INVERSE CLIQUE REGULAR DOMINATION NUMBER IN FUZZY GRAPHS

Fuzzy graphs have a prominent place in the mathematical modelling of the problems due to the simplicity of representing the relationships between topics. Gradually, with the development of science and in encountering with complex problems and the existence of multiple relationships between variables, the need to consider fuzzy graphs with multiple relationships was felt. With the introduction of the graph structures, there was better flexibility than the graph in dealing with problems. By combining a graph structure with a fuzzy graph, a fuzzy graph structure was introduced that increased the decision-making power of complex problems based on uncertainties. The previous definitions restrictions in fuzzy graphs have made us present new definitions in the fuzzy graph structure. The domination of fuzzy graphs has many applications in other sciences including computer science, intelligent systems, psychology, and medical sciences. Hence, in this paper, first we study the dominating set in a fuzzy graph structure from the perspective of the domination number of its fuzzy relationships. Likewise, we determine the domination in terms of neighborhood, degree, and capacity of vertices with some examples. Finally, applications of domination are introduced in fuzzy graph structure.

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Some Results on Point Set Domination of Fuzzy Graphs

Cybernetics and Information Technologies ◽

10.2478/cait-2013-0014 ◽

2013 ◽

Vol 13 (2) ◽

pp. 58-62

Author(s):

S. Vimala ◽

J. S. Sathya

Keyword(s):

Dominating Set ◽

Domination Number ◽

Fuzzy Graph ◽

Minimal Point ◽

Minimum Cardinality ◽

Fuzzy Graphs ◽

Point Set

Abstract Let G be a fuzzy graph. Let γ(G), γp(G) denote respectively the domination number, the point set domination number of a fuzzy graph. A dominating set D of a fuzzy graph is said to be a point set dominating set of a fuzzy graph if for every S⊆V-D there exists a node d∈D such that 〈S ∪ {d}〉 is a connected fuzzy graph. The minimum cardinality taken over all minimal point set dominating set is called a point set domination number of a fuzzy graph G and it is denoted by γp(G). In this paper we concentrate on the point set domination number of a fuzzy graph and obtain some bounds using the neighbourhood degree of fuzzy graphs.

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On (r,s)-Fuzzy Domination in Fuzzy Graphs

New Mathematics and Natural Computation ◽

10.1142/s1793005716500010 ◽

2016 ◽

Vol 12 (01) ◽

pp. 1-10

Author(s):

S. Arumugam ◽

Kiran Bhutani ◽

L. Sathikala

Keyword(s):

Dominating Set ◽

Domination Number ◽

Fuzzy Graph ◽

Fuzzy Subset ◽

Fuzzy Graphs ◽

Finite Set

Let [Formula: see text] be a fuzzy graph on a finite set [Formula: see text] Let [Formula: see text] and [Formula: see text] A fuzzy subset [Formula: see text] of [Formula: see text] is called an [Formula: see text]-fuzzy dominating set ([Formula: see text]-FD set) of G if [Formula: see text] Then [Formula: see text] is called the [Formula: see text]-fuzzy domination number of [Formula: see text] where the minimum is taken over all [Formula: see text]-FD sets [Formula: see text] of [Formula: see text] In this paper we initiate a study of this parameter and other related concepts such as [Formula: see text]-fuzzy irredundance and [Formula: see text]-fuzzy independence. We obtain the [Formula: see text]-fuzzy domination chain which is analogous to the domination chain in crisp graphs.

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Wiener Index of Fuzzy Graph by Using Strong Domination

Asian Journal of Probability and Statistics ◽

10.9734/ajpas/2020/v10i330247 ◽

2021 ◽

pp. 13-24

Author(s):

Saqr H. AL- Emrany ◽

Mahiuob M. Q. Shubatah

Keyword(s):

Domination Number ◽

Wiener Index ◽

New Method ◽

Fuzzy Graph ◽

Fuzzy Graphs ◽

The Relationship

Aims/ Objectives: This paper presents a new method to calculate the Wiener index of a fuzzy graph by using strong domination number s of a fuzzy graph G. This method is more useful than other methods because it saves time and eorts and doesn't require more calculations, if the edges number is very larg (n). The Wiener index of some standard fuzzy graphs are investigated. At last, we nd the relationship between strong domination number s and the average (G) of a fuzzy graph G was studied with suitable examples.

