Domination in bipolar fuzzy soft graphs

Abstract

The concept of domination in graphs is very ancient. Several types of notions of domination in graphs have been discussed by many researchers. In this work, the concept of domination and some notions of domination sets, minimal dominating sets, independence sets, and maximal independence sets are introduced in bipolar fuzzy soft graphs. Additionally, several properties of dominating sets are discussed and some theorems in bipolar fuzzy soft graphs are proved.

Keywords

Domination in bipolar fuzzy soft graphs minimal domination set in bipolar fuzzy soft graphs maximal independence set in bipolar fuzzy soft graphs

1 Introduction

The idea of graph theory was introduced by Euler while he solved Konigsberg bridge problem and graph theory has large variety of applications in various domains [1]. Lotfi A. Zadeh, familiar as father of fuzzy logic, introduced the notion of fuzzy set in 1965 which was defined as the set of distinct objects having membership values in [0, 1], later on many notions were introduced in fuzzy set [2]. The notion of fuzzy graphs was introduced by Rosenfeld in 1975 [3], then after some remarks on fuzzy graphs were given by Battacharya [4]. Some new ideas on fuzzy graph were initiated by Mordeson and Nair [5]. A lot of new terminologies on fuzzy graphs also introduced by akram et al. [6 –9].

The concept of bipolar fuzzy sets was introduced by Zhang [10]. Bipolar fuzzy set is a generalization of fuzzy sets [11] and having membership values ranging in [– 1, 1]. An element having 0 membership value then this indicates that the corresponding property of an element is irrelevant, if an element having membership value ranging in (0, 1], then this element fulfil the corresponding property, in the same way if an element having membership value ranging in (– 1, 0), then this element fulfil the corresponding implicit counter property. Akram initiated the idea of bipolar fuzzy graphs and investigated basic properties of bipolar fuzzy graphs [6 , 12– 14].

The fuzzy soft sets that are the combinations of fuzzy sets and soft sets, which was introduced by P.K. Maji, A.R. Roy and R Biswas in 2002 [15, 16]. The combination of fuzzy sets and rough sets, new models of soft rough set, rough soft set and fuzzy soft set were introduced by using soft fuzzy sets by Feng et al. [17 –19]. Thereafter Naz and Sabir initiated the idea of the bipolar fuzzy soft sets that are combination of bipolar fuzzy sets and soft set [20] and Aslam et al. work on some basic properties and operations on bipolar fuzzy soft sets [21]. Later on Yildiray Celik introduced the concept of bipolar fuzzy soft graphs by applying the concept of bipolar fuzzy soft sets to graphs and introduced several operations and investigate the basic characteristics of bipolar fuzzy soft graphs [22]. Readers are referred to [23 –26] for further applications.

The concept of domination in graph theory was first introduced in the game of chess. The issue began when all squares faced the involvement of a queen, by checking the number of at least the queen’s number of queens [27]. The domination in graph theory was introduced by Berge [28]. It was further explained by Cockayne and Hedetniemi [29, 30].

Aras et al. [34] introduced the structures of bipolar soft locally compact and bipolar soft paracompact spaces. Malik et al. [35] introduced the rough bipolar fuzzy left ideal, rough bipolar fuzzy right ideal, rough bipolar fuzzy two-sided ideal, rough bipolar fuzzy interior ideal, and rough bipolar fuzzy bi-ideal in SGs. Malik et al. [36] proposed that becoming the most prominent reason for the increased rate of disease in an area. Gul et al. [37] present (α, β)-optimistic multi-granulation bipolar fuzzified preference rough sets ((α, β) o-MG-BFPRSs). Al-Shami [38] introduced bipolar soft sets to build an optimal choice application. Several varieties of domination in fuzzy graphs have been well expressed in [31– 33 , 39– 41].

1.1 Contribution

The key contribution of this writing is as follows.

To make some notions of domination sets, minimal dominating sets

To consider independence sets, and maximal independence sets are introduced in bipolar fuzzy soft graphs.

To build a multi-attribute DM (MADM) approach under

To categorize theorems in bipolar fuzzy soft graphs

To define the Numerical application

1.2 Motivation

In this paper, we define the new idea of domination in bipolar fuzzy soft graph, based on different properties and theorems.

bipolar fuzzy soft graphs is implemented to evaluate the claim seeing the DMs’ chance favorite done the submission of bipolar fuzzy soft graphs, which troubles the different things of additional, and to closely indenture the great provider.

The bipolar fuzzy soft graphs is documented to dialectal MCDM difficulties and deliver an organized and reliable claim to philological claim assortment problems.

To define through of all, there are regular reproduces in the writing on basis graphs. Mainly in superior a protracted period, vulnerability in selection-making minutes for as pleasantly many standards has lengthy.

We current the bipolar fuzzy soft graphs within the form Domination, Minimal Domination, Maximal Independence set in Bipolar fuzzy soft graphs.

We describe the application of bipolar fuzzy soft graphs.

We describe the comparison method with existing method.

1.3 Novelty

Many investigators have worked on fuzzy graph mean and variances, but can study the domination of bipolar fuzzy soft graph. We aimed to develop a novel MCDM technique based on domination and apply them to real models. The following novel idea is developed in this study.

The domination of bipolar fuzzy soft graph is introduced.

A novel MCDM is developed based on bipolar fuzzy soft graph.

We have considered unique numerical example with bipolar fuzzy soft graph to show the applicability of our mentioned method.

We have discussed the comparison analysis.

In this research work, the concepts of domination set, minimal dominating set, and maximal independence set in bipolar fuzzy soft graphs will be discussed. Figure 1.1 is given below.

Fig. 1.1

History.

We use some abbreviations in this paper that are given in Table 1.

Table 1

List of abbreviations

Abbreviations	Name
$ϝ S$	Fuzzy set
$B ϝ$	Bipolar fuzzy
$ϝ G$	Fuzzy graph
$B ϝ G$	bipolar fuzzy graph
$ϝ S^{'} S$	Fuzzy soft set
$B ϝ S^{'} S$	Bipolar fuzzy soft set
$B ϝ S^{'} G$	bipolar fuzzy soft graph
$ϝ G$	Fuzzy graph
$D S$	Domination set
$I S$	Independent set

2 Preliminaries

Definition 2.1. For a universal class ∨≠ ∅ and Å is a subset of ∨. A $ϝ S$ Å is represented by Å = {(⋎, γ_Å (⋎)) : γ_Å (⋎) > 0, ⋎ ∈ ∨}, where the function γ_Å (⋎) : ∨ → [0, 1] is the membership value of ⋎ in Å.

Definition 2.2. For a universal class ∨≠ ∅ and Å is a subset of ∨. A $B ϝ$ is of the form Å ={ (⋎, γ⁺ (⋎) , γ^- (⋎)) : ⋎ ∈ ∨ } where γ⁺, γ^- are the functions defined as γ⁺ : ∨ → [0, 1] and γ^- : ∨ → [- 1, 0].

Definition 2.3. Let $Å = (γ_{Å}^{+}, γ_{Å}^{-})$ and $B = (γ_{B}^{+}, γ_{B}^{-})$ be a $B ϝ$ sets on a set ∨, then $Å = (γ_{Å}^{+}, γ_{Å}^{-})$ is called a $B ϝ$ relation on $B = (γ_{B}^{+}, γ_{B}^{-})$ if for all _⋎1, _⋎2∈ ∨ such that $γ_{B}^{+} (⋎_{1}, ⋎_{2}) ⩽ γ_{Å}^{+} (⋎_{1}) \land γ_{Å}^{+} (⋎_{2})$ and $γ_{B}^{-} (⋎_{1}, ⋎_{2}) ⩾ γ_{Å}^{-} (⋎_{1}) \land γ_{Å}^{-} (⋎_{2})$ .

