Nikfar Domination in Neutrosophic Graphs

Many various using of this new-born fuzzy model for solving real-world problems and urgent requirements involve introducing new concept for analyzing the situations which leads to solve them by proper, quick and efficient method based on statistical data. This gap between the model and its solution cause that we introduce nikfar domination in neutrosophic graphs as creative and effective tool for studying a few selective vertices of this model instead of all ones by using special edges. Being special selection of these edges affect to achieve quick and proper solution to these problems. Domination hasn’t ever been introduced. So we don’t have any comparison with another definitions. The most used graphs which have properties of being complete, empty, bipartite, tree and like stuff and they also achieve the names for themselves, are studied as fuzzy models for getting nikfar dominating set or at least becoming so close to it. We also get the relations between this special edge which plays main role in doing dominating with other special types of edges of graph like bridges. Finally, the relation between this number with other special numbers and characteristic of graph like order are discussed.


Introduction
Neutrosophy as a newly-born science is a branch of philosophy that studies the origin, nature and scope of neutralities.In 1965, Zadeh introduced "fuzzy set" by the concept of degree of truth membership [12].In 1986, Atanassow introduced "intuitionistic fuzzy set" by adding the concept of degree of false membership to the fuzzy set [2].In 1995, Smarandache introduced "neutrosophic set" by adding the concept of degree of indeterminate membership to the intuitionistic fuzzy set [10].There are three different types of definitions of a neutrosophic graph [1,4,8].Broumi et al. [4] and Shah-Hussain [8] introduced two different definitions of neutrosophic graph by generalizations of intuitionistic fuzzy graph [9].Akram and shahzadi introduced neutrosophic graph by using concept of neutrosophic set [1].They also highlighted [1] some flaws in the definitions of Broumi et al. [4] and Shah-Hussain [8].They introduced some counterexamples which state the complement of a neutrosophic graph isn't always a neutrosophic graph ( [1], Example 3.5, pp.22, 23) by using Shah-Hussain's definition of neutrosophic graph [8] and we even have much bad situations if we used Broumi et al.'s definition of neutrosophic graph ( [4], Definition 3.1 p. 89) beacuse of not only we don't have complement of a neutrosophic graphs ( [1], Example 3.2, p. 21) but also we don't have join of them( [1], Example 3.3,pp. 21,22).Moreover, they introduced binary operations cartesian product, composition, union, join, cross, lexicographic, strong product and unary operation complement along with proofs which show these operations hold neutrosophic property of graphs [1].In other words, the new graph is produced by these operations, is also a neutrosophic graph.Regarding these points, we use the definition of Akram and Shahzadi ([1], Definition 2.2, pp. 2, 3) as the main framework for our own study.The study behaviors of modeling is of spotlight by using few parameters.Some parameters are so close to others one.if we defined being "so close" concept properly by adding some extra properties more than existence of edge between them, we would achieve the useful tool.This tool would cause solving real-world problems by deleting useless data and focusing on a few one.This leads to the concept of domination in modeling.Domination hasn't ever been introduced on any kind of neutrosophic graphs.Regarding these points, the aim of this paper is to introduce the notion of domination in this new-born fuzzy model.It is a normal question about effects of dominations in neutrosophic graphs.From here comes the main motivation for this and in this regard, we have considered some routine and fundamental framework for studying this concept.

