Optimizing Decision-Making Using Domination Theory in Product Bipolar Fuzzy Graphs

Abstract

The bipolar fuzzy model is a rapidly evolving research area that provides a robust framework for addressing real-world problems, with wide-ranging applications in scientific and technical domains. Within this framework, bipolar fuzzy graphs play a significant role in decision-making and problem-solving, particularly through domination theory, which helps tackle practical challenges. This study explores various operations on product bipolar fuzzy graphs, including union (∪), join (+), intersection (∩), Cartesian product (×), composition (∘), and complement, leading to the generation of new graph structures. Several important results related to complete product bipolar fuzzy graphs under these operations are established. Additionally, we introduce key concepts such as dominating sets, minimal dominating sets, and the domination number $⋎\left(\mathcal{H}\right)$, supported by illustrative examples. This study further investigates the properties of domination in the context of these operations. To demonstrate practical applicability, we present a decision-making problem involving the optimization of bus routes and the strategic placement of bus stations using domination principles. This research contributes to the advancement of bipolar fuzzy graph theory and its practical applications in real-world scenarios.

1. Introduction

In 1783, Euler introduced the concept of graph theory. Graphs play a crucial role in tackling combinatorial problems in different fields, including geometry, algebra, number theory, topology, computer science, and many more. Lofti A. Zadeh introduced fuzzy set theory [1] in 1965 for representing the vagueness, uncertainty, and inexact thinking that occurs in real-life situations. Fuzzy sets play a fundamental role in various research domains. The theory of the bipolar fuzzy (${\mathrm{B}}_f$) set was proposed by Zhang [2]; its degree of membership range is $[-1,1]$, and it is an extension of fuzzy sets. The component whose membership grade is $(0,1]$ shows that the component somewhat fulfills the property, and the component whose membership grade is $[-1,0)$ shows that the component somewhat fulfills the implicit counter-property. The component whose membership grade is 0 indicates that the component is not related to the corresponding property. The thought that motivates such a portrayal is associated with the presence of “bipolar facts” (e.g., positive facts and negative facts) about the particular set. Positive facts indicate what is granted to be possible, while negative facts indicate what is considered to be impossible. ${\mathrm{B}}_f$ sets capture the double-sided positive facts and negative facts of human perceptions, e.g., young and old, healthy and unhealthy, advantages and disadvantages, and so on. ${\mathrm{B}}_f$ sets have numerous applications, including in artificial intelligence, information technology, social science, etc. Based on Zadeh’s [3] fuzzy relations, Kaufmann [4] introduced the idea of the fuzzy graph in 1973. Another elaborated definition of a fuzzy graph and the fuzzy relations between sets was studied by Rosenfeld [5] in 1975. Moreover, some remarkable results on fuzzy graphs were investigated by Bhattacharya [6], and Mordeson and Nair [7] studied certain operations on fuzzy graphs. Later, in 1994, Mordeson and Peng [8] defined certain operations on fuzzy graphs. In 2002, Sunitha and Vijayakumar [9] presented the idea of the complement of a fuzzy graph. In 2011, Akram [10] introduced the concept of bipolar fuzzy graphs. Recently, Akram and Shumaiza [11] presented the idea of multi-criteria decision-making methods with bipolar fuzzy sets.

The notion of the domination set (DS) in graph theory was studied by Ore and Berge [12]. Domination plays a significant role in graph theory. Basically, domination arises in problems involving finding sets of representatives, for example, facility location problems, school bus routing, the selection of a group leader in a class, locating a station along a railway, electrical networks, and computer networks. To find solutions to such problems, the concept of domination is used. Later, E. J. Cockayne and S. T. Hedetniemi [13] advanced this idea and published a paper on domination in graphs. In 1978, R. B. Allan and R. Laskar [14] introduced the notion of independent domination in graphs. The notion of domination in fuzzy graphs using effective edges was initiated by A. Somasundram and S. Somasundaram [15]. M. G. Karunambigai, Akram, and Palanivel [16] introduced the notion of domination in ${\mathrm{B}}_f$ graphs. The idea of edge domination in fuzzy graphs was studied by C. Ponnappan, S. B. Ahamed, and P. Surulinathan [17].

In real-life situations, when a critical problem arises, we use a mathematical model to find the best possible solution, often representing it in the form of a graph. The key elements of the problem are represented as vertices, while the relationships between them are represented as edges. Depending on the role of vertices and edges, fuzzy values are assigned to each vertex and edge. Sometimes, ambiguity arises in the relationship between two objects, where the edge is assigned either the maximum value of its incident vertices or the minimum value of the product of its incident vertices. Such types of graph are called “anti fuzzy graphs” and “product fuzzy graphs”. The idea of product fuzzy (${\mathrm{P}}_f$) graphs was first investigated by Ramaswamy and Poornima in 2009 [18]. In a ${\mathrm{P}}_f$ graph, we replace the sign of the minimum, which is in the definition of fuzzy graphs, with the product and call the resulting structure a ${\mathrm{P}}_f$ graph. The idea of domination in ${\mathrm{P}}_f$ graphs was introduced by Mahioub Shubatah [19] in 2012. Moreover, Haifa and Mahioub Shubatah [20] defined certain operations on ${\mathrm{P}}_f$ graphs and gave some results of domination for some operations on ${\mathrm{B}}_f$ graphs. Recently, a significant amount of research has been conducted on the domination of bipolar fuzzy graphs [21,22,23,24,25,26].