A $B ϝ$ relation Å on ∨ is called symmetric if for all _⋎1, _⋎2∈ ∨ such that γ⁺ (_⋎1, _⋎2) = γ⁺ (_⋎2, _⋎1) and γ^- (_⋎1, _⋎2) = γ^- (_⋎2, _⋎1).

Definition 2.4. For a nonempty set ∨ and γ and τ are fuzzy subset and fuzzy relation respectively on ∨. Then a $ϝ G$ is a pair of mappings defined as τ : ∨ → [0, 1] and γ : ∨ × ∨ → [0, 1] for all _⋎1, _⋎2∈ ∨ such that γ (_⋎1, _⋎2) ⩽ τ (_⋎1) ∧ τ (_⋎2) for all.

Definition 2.5. For a nonempty set ∨. A $B ϝ G$ with over ∨ is represented by a pair $(Å, B)$ where $Å = (γ_{Å}^{+}, γ_{Å}^{-})$ is a $B ϝ$ set in ∨ and $B = (γ_{B}^{+}, γ_{B}^{-})$ is a $B ϝ$ set in ∨× ∨, for all _⋎1, _⋎2∈ ∨ such that $γ_{B}^{+} (⋎_{1}, ⋎_{2}) ⩽ γ_{Å}^{+} (⋎_{1}) \land γ_{Å}^{+} (⋎_{2})$ and $γ_{B}^{-} (⋎_{1}, ⋎_{2}) ⩾ γ_{Å}^{-} (⋎_{1}) \lor γ_{Å}^{-} (⋎_{2})$ .

In $G = (Å, B)$ , Å is said to be a $B ϝ$ vertex and $B$ is said to be a $B ϝ$ edge and the notation (_⋎1, _⋎2) is used for a member ∨× ∨. Thus, $G = (Å, B)$ is a $B ϝ G$ if for all _⋎1, _⋎2∈ ∨ such that $γ_{B}^{+} (⋎_{1}, ⋎_{2}) ⩽ γ_{Å}^{+} (⋎_{1}) \land γ_{Å}^{+} (⋎_{2})$ and $γ_{B}^{-} (⋎_{1}, ⋎_{2}) ⩾ γ_{Å}^{-} (⋎_{1}) \lor γ_{Å}^{-} (⋎_{2})$ .

Definition 2.6. For a universal class ∨ and $ℙ$ being a set of parameters. The power set $ℙ (\lor)$ of ∨. A pair denoted by $(H, ℙ)$ is said to be a soft set over ∨, where H is function defined as $H : ℙ \to ℙ (\lor)$ .

Definition 2.7. For a universal class ∨ and $ℙ$ being a set of parameters and Q is a subset of $ℙ$ . Let I^∨ represent the collection that contain all fuzzy subsets. A brace (H, Q) is said to be a $ϝ S^{'} S$ over ∨ where H is function defined as H : Q → I^∨.

Definition 2.8. For a universal class ∨ and $ℙ$ being a set of parameters and Q is a subset of $ℙ$ . Let $B ϝ (\lor)$ present the set of all $B ϝ$ subset of ∨. If φ present a mapping, such as $φ : Q \to B ϝ (\lor)$ , then the pair (φ, Q) is called a $B ϝ S^{'} S$ over ∨. For any q ∈ Q, φ (q) is the set of q-approximate elements of (φ, Q) and can be defined as $φ (q) = {γ_{φ (q)}^{+} (⋎), γ_{φ (q)}^{-} (⋎) : ⋎ \in \lor, q \in Q}$ , where $γ_{φ (q)}^{+} (⋎)$ represents the degree of ⋎ keeping the parameter q, $γ_{φ (q)}^{-} (⋎)$ presents the degree of ⋎.

Definition 2.9. A bipolar fuzzy soft graph ( $B ϝ S^{'} G$ ) is an ordered 4-tuple $G_{\lor, ℙ} = (G, Γ, K^{'}, ℙ)$ over set ∨ such that:

$ℙ$ is a set of parameters that is $ℙ \neq \emptyset$ ,

$(Γ, ℙ)$ is $B ϝ S^{'} S$ over ∨,

is a $B ϝ S^{'} S$ over ∨ × ∨ ,

(Γ (ρ) , K (ρ)) is a $B ϝ S^{'} G$ for all $ρ \in ℙ$ ,

That is for all _⋎1, _⋎2∈ ∨ × ∨ and for all $ρ \in ℙ$ such that $γ_{K^{'} (ρ)}^{+} (⋎_{1}, ⋎_{2}) ⩽ γ_{Γ (ρ)}^{+} (⋎_{1}) \land γ_{Γ (ρ)}^{+} (⋎_{2})$ and $γ_{K^{'} (ρ)}^{-} (⋎_{1}, ⋎_{2}) ⩾ γ_{Γ (ρ)}^{-} (⋎_{1}) \lor γ_{Γ (ρ)}^{-} (⋎_{2})$ .

Definition 2.10. In a $ϝ G$ , $G = (\lor, Γ, K^{'})$ , An edge of $G$ is referred to as a strong edge if for all _⋎1, _⋎2∈ ∨ such that K′ (_⋎1, _⋎2) = Γ (_⋎1) ∧ Γ (_⋎2).

Definition 2.11. In a $ϝ G$ $G = (\lor, Γ, K^{'})$ , A vertex of _⋎1 is called Neighborhood of vertex _⋎2 if there is a strong edge for joining the vertices _⋎1 and _⋎2.

Definition 2.12. In a $ϝ G$ $G = (\lor, Γ, K^{'})$ , A vertex of _⋎1 is called an isolated vertex if K′ (_⋎1, _⋎2) < Γ (_⋎1) ∧ Γ (_⋎2) for all _⋎1, _⋎2∈ ∨. So that N^d (_⋎1) =∅.

Definition 2.13. In a $ϝ G$ $G = (\lor, Γ, K^{'})$ , A vertex _⋎1 dominates the vertex _⋎2 ∈ ∨ ∖ _⋎1 if for all _⋎1, _⋎2∈ ∨ such that K′ (_⋎1, _⋎2) = Γ (_⋎1) ∧ Γ (_⋎1).

Definition 2.14. In a $ϝ G$ $G = (\lor, Γ, K^{'})$ and $D$ is referred to as a dominating set ( $D S$ ) if for every element $⋎_{2} \in \lor ∖ D$ , we have an element _⋎1 in $D$ such that _⋎1 dominates _⋎2, where $D \subseteq \lor$ .

Definition 2.15. In a $ϝ G$ $G = (\lor, Γ, K^{'})$ , A domination number of $G$ is a $D S$ having minimum cardinality and is presented by $γ (G)$ .

Definition 2.16. In a $ϝ G$ $G = (\lor, Γ, K^{'})$ . A $D S$ is called a minimal $D S$ if there is no proper dominating subset of $D$ .

Definition 2.17. In a $ϝ G$ $G = (\lor, Γ, K^{'})$ , let $D^{I} \subseteq \lor$ is called an Independence set ( $I S$ ) if for all $⋎_{1}, ⋎_{2} \in D^{I}$ such that K′ (_⋎1, _⋎2) = Γ′ (_⋎1) ∧ Γ (_⋎1) that is the set of all isolated vertices of form an $I S$ .

Definition 2.18. In a $ϝ G$ $G = (\lor, Γ, K^{'})$ , An $I S$ $D^{I}$ is called a maximal $I S$ if for any $⋎ \in \lor ∖ D^{I}$ , the set $D^{I} \cup {⋎}$ is not independent.