Preliminaries
We provide some basic background for the paper in this section.Definition 2.1.(Fuzzy Set, [12]) Let V be a given set.The function A : V → [0, 1] is called a fuzzy set on V. Definition 2.2.(Neutrosophic Set, [11]) Let V be a given set.A neutrosophic set A in V is characterized by a truth membership function T A (x), an indeterminate membership function I A (x) and a false membership function F A (x).The functions T A (x), I A (x), and F A (x) are fuzzy sets on V.That is, Notation 2.3.Some special notations frequently appear in this paper.In what follows, we introduce them.Let V be a given set.For the sake of simplicity, we only use the notation E for the representation of the following set on V. E ⊆ {A|A ⊆ V, |A| = 2 It means A has only two elements}, where |A| means cardinality of A. By Analogous to this points, the notation E i is corresponded to V i .Definition 2.4.(Neutrosophic Graph, [11], pp. 2, 3) Let V be a given set.Also, assume E be a given set with respect to V. A neutrosophic graph is a pair Definition 2.5.(Complete Neutrosophic Graph, [11], p. 3) Let G = (A, B) be a neutrosophic graph on a given set V. G is called complete if the following conditions are satisfied: for all {x, y} ∈ E. for all {x, y} ∈ E.
d. order, if was be either of T −order, I−order, and F −order.b.I-acyclic, if there wasn't a I-path P from x to y, with only exception x = y., for all x ∈ V.
c. F-acyclic, if there wasn't a F-path P from x to y, with only exception x = y., for all x ∈ V.
d. acyclic, if it was either of T −acyclic, I−acyclic, and F −acyclic.
Definition 2.12.(Forest) Let G = (A, B) be a neutrosophic graph on a given set V. Then G is called the a. T-forest, if G was T-acyclic and there is a spanning neutrosophic graph F such that for all edge xy out of F, there is a T-path P from x to y, how whose strength greater than T B (xy).
b. I-forest, if G was I-acyclic and there is a spanning neutrosophic graph F such that for all edge xy out of F, there is a I-path P from x to y, how whose strength greater than I B (xy).
c. F-forest, if G was F-acyclic and there is a spanning neutrosophic graph F such that for all edge xy out of F, there is a F-path P from x to y, how whose strength greater than F B (xy). b.I-tree, if G was a I-forest such that there is a I-path P from x to y, for all x, y ∈ V.
c. F-tree, if G was a F-forest such that there is a F-path P from x to y, for all x, y ∈ V.
d. tree, if it was either of T −tree, I−tree, and F −tree.
Notation 2.14.Let V be a given set.For the sake of simplicity, we only use the notation F, p for the representation special spanning neutrosophic graph of a forest and the order a given neutrosophic graph.By Analogous to this points, the notation F i , p i are corresponded to G i .Let us remind you consider three special notations in this paper by three letters.In other words, we have three correspondences for a given set, neutrosophic graph and a forest, we mean p, E i and F i are corresponded to G i , V i and G i , respectively.Final remark is of about writing xy instead of {x, y}.

Main Results
Definition 3.1.(Path) Let G = (A, B) be a neutrosophic graph on V and v 0 , v n be two given vertices such that n ∈ N. Then a.A distinct sequence of vertices T −path, I−path, and F −path, simultaneously.In this case, the min{µ G (P ) T , µ G (P ) I , µ G (P ) F } is called strength of path and is denoted by µ G (P ).

Definition 3.2. (Strength between Two Vertices)
Let G = (A, B) be a neutrosophic graph on V and v i , v j be two given vertices such that i > j and i, j ∈ N. Then a.The max{µ G (P ) T } in G is called the T-strength between v i and v j and is denoted by ) be a neutrosophic graph on V as Figure 1.The various types of some paths of length 3 from v 1 to v 5 are investigated.P 1 : v 1 , v 4 , v 2 , v 5 , P 2 : v 1 , v 3 , v 4 , v 5 , and P 3 : v 1 , v 2 , v 4 , v 5 are the T-path, I-path and F-path of length 3 from v 1 to v 5 , respectively and not other ones.The distinct sequences of vertices P 4 : v 5 , v 4 , v 1 , v 2 is not neither of them.P 5 : v 3 , v 4 , v 5 , v 2 is all of them.In this graph, we determine various types of strength of some paths as follows.µ G (P 1 ) T = 0.4, µ G (P 2 ) I = 0.2, and µ G (P 3 ) F = 0.91.For P 5 , we have µ G (P 5 ) T = 0.4, µ G (P 5 ) I = 0.1, µ G (P 5 ) F = 0.92, and µ G (P 5 ) = 0.1.Finally, we discuss about various types of strength between two vertices v 1 and v In what follows, we will define four properties for edges.Based of these properties, we can construct various kindes of dominations in neutrosophic graphs.Example 3.6.Let G = (A, B) be a neutrosophic graph on V as Figure 1.In the following table, we study the properties of edges.For example, v 2 v 5 has not neither of T −effective, I−effective, F −effective, and effective property.The edge v 3 v 4 has both of T −effective and I−effective property.So it is also effective edge.The edges Proof.(a).Let G = (A, B) be a neutrosophic graph on a given set V. The T-strength of path P from u to v is of the form T It is interesting to note that the converse of Propositions 3.9, does not hold.

