fuzzy soft cycles [30], and fuzzy soft β−coverings All these existing models have the same restriction that one cannot freely select the parameters. To deal with partial membership of objects, the main focus of this study is to introduce a hybrid model by combining fuzzy set theory with type-2 soft sets. (1) e present study introduces the mathematical approaches of vertex-neighbors-based type-2 soft set and vertex-neighbors-based type-2 soft graphs under fuzzy environment.
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