3 Domination, minimal domination, maximal independence set in bipolar fuzzy soft graphs

Definition 3.1. Let $G_{\lor, ℙ} = (G^{'}, Γ, K^{'}, ℙ)$ be a $B ϝ S^{'} G$ , then an edge of $G_{\lor, ℙ}$ is called a strong edge if for all _⋎1, _⋎2∈ ∨ and for all $ρ \in ℙ$ such that $γ_{K^{'} (ρ)}^{+} (⋎_{1}, ⋎_{2}) = γ_{Γ (ρ)}^{+} (⋎_{1}) \land γ_{Γ (ρ)}^{+} (⋎_{2})$ and $γ_{K^{'} (ρ)}^{-} (⋎_{1}, ⋎_{2}) = γ_{Γ (ρ)}^{-} (⋎_{1}) \lor γ_{Γ (ρ)}^{-} (⋎_{2})$ .

In Example 3.4, Edges (_⋎1, _⋎2) and (_⋎1, _⋎4) are strong edges.

Definition 3.2. Let $G_{\lor, ℙ} = (G^{'}, Γ, K^{'}, ℙ)$ be a $B ϝ S^{'} G$ , then a vertex _⋎1 of $G_{\lor, ℙ}$ . Is called a Neighborhood of _⋎2, then N^d (_⋎1) ={ _⋎2 : _⋎2 ∈ ∨ - _⋎1, (_⋎1, _⋎2) is as trongedge }.

In Example 3.4, N^d (_⋎1) ={ _⋎2, _⋎4 }, N^d (_⋎2) ={ _⋎1 }, N^d (_⋎3) =∅, N^d (_⋎4) ={ _⋎1 }.

Definition 3.3. Let $G_{\lor, ℙ} = (G^{'}, Γ, K^{'}, ℙ)$ be a $B ϝ S^{'} G$ , then a vertex _⋎1 of $G_{\lor, ℙ}$ is called an isolated vertex if for all _⋎1, _⋎2∈ ∨ and for all $ρ \in ℙ$ such that $γ_{K^{'} (ρ)}^{+} (⋎_{1}, ⋎_{2}) < γ_{Γ (ρ)}^{+} (⋎_{1}) \land γ_{Γ (ρ)}^{+} (⋎_{2})$ and $γ_{K (ρ)}^{-} (⋎_{1}, ⋎_{2}) > γ_{Γ (ρ)}^{-} (⋎_{1}) \lor γ_{Γ (ρ)}^{-} (⋎_{2})$ . So that N^d (⋎) =∅.

In Example 3.4, As N^d (_⋎3) =∅ so _⋎3 is an isolated vertex.

Example 3.4. Let $G_{\lor, ℙ} = (G^{'}, Γ, K^{'}, ℙ)$ be a $B ϝ S^{'} G$ with underlying $G^{'} = (\lor, E^{'})$ such that ∨ ={ _⋎1, _⋎2, _⋎3, _⋎4 } and E′ ={ (_⋎1_⋎2) , (_⋎2_⋎3) , (_⋎3_⋎4) , (_⋎4_⋎1) }, set of parameters $ℙ = {ρ_{1}, ρ_{2}}$ , $(Γ, ℙ)$ and $(K, ℙ)$ are $B ϝ S^{'} S$ s over vertex and edge sets respectively are in given in Table 2. Clearly H′ (ρ₁) = (Γ (ρ₁) , K (ρ₁)) and H (ρ₂) = (Γ (ρ₂) , K (ρ₂)) are $B ϝ S^{'} G$ s according to parameters ρ₁ and ρ₂ respectively, as present in Fig. 3.1. Hence $G_{\lor, ℙ} = (G^{'}, Γ, K^{'}, ℙ)$ is a $B ϝ S^{'} G$ .

Figure 3.1. is given as below.

Fig. 3.1

$B ϝ S^{'} G$ .

Table 2

Bipolar fuzzy soft vertex and edge sets

Γ	_⋎1	_⋎2	_⋎3	_⋎4
ρ ₁	(0.2, - 0.4)	(0.5, - 0.5)	(0.2, - 0.6)	(0.2, - 0.7)
ρ ₂	(0.1, - 0.8)	(0.6, - 0.4)	(0, 0)	(0.3, - 0.5)
K	(_⋎1_⋎2)	(_⋎2_⋎3)	(_⋎3_⋎4)	(_⋎4_⋎1)
ρ ₁	(0.2, - 0.4)	(0.6, - 0.3)	(0.1, - 0.4)	(0.2, - 0.4)
ρ ₂	(0.1, - 0.4)	(0, 0)	(0, 0)	(0.1, - 0.5)

Figure 3.4 is given as below.

Fig. 3.2

$B ϝ S^{'} G s$

Fig. 3.3

$B ϝ S^{'} G s$ .

Fig. 3.4

$B ϝ S^{'} G s$ .

Figure 3.5 is given as below.

Fig. 3.5

$B ϝ S^{'} G s$ corresponding to parameters ρ₁ and ρ₂.

Definition 3.5. Let $G_{\lor, ℙ} = (G^{'}, Γ, K^{'}, ℙ)$ be a $B ϝ S^{'} G$ over $G^{'} = (\lor, E^{'})$ , then cardinality of $G_{\lor, ℙ}$ is given by

$\begin{matrix} | G_{\lor, ℙ} | = \sum_{ρ_{i} \in ℙ} {(\sum_{⋎_{i} \in \lor} \frac{1 + γ_{Γ (ρ_{i})}^{+} (⋎_{i}) + γ_{Γ (ρ_{i})}^{-} (⋎_{i})}{2}) \\ + (\sum_{(⋎_{i} ⋎_{i}) \in E^{'}} \frac{1 + γ_{K^{'} (ρ_{i})}^{+} (⋎_{i}) + γ_{K^{'} (ρ_{i})}^{-} (⋎_{i})}{2})} \end{matrix}$

Definition 3.6. Let $G_{\lor, ℙ} = (G^{'}, Γ, K^{'}, ℙ)$ be a $B ϝ S^{'} G$ over $G^{'} = (\lor, E^{'})$ , then vertex cardinality of $G_{\lor, ℙ}$ is given by $| \lor | = \sum_{ρ_{i} \in ℙ} (\sum_{⋎_{i} \in \lor} \frac{1 + γ_{Γ (ρ_{i})}^{+} (⋎_{i}) + γ_{Γ (ρ_{i})}^{-} (⋎_{i})}{2})$

Definition 3.7. Let $G_{\lor, ℙ} = (G^{'}, Γ, K^{'}, ℙ)$ be a $B ϝ S^{'} G$ over $G^{'} = (\lor, E^{'})$ , then edge cardinality of $G_{\lor, ℙ}$ is given by $| E^{'} | = \sum_{ρ_{i} \in ℙ} (\sum_{(⋎_{i}, ⋎_{j}) \in E^{'}} \frac{1 + γ_{K^{'} (ρ_{i})}^{+} (⋎_{i}) + γ_{K^{'} (ρ_{i})}^{-} (⋎_{i})}{2})$

In Example 3.4, the vertex cardinality of $G_{\lor, ℙ} = (0.4 + 0.5 + 0.3 + 0.25) + (0.15 + 0.4 + 0.6) = 2.6$ , the edge cardinality of $G_{\lor, ℙ} = (0.4 + 0.65 + 0.35 + 0.4) + (0.35 + 0.3) = 2.45$ and cardinality of $G_{\lor, ℙ} = 2.6 + 2.45 = 5.05$ .

Definition 3.8. Let $G_{\lor, ℙ} = (G^{'}, Γ, K^{'}, ℙ)$ be a $B ϝ S^{'} G$ . For two disjoint vertices _⋎1 and _⋎2 then a vertex _⋎1 dominates the vertex _⋎2 in $G_{\lor, ℙ}$ if for all _⋎1, _⋎2∈ ∨ and for all $ρ \in ℙ$ such that $γ_{K^{'} (ρ)}^{+} (⋎_{1}, ⋎_{2}) = γ_{Γ (ρ)}^{+} (⋎_{1}) \land γ_{Γ (ρ)}^{+} (⋎_{2})$ and $γ_{K (ρ)}^{-} (⋎_{1}, ⋎_{2}) = γ_{Γ (ρ)}^{-} (⋎_{1}) \lor γ_{Γ (ρ)}^{-} (⋎_{2})$ .