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The Results on Vertex Domination in Fuzzy Graphs

10.20944/preprints201804.0085.v2 ◽

2019 ◽

Author(s):

Mohammadesmail Nikfar

Keyword(s):

Domination Number ◽

Fuzzy Graph ◽

Fuzzy Graphs ◽

Graph Property ◽

The Relationship

We do fuzzification the concept of domination in crisp graph by using membership values of nodes, $\alpha$-strong arcs and arcs. In this paper, we introduce a new variation on the domination theme which we call vertex domination. We determine the vertex domination number $\gamma_v$ for several classes of fuzzy graphs, specially complete fuzzy graph and complete bipartite fuzzy graphs. The bounds is obtained for the vertex domination number of fuzzy graphs. Also the relationship between $M$-strong arcs and $\alpha$-strong is obtained. In fuzzy graphs, monotone decreasing property and monotone increasing property is introduced. We prove the vizing's conjecture is monotone decreasing fuzzy graph property for vertex domination. we prove also the Grarier-Khelladi's conjecture is monotone decreasing fuzzy graph property for it. We obtain Nordhaus-Gaddum (NG) type results for these parameters. The relationship between several classes of operations on fuzzy graphs with the vertex domination number of them is studied.

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Graphical Analysis of Covering and Paired Domination in the Environment of Neutrosophic Information

Mathematical Problems in Engineering ◽

10.1155/2021/5518295 ◽

2021 ◽

Vol 2021 ◽

pp. 1-12

Author(s):

Sami Ullah Khan ◽

Abdul Nasir ◽

Naeem Jan ◽

Zhen-Hua Ma

Keyword(s):

Graph Theory ◽

Optimization Problems ◽

Domination Number ◽

Real Life ◽

Graphical Analysis ◽

Fuzzy Graphs ◽

Life Problems ◽

Neutrosophic Graph ◽

Paired Domination ◽

Intuitionistic Fuzzy Graphs

Neutrosophic graph (NG) is a powerful tool in graph theory, which is capable of modeling many real-life problems with uncertainty due to unclear, varying, and indeterminate information. Meanwhile, the fuzzy graphs (FGs) and intuitionistic fuzzy graphs (IFGs) may not handle these problems as efficiently as NGs. It is difficult to model uncertainty due to imprecise information and vagueness in real-world scenarios. Many real-life optimization problems are modeled and solved using the well-known fuzzy graph theory. The concepts of covering, matching, and paired domination play a major role in theoretical and applied neutrosophic environments of graph theory. Henceforth, the current study covers this void by introducing the notions of covering, matching, and paired domination in single-valued neutrosophic graph (SVNG) using the strong edges. Also, many attention-grabbing properties of these concepts are studied. Moreover, the strong covering number, strong matching number, and the strong paired domination number of complete SVNG, complete single-valued neutrosophic cycle (SVNC), and complete bipartite SVNG are worked out along with their fascinating properties.

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Perfect Dominating Set of An Interval-Valued Fuzzy Graphs

Asian Journal of Probability and Statistics ◽

10.9734/ajpas/2021/v11i430273 ◽

2021 ◽

pp. 35-46

Author(s):

Faisal M. AL-Ahmadi ◽

Mahiuob M. Q. Shubatah

Keyword(s):

Network Theory ◽

Dominating Set ◽

Domination Number ◽

Fuzzy Graphs ◽

Relationship Of ◽

The Relationship ◽

Interval Valued

Aims/ Objectives: Perfect domination is very much useful in network theory, Electrical stations and several fields of mathematics. In This paper, perfect domination in an intervalvalued fuzzy graphs is defined and studied. Some bounds on perfect domination number γp(G) are provided for several interval-valued fuzzy graphs, such as complete, wheel and star,.. etc. Furthermore, the relationship of γp(G): with some other known parameters in interval-valued fuzzy graphs investigated with some suitable examples.

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