The motives of this study are as follows:

• Graph theory is a fundamental way of dealing with relations between objects using a figure comprising vertices and a line that joins these vertices. Domination plays a fundamental role in graph theory for solving problems of daily life. To deal with uncertainty and approximate reasoning, we use fuzzy graph models. Due to the bipolar facts that arise in real-life situations, the theory of fuzzy graphs is inadequate. However, depending on the problems, we use models of ${\mathrm{B}}_f$ graphs.
• To handle situations where the membership grade of an edge is either less than or greater than the product of the membership grades of its adjacent vertices, we use the theory of product bipolar fuzzy (${\mathrm{PB}}_f$) graphs, as bipolar fuzzy (${\mathrm{B}}_f$) graph theory does not always yield better results.

The main contributions of this article are as follows:

• The notion of complete ${\mathrm{PB}}_f$ graphs, the order and size of ${\mathrm{PB}}_f$ graphs, dominating sets (DSs), minimal dominating sets (MDSs), domination number (${\mathrm{D}}_n$), and operations like the complement of a ${\mathrm{PB}}_f$ graph, ∪, +, ∩, ×, ∘ on ${\mathrm{PB}}_f$ graphs are introduced in this article. Various results related to DSs, MDSs, and the ${\mathrm{D}}_n$ are discussed in this article, and some results of the domination of the above-mentioned operations are also studied.
• The importance of the given notions is studied with an application involving the placement of stations along a bus route and designing an algorithm to sort out this decision-making problem. Moreover, the comparison between ${\mathrm{B}}_f$ graphs and ${\mathrm{PB}}_f$ graphs is studied in this paper.

The abbreviations that we used in this article are given in

Table 1. List of abbreviations.

Abbreviations	Descriptions
${\mathrm{P}}_f$ graph	Product fuzzy graph
${\mathrm{B}}_f$ graph	Bipolar fuzzy graph
${\mathrm{PB}}_f$ graph	Product bipolar fuzzy graph
DS	Dominating set
DSs	Dominating sets
MDS	Minimal dominating set
MDSs	Minimal dominating sets
${\mathrm{D}}_n$	Domination number

.

2. Preliminaries

In this section, we define some elementary concepts and terminologies. For other concepts and terminologies that are not specified in this paper, we refer the reader to [9,10,19,27,28]. Let W be a non-empty set. A crisp graph ${\mathcal{G}}^{*}$ on W is a pair $(W,E)$, where W is called a vertex set, and $E\subseteq W\times W$ is the set of all edges of ${\mathcal{G}}^{*}$.

Definition 1 

([28]). A $B_f$ graph on W is a pair $\mathcal{J}$ = $(M,N)$, where M = $({\eta }_M^{+},{\eta }_M^{-})$ is a $B_f$ set on W, and N = $({\eta }_N^{+},{\eta }_N^{-})$ is a $B_f$ relation on W with the property that

$${\eta }_N^{+}\left(w_1{w}_2\right)⩽{\eta }_M^{+}\left(w_1\right)\wedge {\eta }_M^{+}\left(w_2\right) and {\eta }_N^{-}\left(w_1{w}_2\right)⩾{\eta }_M^{-}\left(w_1\right)\vee {\eta }_M^{-}\left(w_2\right),$$

for all $w_1,w_2\in W.$

Definition 2 

([29]). Let $\mathcal{J}$ = (M, N) be a $B_f$ graph on $W.$ An edge $w_1{w}_2$ of a $B_f$ graph is called an effective edge if it satisfies the following condition:

$${\eta }_N^{+}\left(w_1{w}_2\right)={\eta }_M^{+}\left(w_1\right)\wedge {\eta }_M^{+}\left(w_2\right) and {\eta }_N^{-}\left(w_1{w}_2\right)={\eta }_C^{-}\left(w_1\right)\vee {\eta }_C^{-}\left(w_2\right),$$

where $w_1,w_2\in W.$ Otherwise, edge $w_1{w}_2$ is called a non-effective edge. We say that $w_1$ dominates $w_2$ in a $B_f$ graph $\mathcal{J}$ if there exists an effective edge between $w_1$ and $w_2.$

Definition 3 

([29]). Let $\mathcal{J}=(M,N)$ be a $B_f$ graph on $W.$ Suppose $\mathcal{D}$ is a subset of W, and if for each $w_2\in W-\mathcal{D}$ there exists $w_1\in \mathcal{D}$ such that $w_1$ dominates $w_2$, then $\mathcal{D}$ is called a DS of a $B_f$ graph $\mathcal{J}.$

Definition 4 

([29]). Let $\mathcal{D}$ be a DS of a $B_f$ graph $\mathcal{J}.$ Then, $\mathcal{D}$ is called an MDS if no proper subset of $\mathcal{D}$ is a DS of a $B_f$ graph $\mathcal{J}.$

Definition 5 

([29]). The minimum cardinality among all MDSs of a $B_f$ graph $\mathcal{J}$ is called a $D_n$ of a $B_f$ graph. It is represented as $⋎\left(\mathcal{J}\right)$ or $⋎.$