Remark 3.9.

Domination is symmetric relation that is if _⋎1 dominates _⋎2 then also _⋎2 dominates _⋎1_⋎1, _⋎2∈ ∨.

{N^d (⋎) : any⋎∈ ∨ } is a collection of all vertices of ∨ that are dominated by ⋎,

If for all _⋎1, _⋎2∈ ∨ and for all $ρ \in ℙ$ such that $γ_{K^{'} (ρ)}^{+} (⋎_{1}, ⋎_{2}) < γ_{F^{'} (ρ)}^{+} (⋎_{1}) \land γ_{F^{'} (ρ)}^{+} (⋎_{2})$ and $γ_{K (ρ)}^{-} (⋎_{1}, ⋎_{2}) > γ_{Γ (ρ)}^{-} (⋎_{1}) \lor γ_{Γ (ρ)}^{-} (⋎_{2})$ then ∨ is only $D S$ of $G_{\lor, ℙ}$ .

Definition 3.10. Let $G_{\lor, ℙ} = (G^{'}, Γ, K^{'}, ℙ)$ be a $B ϝ S^{'} G$ and $D$ is a subset of ∨, then for every $⋎_{2} \in \lor ∖ D$ ∃ an element _⋎1 in $D$ such that _⋎1 dominates _⋎2.

Definition 3.11. Let $G_{\lor, ℙ} = (G^{'}, Γ, K^{'}, ℙ)$ be a $B ϝ S^{'} G$ . A $D S$ $D$ , subset of ∨ is called a minimal $D S$ if for any $⋎ \in D$ , $D ∖ ⋎$ is not a dominating set in $G_{\lor, ℙ}$ .

Definition 3.12. Let $G_{\lor, ℙ} = (G^{'}, Γ, K^{'}, ℙ)$ be a $B ϝ S^{'} G$ , then lower domination number is defined as infimum cardinality of minimal $D S$ s of $G_{\lor, ℙ}$ and it is denoted by $d_{B} (G_{\lor, ℙ})$ .

Definition 3.13. Let $G_{\lor, ℙ} = (G^{'}, Γ, K^{'}, ℙ)$ be a $B ϝ S^{'} G$ , then upper domination number is defined as supremum cardinality of minimal $D S$ s of $G_{\lor, ℙ}$ and it is denoted by $D_{B} (G_{\lor, ℙ})$ .

Example 3.14. Let $G_{\lor, ℙ} = (G^{'}, Γ, K^{'}, ℙ)$ be a $B ϝ S^{'} G$ with underlying $G^{'} = (\lor, E^{'})$ such that ∨ ={ _⋎1, _⋎2, _⋎3, _⋎4 } and E′ ={ (_⋎1_⋎2) , (_⋎2_⋎3) , (_⋎3_⋎4) , (_⋎4_⋎2) }, set of parameters $ℙ = {ρ_{1}, ρ_{2}, ρ_{3}}$ , $(F, ℙ)$ and $(K, ℙ)$ are $B ϝ S^{'} S$ s over vertex and edge sets respectively are in given in Table 3. Clearly H′ (ρ₁) = (Γ (ρ₁) , K (ρ₁)), H′ (ρ₂) = (Γ (ρ₂) , K (ρ₂)) and H′ (ρ₃) = (Γ (ρ₃) , K (ρ₃)) are BϝS’Gs according to parameters ρ₁, ρ₂ and ρ₃ respectively, as present in Fig. 3.2. Table 3 is given in Bipolar fuzzy soft vertex and Edge sets.

Figure 3.2 is given in below.

Table 3

Bipolar fuzzy soft vertex and edge sets

Γ	_⋎1	_⋎2	_⋎3	_⋎4
ρ ₁	(0.4, –0.4)	(0.3, –0.2)	(0.4, –0.5)	(0.7, –0.1)
ρ ₂	(0.5, –0.5)	(0.4, - 0.3)	(0.5, - 0.6)	(0.8, - 0.2)
ρ ₃	(0.3, - 0.2)	(0.1, - 0.4)	(0.1, - 0.8)	(0.1, - 0.8)
K	(_⋎1_⋎2)	(_⋎ - * - +2_⋎3)	(_⋎3_⋎4)	(_⋎4_⋎2)
ρ ₁	(0.3, - 0.2)	(0.3, - 0.2)	(0.4, - 0.1)	(0.3, - 0.1)
ρ ₂	(0.4, - 0.3)	(0.4, - 0.3)	(0.5, - 0.2)	(0.4, - 0.2)
ρ ₃	(0.1, - 0.2)	(0.1, - 0.4)	(0.1, - 0.8)	(0.1, - 0.4)

In each parametrized graphs, the minimal $D S$ s are $D_{1} = {⋎_{1}, ⋎_{4}}$ , $D_{2} = {⋎_{1}, ⋎_{3}}$ , $D_{3} = {⋎_{2}}$ shown in Fig. 2, and cardinality of $D_{1}$ , $D_{2}$ and $D_{3}$ corresponding to parameter ρ₁ are $\begin{matrix} | D_{1} | = (\frac{1 + 0.4 - 0.4}{2} + \frac{1 + 0.7 - 0.1}{2}) = 1.3, | D_{2} | \\ = (\frac{1 + 0.4 - 0.4}{2} + \frac{1 + 0.4 - 0.5}{2}) = 0.95, | D_{3} | \\ = (\frac{1 + 0.3 - 0.2}{2}) = 0.55 \end{matrix}$

In the same way, cardinality of $D_{1}$ , $D_{2}$ and $D_{3}$ corresponding to parameter ρ₂ are 1.3, 0.95, 0.55 respectively and corresponding to parameter ρ₂ are 0.7, 0.7, 0.35 respectively. Now corresponding to parameter ρ₁, lower domination number that is $d_{B} (G_{\lor, ρ_{1}}) = 0.55$ and upper domination number that is $D_{B} (G_{\lor, ρ_{1}}) = 1.3$ , corresponding to parameter ρ₂, lower domination number that is $d_{B} (G_{\lor, ρ_{1}}) = 0.55$ and upper domination number that is $D_{B} (G_{\lor, ρ_{1}}) = 1.3$ and corresponding to parameter ρ₃, lower domination number that is $d_{B} (G_{\lor, ρ_{1}}) = 0.35$ and upper domination number that is $D_{B} (G_{\lor, ρ_{1}}) = 0.7$ .

Theorem 3.15. A $D S$ $D$ is minimal if and only if one of the below conditions are satisfied.

$N^{d} (⋎) \cap D = \emptyset$ .

For each $⋎_{1} \in D$ , there is a vertex $⋎_{2} \in \lor ∖ D$ such that $N^{d} (⋎_{2}) \cap D = ⋎_{1}$ .

Proof. Consider $G_{\lor, ℙ} = (G^{'}, Γ, K^{'}, ℙ)$ be a $B ϝ S^{'} G$ and $D$ is a minimal $D S$ , for any $⋎ \in D$ , $D ∖ {⋎_{1}}$ is not a $D S$ in $G_{\lor, ℙ}$ then there exist $⋎_{2} \in \lor ∖ (D ∖ {⋎_{1}})$ that is not a dominated by any vertex in $D ∖ {⋎_{1}}$ of $G_{\lor, ℙ} (ρ_{i})$ for all $ρ_{j} \in ℙ$ for j = 1, 2, 3, …, m. For _⋎1 = _⋎2 then $N^{d} (⋎_{1}) \subseteq \lor ∖ D$ that gives $N^{d} (⋎_{1}) \cap D = \emptyset$ , for all $ρ_{j} \in ℙ$ for j = 1, 2, 3, …, m and for _⋎1 ≠ _⋎2 then _⋎2 is not dominated by $D ∖ {⋎_{1}}$ , but is dominated by $D$ that is _⋎2 is only dominated by _⋎1 of $G_{\lor, ℙ} (ρ_{i})$ for all $ρ_{j} \in ℙ$ for j = 1, 2, 3, …, m, also $N^{d} (⋎_{2}) \cap D = {⋎_{1}}$ in $G_{\lor, ℙ} (ρ_{i})$ for all $ρ_{j} \in ℙ$ for j = 1, 2, 3, …, m. Hence for each $⋎_{1} \in D$ , there is a vertex $⋎_{2} \in \lor ∖ D$ such that $N^{d} (⋎_{2}) \cap D = {⋎_{1}}$ in $G_{\lor, ℙ}$ .