Definition 2 . 6 .
(Empty Neutrosophic Graph) Let G = (A, B) be a neutrosophic graph on a given set V. G is called empty if the following conditions are satisfied: T B (xy) = I B (xy) = F B (xy) = 0.

Definition 2 . 9 .
(Bridge) Let G = (A, B) be a neutrosophic graph on a given set V. Then an edge xy in G is called the a. T-bridge, if the strengths of each T-path P from x to y, not involving xy, were less than T B (xy). b.I-bridge, if the strengths of each T-path P from x to y, not involving xy, were less than T B (xy). c.F-bridge, if the strengths of each T-path P from x to y, not involving xy, were less than T B (xy). d. bridge, if it was either of T −bridge, I−bridge, and F −bridge.Definition 2.10.(Acyclic) Let G = (A, B) be a neutrosophic graph on a given set V. Then G is called the a. T-acyclic, if there wasn't a T-path P from x to y, with only exception x = y., for all x ∈ V.
d. forest, if it was either of neutrosophic T −forest, neutrosophic I−forest, and neutrosophic F −forest.Definition 2.13.(Tree) Let G = (A, B) be a neutrosophic graph on a given set V. Then G is called the a. T-tree, if G was a T-forest such that there is a T-path P from x to y, for all x, y ∈ V.

Figure 1 :
Figure 1: The strength in the neutrosophic graph G = (A, B)
It means that the edge uv is T-effective.All edges are T-effective and each vertex is adjacent to all other vertices.So D = {u} is a T-effective dominating set and Σ xy is a T-effective edge T B (xy) = Σ xy is a edge T B (xy) for each u ∈ V.The result follows.By analogues to the proof of (a), the result is obviously hold for (b), (c), and (d).Proposition 3.9.Let G = (A, B) be an empty neutrosophic graph on a given set V. Then γ v (G) T = γ v (G) I = γ v (G) F = γ v (G) = p where p denotes the order of G. Proof.Let G be an empty neutrosophic graph on a given set V. Hence V is only T-effective dominating set in G and there is also no T-effective edge.So by Definition 3.7(a), we have γ v (G) T = min D∈S [Σ u∈D T A (u)] = Σ u∈V T A (u) = p.Therefore γ v (G) T = p.By analogues to the proof of γ v (G) T = p and Definition 3.7, the result is obviously hold for γ v (G) I , γ v (G) F and γ v (G).

Example 3 . 10 .
We show that the converse of Propositions 3.9, does not hold.Let G = (σ, µ) be a fuzzy graph as Figure2.The edges {v 2 v 5 , v 2 v 4 , v 3 v 4 , v 1 v 3 } are T-effective, I-effective, F-effective, and effective
Domination as a theoretical area in graph theory was formalized by Berge in 1958, in the chapter 4 with title " The fundamental Numbers of the theory of Graphs" ([3], Theorem 7, p.40) and Ore ([7], Chapter 13 , pp. 206, 207) in 1962.Since 1977, when Cockayne and Hedetniemi ([6], Section 3, p. 249-251) presented a survey of domination results, domination theory has received considerable attention.A set S of vertices of G ([5], Chap.10, p. 302) is a dominating set if every vertex in V (G) − S is adjacent to at least one vertex in S. The minimum cardinality among the dominating sets of G is called the domination number of G and is denoted by γ(G).A dominating set of cardinality γ(G) is then referred to as minimum dominationg set.Dominating sets appear to have their origins ([3], Example 2, p. 41) in the game of chess, where the goal is to cover or dominate various squares of a chessboard by certain chess pieces.