Definition 6 

([30]). A ${\mathit{PB}}_f$ graph on W is a pair $\mathcal{H}=(C,D)$, where $C=({\eta }_C^{+},{\eta }_C^{-})$ is a $B_f$ set on W, and $D=({\eta }_D^{+},{\eta }_D^{-})$ is a $B_f$ relation on W with the property that

$${\eta }_D^{+}\left(w_1{w}_2\right)\le {\eta }_C^{+}\left(w_1\right)\times {\eta }_C^{+}\left(w_2\right) and {\eta }_D^{-}\left(w_1{w}_2\right)\ge -({\eta }_C^{-}\left(w_1\right)\times {\eta }_C^{-}\left(w_2\right)),$$

for all $w_1,w_2\in W.$ Thus, $\mathcal{H}=(C,D)$ is a ${\mathit{PB}}_f$ graph of ${\mathcal{G}}^{*}=(W,E)$ if

$${\eta }_D^{+}\left(w_1{w}_2\right)\le {\eta }_C^{+}\left(w_1\right)\times {\eta }_C^{+}\left(w_2\right)and {\eta }_D^{-}\left(w_1{w}_2\right)\ge -({\eta }_C^{-}\left(w_1\right)\times {\eta }_C^{-}\left(w_2\right)),$$

for all $w_1{w}_2\in E.$

Definition 7 

([30]). A ${\mathit{PB}}_f$ graph of ${\mathcal{G}}^{*}$ is called a strong ${\mathit{PB}}_f$ graph if

$${\eta }_D^{+}\left(w_1{w}_2\right)={\eta }_C^{+}\left(w_1\right)\times {\eta }_C^{+}\left(w_2\right) and {\eta }_D^{-}\left(w_1{w}_2\right)=-({\eta }_C^{-}\left(w_1\right)\times {\eta }_C^{-}\left(w_2\right)),$$

for all $w_1{w}_2\in E.$

3. Domination in Product Bipolar Fuzzy Graphs

In this section, we define the notions of DSs, MDSs, and the ${\mathrm{D}}_n$ in a ${\mathrm{PB}}_f$ graph. We also evaluate the DS, MDS, and ${\mathrm{D}}_n$ of a ${\mathrm{PB}}_f$ graph.

Definition 8. 

Let $\mathcal{H}$ = $(C,D)$ be a ${\mathit{PB}}_f$ graph of ${\mathcal{G}}^{*}.$ The order of a ${\mathit{PB}}_f$ graph $\mathcal{H}$ is p, where

$$p=\sum _{w\in W}\left({\displaystyle \frac{1+{\eta }_C^{+}\left(w\right)+{\eta }_C^{-}\left(w\right)}{2}}\right).$$

The size of $\mathcal{H}$ is q, where

$$q=\sum _{w_1{w}_2\in E}\left({\displaystyle \frac{1+{\eta }_D^{+}\left(w_1{w}_2\right)+{\eta }_D^{-}\left(w_1{w}_2\right)}{2}}\right).$$

If $W^{\prime }\subseteq W$, then the cardinality of $W^{\prime }$ is defined as

$$|W^{\prime }|=\sum _{w^{\prime }\in W^{\prime }}\left({\displaystyle \frac{1+{\eta }_C^{+}\left(w^{\prime }\right)+{\eta }_C^{-}\left(w^{\prime }\right)}{2}}\right).$$

Definition 9. 

Let $\mathcal{H}$ = $(C,D)$ be a ${\mathit{PB}}_f$ graph on $W.$ If

$${\eta }_D^{+}\left(w_1{w}_2\right)={\eta }_C^{+}\left(w_1\right)\times {\eta }_C^{+}\left(w_2\right) and {\eta }_D^{-}\left(w_1{w}_2\right)=-({\eta }_C^{-}\left(w_1\right)\times {\eta }_C^{-}\left(w_2\right)),$$

for all $w_1,w_2\in W$, then a ${\mathit{PB}}_f$ graph is called a complete ${\mathit{PB}}_f$ graph. It is represented as ${\mathcal{K}}_C.$

Remark 1. 

Every ${\mathit{PB}}_f$ graph is a $B_f$ graph. Let $\mathcal{H}$ = $(C,D)$ be a ${\mathit{PB}}_f$ graph on W. Then, ${\eta }_C^{+}\left(w_1\right)$ and ${\eta }_C^{+}\left(w_2\right)$ are less than or equal to 1; likewise, ${\eta }_C^{-}\left(w_1\right)$ and ${\eta }_C^{-}\left(w_2\right))$ are greater than or equal to −1. It follows that

$$\begin{gathered} {\eta }_D^{+}\left(w_1{w}_2\right)\le {\eta }_C^{+}\left(w_1\right)\times {\eta }_C^{+}\left(w_2\right)\le {\eta }_C^{+}\left(w_1\right)\wedge {\eta }_C^{+}\left(w_2\right) and \\ {\eta }_D^{-}\left(w_1{w}_2\right)\ge -({\eta }_C^{-}\left(w_1\right)\times {\eta }_C^{-}\left(w_2\right))\ge {\eta }_C^{-}\left(w_1\right)\vee {\eta }_C^{-}\left(w_2\right), \end{gathered}$$

for all $w_1,w_2\in W.$ Hence, $\mathcal{H}$ = $(C,D)$ is a $B_f$ graph. Thus, every ${\mathit{PB}}_f$ graph is a $B_f$ graph.

Definition 10. 

Let $\mathcal{H}$ = (C, D) be a ${\mathit{PB}}_f$ graph on $W.$ An edge $w_1{w}_2$ of a ${\mathit{PB}}_f$ graph is called an effective edge if it satisfies the following condition:

$${\eta }_D^{+}\left(w_1{w}_2\right)={\eta }_C^{+}\left(w_1\right)\times {\eta }_C^{+}\left(w_2\right) and {\eta }_D^{-}\left(w_1{w}_2\right)=-({\eta }_C^{-}\left(w_1\right)\times {\eta }_C^{-}\left(w_2\right)),$$

where $w_1,w_2\in W.$ Otherwise, edge $w_1{w}_2$ is called a non-effective edge.