Conversely, Consider for given conditions (i), (ii), one of the condition holds for $D S$ $D$ , If $D$ is not a minimal $D S$ in $G_{\lor, ℙ} (ρ_{i})$ for all $ρ_{j} \in ℙ$ for j = 1, 2, 3, …, m, then there exist a _⋎1 of $D$ such that $D ∖ {⋎_{1}}$ is a $D S$ in $G_{\lor, ℙ} (ρ_{i})$ for all $ρ_{j} \in ℙ$ for j = 1, 2, 3, …, m. So _⋎1 is dominated by minimum one _⋎i in $D ∖ {⋎_{1}}$ in $G_{\lor, ℙ} (ρ_{i})$ for all $ρ_{j} \in ℙ$ for j = 1, 2, 3, …, m. Then N^d (_⋎1) is not a subset of $D ∖ {⋎_{1}}$ in $G_{\lor, ℙ} (ρ_{i})$ for all $ρ_{j} \in ℙ$ for j = 1, 2, 3, …, m. Thus condition (i) does not hold. Also, if $D ∖ {⋎_{1}}$ is a $D S$ in $G_{\lor, ℙ} (ρ_{i})$ for all $ρ_{j} \in ℙ$ for j = 1, 2, 3, …, m, then for each $⋎_{2} \in \lor ∖ (D ∖ {⋎_{1}})$ is dominated by minimum exactly one vertex in $D ∖ {⋎_{1}}$ in $G_{\lor, ℙ} (ρ_{i})$ for all $ρ_{j} \in ℙ$ for j = 1, 2, 3, …, m. Hence $N^{d} (⋎_{2}) \cap D \neq {⋎_{1}}$ in $G_{\lor, ℙ} (ρ_{i})$ for all $ρ_{j} \in ℙ$ for j = 1, 2, 3, …, m. So condition (ii) does not hold, which contradict our supposition. Therefore $D$ should be a minimal $D S$ in $G_{\lor, ℙ} (ρ_{i})$ for all $ρ_{j} \in ℙ$ for j = 1, 2, 3, …, m. Thus $D S$ $D$ is minimal.

Theorem 3.16. Let $G_{\lor, ℙ} = (G^{'}, Γ, K^{'}, ℙ)$ be a BϝS’G that has no isolated vertex. If $D$ is a minimal $D S$ of $G_{\lor, ℙ}$ then $\lor ∖ D$ is a $D S$ of $G_{\lor, ℙ}$ .

Proof. Consider $D$ is a minimal $D S$ of $G_{\lor, ℙ}$ and $⋎_{1} \in D$ . As $G_{\lor, ℙ}$ has no isolated vertex then there exist a _⋎2∈ N^d { _⋎1 } in $G_{\lor, ℙ} (ρ_{i})$ for all $ρ_{j} \in ℙ$ for j = 1, 2, 3, …, m, By adopting the same method that is utilized for Theorem 3.15, we have $⋎_{2} \in \lor ∖ D$ . Hence for every element of $D$ is dominated by some element of $\lor ∖ D$ in $G_{\lor, ℙ} (ρ_{i})$ for all $ρ_{j} \in ℙ$ for j = 1, 2, 3, …, m. So $\lor ∖ D$ is a $D S$ of $G_{\lor, ℙ}$ .

Theorem 3.17. Every superset of $D$ being a dominating set in $G_{\lor, ℙ}$ is a $D S$ of $G_{\lor, ℙ}$ .

Proof. Evidently we can proof it.

Remark 3.18. Every subset of $D$ being a dominating set, need not to be a $D S$ in $G_{\lor, ℙ}$ .

Example 3.19. Let $G_{\lor, ℙ} = (G^{'}, Γ, K^{'}, ℙ)$ be a BϝS’G with underlying $G^{'} = (\lor, E^{'})$ such that ∨ ={ _⋎1, _⋎2, _⋎3, _⋎4, _⋎5, _⋎6, _⋎7 } and E′ = {(_⋎1_⋎2) , (_⋎2_⋎3) , (_⋎3_⋎4) , (_⋎4_⋎5), (_⋎5_⋎6) , (_⋎6_⋎3) , (_⋎3_⋎7) , (_⋎7_⋎1)}, set of parameters $ℙ = {ρ_{1}, ρ_{2}}$ , $(F, ℙ)$ and $(K, ℙ)$ are $B ϝ S^{'} S$ s over vertex and edge sets respectively are in given in Table 4. Clearly H′ (ρ₁) = (Γ (ρ₁) , K (ρ₁)) and H′ (ρ₂) = (Γ (ρ₂) , K (ρ₂)) are BϝS’Gs according to parameters ρ₁ and ρ₂ respectively, as present in Fig. 3.3.

In each parametrized graphs, obviously $D_{1} = {⋎_{1}, ⋎_{2}, ⋎_{3}, ⋎_{4}, ⋎_{5}}$ , $D_{2} = {⋎_{1}, ⋎_{2}, ⋎_{3}, ⋎_{5}}$ , $D_{3} = {⋎_{1}, ⋎_{3}}$ and $D_{4} = {⋎_{1}, ⋎_{3}, ⋎_{5}}$ shown in Fig. 3.1, $D_{2} = {⋎_{1}, ⋎_{2}, ⋎_{3}, ⋎_{5}}$ is a $D S$ , the superset of $D_{2}$ , $D_{1} = {⋎_{1}, ⋎_{2}, ⋎_{3}, ⋎_{4}, ⋎_{5}}$ is also a $D S$ , $D_{3} = {⋎_{1}, ⋎_{3}}$ being a subset of $D_{1}$ is not a $D S$ and $D_{4} = {⋎_{1}, ⋎_{3}, ⋎_{5}}$ be a subset of $D_{1}$ is a $D S$ . Hence in BϝS’Gs the superset of $D S$ is also dominating and subset of a $D S$ may or may not be $D S$ . Table 4 is written in Bipolar fuzzy soft vertex and Edge sets.

Table 4

Bipolar fuzzy soft vertex and edge sets

Γ	_⋎1	_⋎2	_⋎3	_⋎4	_⋎5	_⋎6	_⋎7
ρ ₁	(0.2, - 0.4)	(0.5, - 0.4)	(0.7, - 0.3)	(0.6, - 0.5)	(0.3, - 0.6)	(0.4, - 0.7)	(0.6, - 0.2)
ρ ₂	(0.3, - 0.2)	(0.6, - 0.5)	(0.7, - 0.6)	(0.5, - 0.4)	(0.3, - 0.6)	(0.5, - 0.4)	(0.7, - 0.7)
K	(_⋎1_⋎2)	(_⋎2_⋎3)	(_⋎3_⋎4)	(_⋎4_⋎5)	(_⋎5_⋎6)	(_⋎6_⋎3)	(_⋎3_⋎7)	(_⋎7_⋎1)
ρ ₁	(0.2, - 0.4)	(0.5, - 0.3)	(0.6, - 0.3)	(0.3, - 0.5)	(0.3, - 0.6)	(0.4, - 0.3)	(0.6, - 0.2)	(0.2, - 0.2)
ρ ₂	(0.3, - 0.2)	(0.6, - 0.5)	(0.5, - 0.4)	(0.3, - 0.4)	(0.3, - 0.4)	(0.5, - 0.4)	(0.7, - 0.6)	(0.3, - 0.2)

Figure 3.3. is given as below

Definition 3.20. Let $G_{\lor, ℙ} = (G^{'}, Γ, K^{'}, ℙ)$ be a BϝS’G. For two disjoint vertices _⋎1 and _⋎2 then a vertices _⋎1 and _⋎2 are independent in $G_{\lor, ℙ}$ if for all _⋎1, _⋎2∈ ∨ and for all $ρ \in ℙ$ such that $γ_{K^{'} (ρ)}^{+} (⋎_{1}, ⋎_{2}) < γ_{Γ (ρ)}^{+} (⋎_{1}) \land γ_{Γ (ρ)}^{+} (⋎_{2})$ and $γ_{K (ρ)}^{-} (⋎_{1}, ⋎_{2}) > γ_{Γ (ρ)}^{-} (⋎_{1}) \lor γ_{Γ (ρ)}^{-} (⋎_{2})$ .