Example 1. 

Consider the crisp set $W=\{w_1,w_2,w_3,w_4\}$. Let C be a $B_f$ set on W and D be a $B_f$ relation in W as specified in

Table 2. $B_f$ set C and $B_f$ relation D.

w	(${\mathit{\eta }}_{\mathit{C}}^{+},{\mathit{\eta }}_{\mathit{C}}^{-}$)	w	(${\mathit{\eta }}_{\mathit{D}}^{+},{\mathit{\eta }}_{\mathit{D}}^{-}$)
$w_1$	$(0.75,-0.1)$	$w_1{w}_2$	$(0.3,-0.02)$
$w_2$	$(0.5,-0.3)$	$w_1{w}_3$	$(0.1875,-0.01)$
$w_3$	$(0.25,-0.1)$	$w_2{w}_3$	$(0.125,-0.03)$
$w_4$	$(0.2,-0.2)$	$w_3{w}_4$	$(0.05,-0.02)$

. The ${\mathit{PB}}_f$ graph $\mathcal{H}$ is given in Figure 1.

It is clear that $w_2{w}_3$ is an effective edge of a ${\mathit{PB}}_f$ graph $\mathcal{H}$ because

$$\begin{array}{ccc}\hfill {\eta }_D^{+}\left(w_2{w}_3\right)& =& 0.125and{\eta }_D^{-}\left(w_2{w}_3\right)=-0.03,\hfill \\ \hfill {\eta }_C^{+}\left(w_2\right)\times {\eta }_C^{+}\left(w_3\right)& =& 0.5\times 0.25=0.125and{\eta }_C^{-}\left(w_2\right)\times {\eta }_C^{-}\left(w_3\right)=-0.3\times -0.1=0.03.So,\hfill \\ \hfill 0.125& =& 0.125and-0.03=-0.03.\hfill \end{array}$$

Similarly, $w_1{w}_3$ and $w_3{w}_4$ are effective edges of a ${\mathit{PB}}_f$ graph $\mathcal{H}.$ However, $w_1{w}_2$ is a non-effective edge of a ${\mathit{PB}}_f$ graph because

$$\begin{array}{ccc}\hfill {\eta }_D^{+}\left(w_1{w}_2\right)& =& 0.3and{\eta }_D^{-}\left(w_1{w}_2\right)=-0.02,\hfill \\ \hfill {\eta }_C^{+}\left(w_1\right)\times {\eta }_C^{+}\left(w_2\right)& =& 0.75\times 0.5=0.375and{\eta }_C^{-}\left(w_1\right)\times {\eta }_C^{-}\left(w_2\right)=-0.1\times -0.3=0.03.So,\hfill \\ \hfill 0.3& \ne & 0.375and-0.02\ne -0.03.\hfill \end{array}$$

Definition 11. 

Let $w_1,w_2\in W$ be two vertices of a ${\mathit{PB}}_f$ graph $\mathcal{H}$. We say that $w_1$ dominates $w_2$ in a ${\mathit{PB}}_f$ graph $\mathcal{H}$ if there exists an effective edge between $w_1$ and $w_2.$

From Example 1, it is clear that $w_2$ dominates $w_3$ because $w_2$ and $w_3$ have an effective edge between them. Similarly, $w_1$ dominates $w_3$, and $w_3$ dominates $w_4.$

Definition 12. 

Let $\mathcal{H}$ = $(C,D)$ be a ${\mathit{PB}}_f$ graph on $W.$ The neighborhood of $w_1\in W$ in $\mathcal{H}$ is defined as follows:

$$N\left(w_1\right)=\{w_2:w_2\in W and w_1{w}_2 is an effective edge in \mathcal{H}\}.$$

The closed neighborhood of $w_1$ in $\mathcal{H}$ is defined as follows:

$$N\left[w_1\right]=N\left(w_1\right)\cup \left\{w_1\right\}.$$

Example 2. 

From Figure 1, the following are clear:

1. 

$N\left(w_1\right)$ = $\left\{w_3\right\}$, $N\left(w_2\right)$ = $\left\{w_3\right\}$, $N\left(w_3\right)$ = $\{w_1,w_2,w_4\}$, and $N\left(w_4\right)$ = $\left\{w_3\right\}.$

2. 

$N\left[w_1\right]$ = $\{w_1,w_3\}$, $N\left(w_2\right)$ = $\{w_2,w_3\}$, $N\left(w_3\right)$ = $\{w_1,w_2,w_3,w_4\}$, and $N\left(w_4\right)$ = $\{w_3,w_4\}.$

Definition 13. 

Let $\mathcal{H}=(C,D)$ be a ${\mathit{PB}}_f$ graph on $W.$ Suppose $\mathcal{D}$ is a subset of W, and if for each $w_2\in W-\mathcal{D}$ there exists $w_1\in \mathcal{D}$ such that

$${\eta }_D^{+}\left(w_1{w}_2\right)={\eta }_C^{+}\left(w_1\right)\times {\eta }_C^{+}\left(w_2\right) and {\eta }_D^{-}\left(w_1{w}_2\right)=-({\eta }_C^{-}\left(w_1\right)\times {\eta }_C^{-}\left(w_2\right)),$$

then $\mathcal{D}$ is called a DS of a ${\mathit{PB}}_f$ graph $\mathcal{H}.$

Example 3. 