Definition 3.21. Let $G_{\lor, ℙ} = (G^{'}, Γ, K^{'}, ℙ)$ be a BϝS’G and $D$ is a subset of ∨, then $D$ is called an $I S$ if for all _⋎1, _⋎2∈ ∨ and for all $ρ \in ℙ$ such that $γ_{K^{'} (ρ)}^{+} (⋎_{1}, ⋎_{2}) < γ_{Γ (ρ)}^{+} (⋎_{1}) \land γ_{Γ (ρ)}^{+} (⋎_{2})$ and $γ_{K (ρ)}^{-} (⋎_{1}, ⋎_{2}) > γ_{Γ (ρ)}^{-} (⋎_{1}) \lor γ_{Γ (ρ)}^{-} (⋎_{2})$ .

Definition 3.22. Let $G_{\lor, ℙ} = (G^{'}, Γ, K^{'}, ℙ)$ be a BϝS’G. An independent $D$ , subset of ∨ is called a maximal $I S$ if for any $⋎ \in \lor ∖ D$ , the set $D \cup {⋎}$ is not an $I S$ in $G_{\lor, ℙ}$ .

Definition 3.23. Let $G_{\lor, ℙ} = (G^{'}, Γ, K^{'}, ℙ)$ be a BϝS’G, then lower independence number of is defined as infimum cardinality of maximal $I S$ s of $G_{\lor, ℙ}$ and it is denoted by $i_{B} (G_{\lor, ℙ})$ .

Definition 3.24. Let $G_{\lor, ℙ} = (G^{'}, Γ, K^{'}, ℙ)$ be a BϝS’G, then upper independence number of is defined as supremum cardinality of minimal $I S$ s of $G_{\lor, ℙ}$ and it is denoted by $I_{B} (G_{\lor, ℙ})$ .

Example 3.25. Let $G_{\lor, ℙ} = (G^{'}, Γ, K^{'}, ℙ)$ be a BϝS’G with underlying $G^{'} = (\lor, E^{'})$ such that ∨ ={ _⋎1, _⋎2, _⋎3, _⋎4, _⋎5, _⋎6, _⋎7 } and E′ = {(_⋎1_⋎2) , (_⋎2_⋎3) , (_⋎3_⋎4) , (_⋎4_⋎5),

(_⋎5_⋎6) , (_⋎6_⋎3) , (_⋎3_⋎7) , (_⋎7_⋎1)}, set of parameters $ℙ = {ρ_{1}, ρ_{2}}$ , $(Γ, ℙ)$ and $(Γ, ℙ)$ are $B ϝ S^{'} S$ s over vertex and edge sets respectively are in given in Table 5. Clearly H′ (ρ₁) = (Γ (ρ₁) , K (ρ₁)) and H′ (ρ₂) = (Γ (ρ₂) , K (ρ₂)) are BϝS’Gs according to parameters ρ₁ and ρ₂ respectively, as present in Fig. 3.4.

Table 5

Bipolar fuzzy soft vertex and Edge sets

Γ	_⋎1	_⋎2	_⋎3	_⋎4	_⋎5	_⋎6	_⋎7	_⋎8	_⋎9
ρ ₁	(0.2, - 0.4)	(0.5, - 0.4)	(0.7, - 0.3)	(0.6, - 0.5)	(0.3, - 0.6)	(0.4, - 0.7)	(0.6, - 0.2)	(0.6, - 0.2)	(0.6, - 0.2)
ρ ₂	(0.3, - 0.2)	(0.6, - 0.5)	(0.7, - 0.6)	(0.5, - 0.4)	(0.3, - 0.6)	(0.5, - 0.4)	(0.7, - 0.7)	(0.7, - 0.7)	(0.7, - 0.7)
ρ ₃	(0.3, - 0.2)	(0.6, - 0.5)	(0.7, - 0.6)	(0.5, - 0.4)	(0.3, - 0.6)	(0.5, - 0.4)	(0.7, - 0.7)	(0.7, - 0.7)	(0.7, - 0.7)
K′	(_⋎1_⋎2)	(_⋎2_⋎3)	(_⋎3_⋎4)	(_⋎4_⋎5)	(_⋎5_⋎6)	(_⋎6_⋎7)	(_⋎7_⋎8)	(_⋎8_⋎4)	(_⋎4_⋎9)	(_⋎9_⋎1)
ρ ₁	(0.2, - 0.4)	(0.5, - 0.3)	(0.6, - 0.3)	(0.3, - 0.5)	(0.3, - 0.6)	(0.4, - 0.3)	(0.6, - 0.2)	(0.2, - 0.2)	(0.6, - 0.2)	(0.2, - 0.2)
ρ ₂	(0.3, - 0.2)	(0.6, - 0.5)	(0.5, - 0.4)	(0.3, - 0.4)	(0.3, - 0.4)	(0.5, - 0.4)	(0.7, - 0.6)	(0.3, - 0.2)	(0.7, - 0.6)	(0.3, - 0.2)
ρ ₃	(0.3, - 0.2)	(0.6, - 0.5)	(0.7, - 0.6)	(0.5, - 0.4)	(0.3, - 0.6)	(0.5, - 0.4)	(0.7, - 0.7)	(0.7, - 0.7)	(0.7, - 0.7)	(0.7, - 0.7)

In each parametrized graphs, obviously there are many $I S$ in figure. It we suppose the $I$ $D = {⋎_{2}, ⋎_{3}, ⋎_{4}, ⋎_{5}}$ , then $D$ is said to be maximal $I S$ in $G_{\lor, ℙ}$ if for every $\lor - D = {⋎_{1}, ⋎_{6}, ⋎_{7}, ⋎_{8}, ⋎_{9}}$ , the set $D \cup {⋎_{j}}, (j = 1, 6, 7, 8, 9)$ is not an $I S$ .

Theorem 3.26. Let $G_{\lor, ℙ} = (G^{'}, Γ, K^{'}, ℙ)$ be a BϝS’G, then $D$ subset of ∨ is maximal independent if and only if it is independent and $D S$ .

Proof. Let $G_{\lor, ℙ} = (G^{'}, Γ, K^{'}, ℙ)$ be a BϝS’G, then $D$ subset of ∨ is maximal independent of $G_{\lor, ℙ}$ , then by definition of maximal $I S$ , for every vertex _⋎i of $\lor ∖ D$ , the set $D \cup {⋎_{i}}$ is not $I S$ in $G_{\lor, ℙ} (ρ_{i})$ for all $ρ_{j} \in ℙ$ for j = 1, 2, 3, …, m. In the same pattern, for every vertex _⋎2 of $\lor ∖ D$ , there exist _⋎1 of $D$ such that _⋎1 dominates _⋎2 in $G_{\lor, ℙ} (ρ_{i})$ for all $ρ_{j} \in ℙ$ for j = 1, 2, 3, …, m. Thus $D$ is a $D S$ in $G_{\lor, ℙ} (ρ_{i})$ for all $ρ_{j} \in ℙ$ for j = 1, 2, 3, …, m. Hence $D$ is both independent and $D S$ in BϝS’G $G_{\lor, ℙ} (ρ_{i})$ .

Conversely, if we suppose $D$ is independent and $D S$ in BϝS’G, we will prove that $D$ is maximal $I S$ . To prove it suppose on contrary that $D$ is not maximal $I S$ in $G_{\lor, ℙ} (ρ_{i})$ for all $ρ_{j} \in ℙ$ for j = 1, 2, 3, …, m, then there does not exist a vertex _⋎i of $\lor ∖ D$ , the set $D \cup {⋎_{i}}$ is $I S$ in $G_{\lor, ℙ} (ρ_{i})$ for all $ρ_{j} \in ℙ$ for j = 1, 2, 3, …, m. So for every vertex _⋎2 of $\lor ∖ D$ , there does not exist _⋎1 of $D$ such that _⋎1 dominates _⋎2, that is $D$ is not a $D S$ , which contradicts our supposition. Hence $D$ is maximal $I S$ .

Theorem 3.27. Let $G_{\lor, ℙ} = (G^{'}, Γ, K^{'}, ℙ)$ be a $B ϝ S^{'} G s$ , then every maximal $I S$ is a minimal $D S$ .

Proof. Let $G_{\lor, ℙ} = (G^{'}, Γ, K^{'}, ℙ)$ be a $B ϝ S^{'} G s$ , then $D$ subset of ∨ is maximal independent of $G_{\lor, ℙ}$ . Since we already prove that $D$ is maximal $I S$ iff it is $I S$ and $D S$ , if we suppose that $D$ being not minimal.

Figure 3.4 is given as below

$D S$ in $G_{\lor, ℙ} (ρ_{i})$ for all $ρ_{j} \in ℙ$ for j = 1, 2, 3, …, m, then there exist minimum one vertex _⋎2 of $D$ , So that $D ∖ {u}$ is a $D S$ that gives $D ∖ {u}$ dominates $\lor ∖ (D ∖ {⋎})$ . So there exist minimum one vertex in $D$ that dominate ⋎, which contradicts our supposition. Therefore $D$ is a minimal $D S$ in $G_{\lor, ℙ} (ρ_{i})$ for all $ρ_{j} \in ℙ$ for j = 1, 2, 3, …, m. Hence $D$ is minimal $D S$ .

Bipolar fuzzy soft vertex and Edge sets is written in Table 5.

3.1 Proposed method of domination in bipolar fuzzy soft graphs

In this section, we define the proposed method,

Calculate the bipolar fuzzy soft graphs

Calculate the different value in figure.

4 Application of domination in Bipolar fuzzy soft graphs

Domination have a major role to identify the sets with unique properties that are applicable in solving problems. It is also useful in transportation department. In this department, traffic spot lights are fixed on particular terminals/road intersections /crossings. These spot lights are very helpful to minimize the possibilities of road accidents. For this purpose, they have all the data of all terminals/road intersections /crossings with huge rush for a particular city. In this model, the idea of domination is useful to reduce the complications that are given below.

Consider Makkuana(Mk) and Sadhar city(Sc) as the entrance and exit from East to West of any city. People often travel from Mk and Sc by means of any available transport. There are many terminals/road intersections /crossings between Mk and Sc. To construct a model of domination of bipolar fuzzy soft graphs the terminals/road intersections/crossings are considered as vertices and the routes between them are considered as edges. For set of vertices

∨ = {Tezab mill chowk (T), Kohinoor Chowk (K), Madina Town (M), Saleemi Chowk (S), Fish Formlinebreak (F), Kashmir Pull (P)}

In bipolar fuzzy soft graphs, we will take positive membership values as possibility of accidents according to followed route to reach on desire terminal and negative membership values will depends on availability of traffic setup for that route. The values of traffic setup is taken negative due to good traffic setup will reduce the possibility of accidents on that route. Also we will discuss the possibilities of traffic accidents on our roads in two different parameters that is parameter ρ₁ is in 09 : 00AM to 03 : 00PM and parameter ρ₂ is in 03 : 00PM to 09 : 00PM.

For an instance, one have to cross six terminals/road intersections/crossings. While crossing these one faces a huge traffic rush. To settle this issue transportation department cannot fix traffic spot on all of these terminals/road intersections/crossings. As the flow of traffic is going on all these routes so all the edges have same positive and negative membership values in both parameters ρ₁ and ρ₂.

In the selection of terminals/road intersections/crossings which one is most suitable for the fixing of traffic spot lights, the idea of domination play a vital role. From Fig. 3.5, the encircled vertices having strong neighbors (strong edges) are with their adjacent vertices are those that facing a huge rush. So these terminals/road intersections/crossings are most suitable for the fixing of traffic spot lights. If the vertices have having same positive and negative membership values in both the parameters, because both have same possibilities of accidents and availability of traffic setup. If we select both then the dominating set is not minimal. The purpose is to fix minimum number of light with maximum benefits. So we choose only one of them. Hence {F, T, S} is dominating set.

The $B ϝ S^{'} G s$ according to parameters ρ₁ and ρ₂ are given in Fig. 3.5.

Table 6 is given in Bipolar fuzzy soft vertex and Edge sets.

Table 6
Bipolar fuzzy soft vertex and Edge sets

Γ K T F S M P

ρ ₁ (0.4, - 0.2) (0.3, - 0.6) (0.4, - 0.3) (0.2, - 0.4) (0.5, - 0.4) (0.7, - 0.6)

ρ ₂ (0.5, - 0.4) (0.4, - 0.5) (0.3, - 0.5) (0.5, - 0.7) (0.7, - 0.4) (0.6, - 0.5)

K (K, F) (K, T) (T, F) (F, S) (T, S)

ρ ₁ (0.7, - 0.3) (0.7, - 0.3) (0.7, - 0.3) (0.7, - 0.3) (0.7, - 0.3)

ρ ₂ (0.7, - 0.3) (0.7, - 0.3) (0.7, - 0.3) (0.7, - 0.3) (0.7, - 0.3)

K (T, P) (F, M) (S, P) (S, M) (P, M)

ρ ₁ (0.7, - 0.3) (0.7, - 0.3) (0.7, - 0.3) (0.7, - 0.3) (0.7, - 0.3)

ρ ₂ (0.7, - 0.3) (0.7, - 0.3) (0.7, - 0.3) (0.7, - 0.3) (0.7, - 0.3)

Γ	K	T	F	S	M	P
ρ ₁	(0.4, - 0.2)	(0.3, - 0.6)	(0.4, - 0.3)	(0.2, - 0.4)	(0.5, - 0.4)	(0.7, - 0.6)
ρ ₂	(0.5, - 0.4)	(0.4, - 0.5)	(0.3, - 0.5)	(0.5, - 0.7)	(0.7, - 0.4)	(0.6, - 0.5)
K	(K, F)	(K, T)	(T, F)	(F, S)	(T, S)
ρ ₁	(0.7, - 0.3)	(0.7, - 0.3)	(0.7, - 0.3)	(0.7, - 0.3)	(0.7, - 0.3)
ρ ₂	(0.7, - 0.3)	(0.7, - 0.3)	(0.7, - 0.3)	(0.7, - 0.3)	(0.7, - 0.3)
K	(T, P)	(F, M)	(S, P)	(S, M)	(P, M)
ρ ₁	(0.7, - 0.3)	(0.7, - 0.3)	(0.7, - 0.3)	(0.7, - 0.3)	(0.7, - 0.3)
ρ ₂	(0.7, - 0.3)	(0.7, - 0.3)	(0.7, - 0.3)	(0.7, - 0.3)	(0.7, - 0.3)

Figure 3.5 is given as below

4.1 Comparison method with existing method

In this subsection, we define the Fig. 3.6 is given as below

Figure 3.6 is given below.