Consider the ${\mathit{PB}}_f$ graph $\mathcal{H}$ on W = $\{w_1,w_2,w_3,w_4\}$, as shown in Figure 1. The following are the subsets of W:

$$\begin{array}{c}\hfill \{\varphi ,\left\{w_1\right\},\left\{w_2\right\},\left\{w_3\right\},\left\{w_4\right\},\{w_1,w_2\},\{w_1,w_3\},\{w_1,w_4\},\{w_2,w_3\},\{w_2,w_4\},\\ \hfill \{w_3,w_4\},\{w_1,w_2,w_3\},\{w_1,w_2,w_4\},\{w_1,w_3,w_4\},\{w_2,w_3,w_4\},W\}.\end{array}$$

1. 

Let ${\mathcal{D}}_1$ = $\{w_1,w_3\},$ $W-{\mathcal{D}}_1$ = $\{w_2,w_4\},$ $w_3$ dominate $w_2$, and $w_3$ dominate $w_4$. So, by Definition 13, ${\mathcal{D}}_1$ = $\{w_1,w_3\}$ is a DS of a ${\mathit{PB}}_f$ graph $\mathcal{H}.$

2. 

Let ${\mathcal{D}}_2$ = $\{w_1,w_2\},$ $W-{\mathcal{D}}_2$ = $\{w_3,w_4\},$ and $w_1$ and $w_2$ dominate $w_3$, but neither $w_1$ nor $w_2$ dominate $w_4.$ So, ${\mathcal{D}}_2$ = $\{w_1,w_2\}$ is not a DS of a ${\mathit{PB}}_f$ graph $\mathcal{H}.$

3. 

Similarly, $\left\{w_3\right\}$ is a DS of a ${\mathit{PB}}_f$ graph $\mathcal{H}$, but $\left\{w_1\right\}$, $\left\{w_2\right\}$, and $\left\{w_4\right\}$ are not DSs of a ${\mathit{PB}}_f$ graph $\mathcal{H}.$

Definition 14. 

Let $\mathcal{D}$ be a DS of a ${\mathit{PB}}_f$ graph $\mathcal{H}.$ Then, $\mathcal{D}$ is called an MDS if no proper subset of $\mathcal{D}$ is a DS of a ${\mathit{PB}}_f$ graph $\mathcal{H}.$

Example 4. 

From Example 3, it is clear that $\{w_1,w_3\}$ is not an MDS because $\left\{w_3\right\}$ is a DS of $\mathcal{H}.$ The DS $\left\{w_3\right\}$ is an MDS because there does not exist any proper subset of $\left\{w_3\right\}$ that is dominating in $\mathcal{H}.$

Table 3. DSs and MDSs.

DSs	MDSs
$\left\{w_3\right\}$, $\{w_2,w_3\}$, $\{w_1,w_3\}$	$\left\{w_3\right\}$
$\{w_3,w_4\}$, $\{w_1,w_2,w_3\}$	$\{w_1,w_2,w_4\}$
$\{w_1,w_2,w_4\}$, $\{w_2,w_3,w_4\}$
$\{w_1,w_3,w_4\}$, $\{w_1,w_2,w_3,w_4\}$

shows all DSs and MDSs of a ${\mathrm{PB}}_f$ graph $\mathcal{H}$ (Figure 1).

Theorem 1. 

A DS $\mathcal{D}$ of $\mathcal{H}$ is an MDS if and only if every element $w\in \mathcal{D}$ satisfies at least one of the following conditions:

1. 

w does not dominate the elements in $\mathcal{D}.$

2. 

There exists $w^{\prime }\in W-\mathcal{D}$ such that $w\in \mathcal{D}$ is the only element in $\mathcal{D}$ that dominates $w^{\prime }.$

Proof. 

Suppose $\mathcal{D}$ is an MDS of $\mathcal{H}$ and $w\in \mathcal{D}.$ Now, we want to prove that every element $w\in \mathcal{D}$ satisfies at least one of conditions 1 and 2. $\mathcal{D}$ is an MDS, so $\mathcal{D}-\left\{w\right\}$ is not a DS of $\mathcal{H}$. Then, there exists $w^{\prime }\in W-(\mathcal{D}-\left\{w\right\})$ that is not dominated by any element of $\mathcal{D}-\left\{w\right\}.$ Here, two cases arise:

• Case (i) If $w^{\prime }=w$, then $w^{\prime }$ does not dominate the elements in $\mathcal{D}.$ Suppose $w^{\prime }$ dominates the elements in $\mathcal{D}$. Then, $\mathcal{D}-\left\{w\right\}$ is a DS of $\mathcal{H}.$ This is a contradiction, so no element in $\mathcal{D}$ dominates w. Hence, property 1 holds.
• Case (ii) If $w^{\prime }\ne w,$ then from above we know that $w^{\prime }\in W-(\mathcal{D}-\left\{w\right\}).$ Since $\mathcal{D}$ is a DS of $\mathcal{H}$, then there exists $w_1\in \mathcal{D}$ such that $w_1$ dominates $w^{\prime }.$ However, $w^{\prime }$ is not dominated by any element of $\mathcal{D}-\left\{w\right\}.$ Thus, $w_1=w;$ therefore $w\in \mathcal{D}$ is the only element in $\mathcal{D}$ that dominates $w^{\prime }.$ Hence, property 2 holds.

Conversely, suppose $\mathcal{D}$ is a DS, and every element $w\in \mathcal{D}$ satisfies at least one of conditions 1 and 2. Now, we want to prove that $\mathcal{D}$ is an MDS of $\mathcal{H}$:

• Suppose $\mathcal{D}$ is not an MDS of $\mathcal{H}.$ Then, there exists a $w\in \mathcal{D}$ such that $\mathcal{D}-\left\{w\right\}$ is a DS. Hence, w dominates the elements in $\mathcal{D}-\left\{w\right\}.$ Condition 1 does not hold.
• If $\mathcal{D}-\left\{w\right\}$ is a DS, then for each element in $W-(\mathcal{D}-\{w\left\}\right)$, there exists an element in $\mathcal{D}-\left\{w\right\}$ that dominates the elements in $W-(\mathcal{D}-\{w\left\}\right).$ Condition 2 does not hold.

This is a contradiction; hence $\mathcal{D}$, is an MDS of $\mathcal{H}.$    □

We can also illustrate Theorem 1 in another way, which is the following.

Theorem 2. 

A DS $\mathcal{D}$ of $\mathcal{H}$ is an MDS if and only if for each $w\in \mathcal{D}$, one of the following two conditions holds:

1. 

$N\left(w\right)\cap \mathcal{D}$ = $\varphi .$

2. 

There is a vertex $w^{\prime }\in W-\mathcal{D}$ such that $N\left(w^{\prime }\right)\cap \mathcal{D}$ = $w.$

Proof. 

Let $\mathcal{D}$ be an MDS and $w\in \mathcal{D}.$ Then, ${\mathcal{D}}_w$ = $\mathcal{D}-\left\{w\right\}$ is not a DS, and hence, there exists $w^{\prime }\in W-{\mathcal{D}}_w$ such that $w^{\prime }$ is not dominated by any element of ${\mathcal{D}}_w.$

• If $w^{\prime }=w$, then w is not dominated by any element in $\mathcal{D}$, which implies that $N\left(w\right)\in \mathcal{D}$ = $\varphi .$
• If $w^{\prime }\ne w$, then $w^{\prime }$ is not dominated by any element in ${\mathcal{D}}_w$, but in $\mathcal{D}$, element $w^{\prime }$ only dominates $w\in \mathcal{D}.$ Hence, $N\left(w^{\prime }\right)\cap \mathcal{D}$ = $w.$ The converse is obvious.   □

Definition 15. 

Let $\mathcal{H}$ = $(C,D)$ be a ${\mathit{PB}}_f$ graph on $W.$ A vertex $w_1$ of a ${\mathit{PB}}_f$ graph is said to be an isolated vertex if ${\eta }_D^{+}\left(w_1{w}_2\right)<{\eta }_C^{+}\left(w_1\right)\times {\eta }_C^{+}\left(w_2\right)$ and ${\eta }_D^{-}\left(w_1{w}_2\right)>-({\eta }_C^{-}\left(w_1\right)\times {\eta }_C^{-}\left(w_2\right))$ for all $w_2\in W-\left\{w_1\right\},$ that is, $N\left(w_1\right)=\varphi .$

An isolated vertex does not dominate any other vertex in $\mathcal{H}.$ In Figure 1, there exists no isolated vertex.

Theorem 3. 

Let $\mathcal{H}$ be a ${\mathit{PB}}_f$ graph on W without isolated vertices. If $\mathcal{D}$ is an MDS of $\mathcal{H}$, then $W-\mathcal{D}$ is a DS of $\mathcal{H}.$

Proof. 

Let $\mathcal{D}$ be an MDS of $\mathcal{H}$ and $w\in \mathcal{D}.$ Since $\mathcal{H}$ has no isolated vertices, there is an element $w_1\in N\left(w\right).$ It follows from Theorem 2 that $w_1\in W-\mathcal{D}.$ Element w was arbitrary; the result holds for every $w\in \mathcal{D}.$ Thus, every element of D is dominated by some element of $W-\mathcal{D}.$ Hence, $W-\mathcal{D}$ is a DS of $\mathcal{H}.$    □

Definition 16. 

The minimum cardinality among all MDSs of a ${\mathit{PB}}_f$ graph $\mathcal{H}$ is called a $D_n$ of a ${\mathit{PB}}_f$ graph $\mathcal{H}.$ It is represented as $⋎\left(\mathcal{H}\right)$ or $⋎.$

Definition 17. 

The maximum cardinality among all MDSs of a ${\mathit{PB}}_f$ graph $\mathcal{H}$ is called an upper $D_n$ of a ${\mathit{PB}}_f$ graph $\mathcal{H}$. It is represented as $\Gamma \left(\mathcal{H}\right)$ or $\Gamma .$

Definition 18. 

A DS $\mathcal{D}$ of cardinality ⋎ is called a minimum DS or ⋎-set.

Example 5. 

Consider a ${\mathit{PB}}_f$ graph $\mathcal{H}$, as shown in Figure 1. All MDSs of a ${\mathit{PB}}_f$ graph (Figure 1) are given in Table 3.

1. 