Fig. 3.6

Comparsion technique.

For a $ϝ G$ $G = (\lor, Γ, K^{'})$ with vertex set ∨ ={ _⋎1, _⋎2, _⋎3, _⋎4 } and E′ ={ (_⋎1_⋎2) , (_⋎1_⋎3) , (_⋎1_⋎4) , (_⋎2_⋎3) , (_⋎2_⋎4) } as present in Fig. 3.7.

Fig. 3.7

Flow charts.

In this figure, sets {_⋎2}, {_⋎1, _⋎2}, {_⋎2, _⋎3}, {_⋎1, _⋎3, _⋎4}, etc. are fuzzy dominating sets.

5 Conclusion

The theory of domination is highly intriguing and has a wide range of applications and properties in graphs. In this study, we have introduced the concepts of domination, independence, minimal domination, and maximal independence in BϝS’Gs, and have presented some results. For future research, we can define and discuss additional domination parameters.

References

Tutte

W.T.

, Graph theory (Vol. 21). Cambridge university press, 2001.

Zadeh

L.A.

, Fuzzy sets, Information and Control 8(3) (1965), 338–353.

Rosenfeld

, Fuzzy graphs, In Fuzzy sets and their applications to cognitive and decision processes (pp. 77–95). Academic press, (1975).

Bhattacharya

, Some remarks on fuzzy graphs, Pattern Recognition Letters 6(5) (1987), 297–302.

Mordeson

J.N.

, Nair

P.S.

, Mordeson

J.N.

and Nair

P.S.

, Fuzzy graphs, Fuzzy Mathematics: An Introduction for Engineers and Scientists (2001), 21–65.

Akram

, Bipolar fuzzy graphs with applications, Knowledge-Based Systems 39 (2013), 1–8.

Akram

, Interval-valued fuzzy line graphs, Neural Computing and Applications 21 (2012), 145–150.

Akram

and Davvaz

, Strong intuitionistic fuzzy graphs, Filomat 26(1) (2012), 177–196.

Akram

, Interval-valued fuzzy line graphs, Neural Computing and Applications 21 (2012), 145–150.

10.

Zhang

W.R.

and Zhang

, YinYang bipolar logic and bipolar fuzzy logic, Information Sciences 165(3-4) (2004), 265–287.

11.

Zadeh

L.A.

, Fuzzy sets, Information and Control 8(3) (1965), 338–353.

12.

Akram

, Bipolar fuzzy graphs with applications, Knowledge-Based Systems 39 (2013), 1–8.

13.

Akram

, Bipolar fuzzy graphs with applications, Knowledge-Based Systems 39 (2013), 1–8.

14.

Akram

and Karunambigai

M.G.

, Metric in bipolar fuzzy graphs, World Applied Sciences Journal 14(12) (2011), 1920–1927.

15.

Feng

, Liu

, Leoreanu-Fotea

and Jun

Y.B.

, Soft sets and soft rough sets, Information Sciences 181(6) (2011), 1125–1137.

16.

Feng

, Jun

Y.B.

, Liu

and Li

, An adjustable approach to fuzzy soft set based decision making, Journal of Computational and Applied Mathematics 234(1) (2010), 10–20.

17.

Feng

, Li

, Davvaz

and Ali

M.I.

, Soft sets combined with fuzzy sets and rough sets: a tentative approach, Soft Computing 14 (2010), 899–911.

18.

Jiang

, Tang

, Chen

, Wang

and Tang

, Extending soft sets with description logics, Computers & Mathematics with Applications 59(6) (2010), 2087–2096.

19.

Gong

, Xiao

and Zhang

, The bijective soft set with its operations, Computers & Mathematics with Applications 60(8) (2010), 2270–2278.

20.

Naz

and Shabir

, On fuzzy bipolar soft sets, their algebraic structures and applications, Journal of Intelligent & Fuzzy Systems 26(4) (2014), 1645–1656.

21.

Abdullah

, Aslam

and Ullah

, Bipolar fuzzy soft sets and its applications in decision making problem, Journal of Intelligent & Fuzzy Systems 27(2) (2014), 729–742.

22.

Celik

Y.I.L.D.I.R.A.Y.

, On bipolar fuzzy soft graphs, Creative Mathematics and Informatics 27(2) (2018), 123–132.

23.

Aygünoglu

and Aygün

, Introduction to fuzzy softgroups, Introduction to fuzzy soft groups, Computers & Mathematics with Applications 58(6) (2009), 1279–1286.

24.

Akram

and Nawaz

, On fuzzy soft graphs, Italian Journal of Pure and Applied Mathematics 34 (2015), 497–514.

25.

Akram

, Feng

, Borumand Saeid

and Leoreanu-Fotea

, A new multiple criteria decision-making method based on bipolar fuzzy soft graphs, Iranian Journal of Fuzzy Systems 15(4) (2018), 73–92.

26.

Amin

, Fahmi

and Aslam

, Approaches to multiple attribute group decision making based on triangular cubic linguistic uncertain fuzzy aggregation operators, Soft Computing 24 (2020), 11511–11533.

27.

Pal

, Samanta

, Ghorai

, Modern trends in fuzzy graph theory (pp. 7–93). Berlin: Springer, 2020.

28.

Berge

, Graphs and Hypergraphs, volume 6 of, (1973).

29.

Cockayne

E.J.

and Hedetniemi

S.T.

, Towards a theory of domination in graphs, Networks 7(3) (1977), 247–261.

30.

Cockayne

E.J.

, Dawes

R.M.

and Hedetniemi

S.T.

, Total domination in graphs, Networks 10(3) (1980), 211–219.

31.

Manjusha

O.T.

, Set Domination in Fuzzy Graphs Using Strong Arcs, Pan-American Journal of Mathematics 1 (2022), 9.

32.

Manjusha

O.T.

and Sunitha

M.S.

, Strong domination in fuzzy graphs, Fuzzy Information and Engineering 7(3) (2015), 369–377.

33.

Merouane

H.B.

and Chellali

, On secure domination in graphs, Information Processing Letters 115(10) (2015), 786–790.

34.

Aras

C.G.

, Al-shami

T.M.

, Mhemdi

, Bayramov

, Local compactness and paracompactness on bipolar soft topological spaces, Journal of Intelligent & Fuzzy Systems (Preprint) (2022), 1–9.

35.

Malik

, Shabir

, Al-shami

T.M.

, Gul

, Arar

, Hosny

, Rough bipolar fuzzy ideals in semigroups, Complex & Intelligent Systems (2023), 1–16.

36.

Malik

, Shabir

, Al-shami

T.M.

, Gul

and Mhemdi

, Medical decision-making techniques based on bipolar soft information, AIMS Mathematics 8(8) (2023), 18185–18205.

37.

Gul

, Shabir

, Al-shami

T.M.

and Hosny

, A Comprehensive study on (α, β)-multi-granulation bipolar fuzzy rough sets under bipolar fuzzy preference relation, AIMS Mathematics 8(11) (2023), 25888–25921.

38.

Al-Shami

T.M.

, Bipolar soft sets: relations between them and ordinary points and their applications, Complexity 2021 (2021), 1–14.

39.

Shabir

and Gul

, Modified rough bipolar soft sets, Journal of Intelligent & Fuzzy Systems 39(3) (2020), 4259–4283.

40.

Gul

, Shabir

, Mashwani

W.K.

and Ullah

, Novel Bipolar Soft Rough-Set Approximations and Their Application in Solving Decision-Making Problems, International Journal of Fuzzy Logic and Intelligent Systems 22(3) (2022), 303–324.

41.

Gul

, Shabir

, Naz

and Aslam

, A novel approach toward roughness of bipolar soft sets and their applications in MCGDM, IEEE Access 9 (2021), 135102–135120.