Let ${\mathcal{D}}_1$ = $\left\{w_3\right\}$ and ${\mathcal{D}}_2$ = $\{w_1,w_2,w_4\}$, so $|{\mathcal{D}}_1|$ = $\left({\displaystyle \frac{1+0.25-0.1}{2}}\right)$ = $0.575$ and $|{\mathcal{D}}_2|$ = $({\displaystyle \frac{1+0.75-0.1}{2}}+{\displaystyle \frac{1+0.5-0.3}{2}}+{\displaystyle \frac{1+0.2-0.2}{3}})$ = $1.925$. Hence, the $D_n$ of $\mathcal{H}$ is $⋎\left(\mathcal{H}\right)$ = min$\{0.575,1.925\}$ = $0.575$.

2. 

The upper $D_n$ of a ${\mathit{PB}}_f$ graph $\mathcal{H}$ is $\Gamma \left(\mathcal{H}\right)$ = max$\{0.575,1.925\}$ = $1.925$.

3. 

Furthermore, $\left\{w_3\right\}$ is the ⋎-set.

Theorem 4. 

For any ${\mathit{PB}}_f$ graph without isolated vertices, $⋎\le p/2.$

Proof. 

Let $\mathcal{D}$ be an MDS of $\mathcal{H}$. By Theorem 3, we know that $W-\mathcal{D}$ is a DS of $\mathcal{H}$. Then, $⋎\le \left|\mathcal{D}\right|$ and $⋎\le |W-\mathcal{D}|$. However, $2⋎\le$$\left|\mathcal{D}\right|+|W-\mathcal{D}|$. Since $\left|\mathcal{D}\right|+|W-\mathcal{D}|$ = p, $2⋎\le p$, which implies that $⋎\le p/2$.    □

Remark 2. 

Let $\mathcal{H}$ = $(C,D)$ be a ${\mathit{PB}}_f$ graph on W:

• If $w_1$ dominates $w_2$, then $w_2$ dominates $w_1$, where $w_1,w_2\in W.$ Hence, domination is a symmetric relation on $W.$
• If ${\eta }_D^{+}\left(w_1{w}_2\right)<{\eta }_C^{+}\left(w_1\right)\times {\eta }_C^{+}\left(w_2\right)$ and ${\eta }_D^{-}\left(w_1{w}_2\right)>-({\eta }_C^{-}\left(w_1\right)\times {\eta }_C^{-}\left(w_2\right))$ for all $w_1,w_2\in W,$ then W is the only DS of $\mathcal{H}.$
• $⋎\left(\mathcal{H}\right)$ = p if and only if ${\eta }_D^{+}\left(w_1{w}_2\right)<{\eta }_C^{+}\left(w_1\right)\times {\eta }_C^{+}\left(w_2\right)$ and ${\eta }_D^{-}\left(w_1{w}_2\right)>-({\eta }_C^{-}\left(w_1\right)\times {\eta }_C^{-}\left(w_2\right))$ for all $w_1,w_2\in W.$
• For any $w_1\in W,$ $N\left(w_1\right)$ is precisely the set of all $w_2\in W$ that are dominated by $w_1.$

4. Operations on Product Bipolar Fuzzy Graphs

In this section, we define certain operations on ${\mathrm{PB}}_f$ graphs, including the complement of a ${\mathrm{PB}}_f$ graph, ∪, +, ∩, ×, and ∘ of two ${\mathrm{PB}}_f$ graphs to yield new ${\mathrm{PB}}_f$ graphs. The complement of a ${\mathrm{PB}}_f$ graph does not acquire the characteristics of the complement of a crisp graph. We give several results of domination in these operations and also present relations of the complete ${\mathrm{PB}}_f$ graph of the products of ${\mathrm{PB}}_f$ graphs.

4.1. Complement of a Product Bipolar Fuzzy Graph

Definition 19. 

Let $\mathcal{H}$ = $(C,D)$ be a ${\mathit{PB}}_f$ graph on $W.$ The complement of a ${\mathit{PB}}_f$ graph $\mathcal{H}$ = $(C,D)$ is a ${\mathit{PB}}_f$ graph $\overline{\mathcal{H}}$ = $(\overline{C},\overline{D})$ with the following:

(i) 

$\overline{C}$ = C.

(ii) 

$({\eta }_{\overline{C}}^{+}\left(w\right),{\eta }_{\overline{C}}^{-}\left(w\right))$ = $({\eta }_C^{+}\left(w\right),{\eta }_C^{-}\left(w\right))$ for all $w\in W.$

(iii) 

${\eta }_{\overline{D}}^{+}\left(w_1{w}_2\right)$ = ${\eta }_C^{+}\left(w_1\right)\times {\eta }_C^{+}\left(w_2\right)$−${\eta }_D^{+}\left(w_1{w}_2\right)$ and ${\eta }_{\overline{D}}^{-}\left(w_1{w}_2\right)$ = $-({\eta }_C^{-}\left(w_1\right)\times {\eta }_C^{-}\left(w_2\right))$−${\eta }_D^{-}\left(w_1{w}_2\right)$

for all $w_1,w_2\in W.$

Example 6. 

Consider a ${\mathit{PB}}_f$ graph $\mathcal{H}$ = $(C,D)$ on W as shown in Figure 2.

The membership grade of vertex set $\overline{C}$ = $({\eta }_{\overline{C}}^{+},{\eta }_{\overline{C}}^{-})$ and edge set $\overline{D}$ = $({\eta }_{\overline{D}}^{+},{\eta }_{\overline{D}}^{-})$ of $\overline{\mathcal{H}}$ are specified in

Table 4. Membership grades of $\overline{C}$ and $\overline{D}$.

w	(${\mathit{\eta }}_{\overline{\mathit{C}}}^{+},{\mathit{\eta }}_{\overline{\mathit{C}}}^{-}$)	w	(${\mathit{\eta }}_{\overline{\mathit{D}}}^{+},{\mathit{\eta }}_{\overline{\mathit{D}}}^{-}$)
$w_1$	$(0.2,-0.4)$	$w_1{w}_3$	$(0.06,-0.24)$
$w_2$	$(0.3,-0.7)$	$w_1{w}_4$	$(0.01,-0.02)$
$w_3$	$(0.3,-0.6)$	$w_2{w}_3$	$(0.01,-0.02)$
$w_4$	$(0.4,-0.5)$	$w_2{w}_4$	$(0.12,-0.35)$

.

The complement of a ${\mathrm{PB}}_f$ graph of Figure 2 is given in Figure 3.

Remark 3. 

Let $\mathcal{H}$ = $(C,D)$ be a ${\mathit{PB}}_f$ graph on W:

• The complement of a complete ${\mathit{PB}}_f$ graph is always a null graph.
• $⋎\left(\overline{{\mathcal{K}}_C}\right)$ = $p.$
• If $\mathcal{H}$ is a ${\mathit{PB}}_f$ graph, then $\overline{\overline{\mathcal{H}}}=\mathcal{H}.$

  $$\begin{array}{ccc}\hfill \overline{\left(\overline{D}\right)}\left(w_1{w}_2\right)& =& \overline{({\eta }_{\overline{D}}^{+},{\eta }_{\overline{D}}^{-})}\left(w_1{w}_2\right),\hfill \\ & =& (\overline{\left({\eta }_{\overline{D}}^{+}\right)}\left(w_1{w}_2\right),\overline{\left({\eta }_{\overline{D}}^{-}\right)}\left(w_1{w}_2\right)),\hfill \\ & =& ({\eta }_C^{+}\left(w_1\right)\times {\eta }_C^{+}\left(w_2\right)-{\eta }_{\overline{D}}^{+}\left(w_1{w}_2\right),-({\eta }_C^{-}\left(w_1\right)\times {\eta }_C^{-}\left(w_2\right))-{\eta }_{\overline{D}}^{-}\left(w_1{w}_2\right)),\hfill \\ & =& ({\eta }_C^{+}\left(w_1\right)\times {\eta }_C^{+}\left(w_2\right)-[{\eta }_C^{+}\left(w_1\right)\times {\eta }_C^{+}\left(w_2\right)-{\eta }_D^{+}\left(w_1{w}_2\right)],\hfill \\ & & -({\eta }_C^{-}\left(w_1\right)\times {\eta }_C^{-}\left(w_2\right))-[-({\eta }_C^{-}\left(w_1\right)\times {\eta }_C^{-}\left(w_2\right))-{\eta }_D^{-}\left(w_1{w}_2\right)]),\hfill \\ & =& ({\eta }_D^{+}\left(w_1{w}_2\right),{\eta }_D^{-}\left(w_1{w}_2\right)),\hfill \\ \hfill \overline{\left(\overline{D}\right)}\left(w_1{w}_2\right)& =& D\left(w_1{w}_2\right),\hfill \\ \hfill so,\overline{\overline{\mathcal{H}}}& =& \mathcal{H}.\hfill \end{array}$$

Theorem 5. 

Let $\mathcal{H}$ be a ${\mathit{PB}}_f$ graph on W. Then, $⋎+\overline{⋎}\le 2p$, where $\overline{⋎}$ is the $D_n$ of $\overline{\mathcal{H}}$ and $⋎+\overline{⋎}=2p$ if and only if $0<{\eta }_D^{+}\left(w_1{w}_2\right)<{\eta }_C^{+}\left(w_1\right)\times {\eta }_C^{+}\left(w_2\right)$ and $0>{\eta }_D^{-}\left(w_1{w}_2\right)>-({\eta }_C^{-}\left(w_1\right)\times {\eta }_C^{-}\left(w_2\right))$ for all $w_1,w_2\in W.$

Proof. 

The inequality is trivial. Suppose $⋎+\overline{⋎}=2p$. Then, we want to prove that $0<{\eta }_D^{+}(w_1,w_2)<{\eta }_C^{+}\left(w_1\right)\times {\eta }_C^{+}\left(w_2\right)$ and $0>{\eta }_D^{-}(w_1,w_2)>-({\eta }_C^{-}\left(w_1\right)\times {\eta }_C^{-}\left(w_2\right))$ for all $w_1,w_2\in W.$$⋎+\overline{⋎}=2p$ means that $⋎=p$ and $\overline{⋎}=p.$ We know that $⋎=p$ if and only if ${\eta }_D^{+}\left(w_1{w}_2\right)<{\eta }_C^{+}\left(w_1\right)\times {\eta }_C^{+}\left(w_2\right)$ and ${\eta }_D^{-}\left(w_1{w}_2\right)>-({\eta }_C^{-}\left(w_1\right)\times {\eta }_C^{-}\left(w_2\right))$ for all $w_1,w_2\in W$ and $\overline{⋎}=p$ if and only if ${\eta }_{\overline{D}}^{+}\left(w_1{w}_2\right)$ = ${\eta }_C^{+}\left(w_1\right)\times {\eta }_C^{+}\left(w_2\right)-{\eta }_D^{+}\left(w_1{w}_2\right)<{\eta }_C^{+}\left(w_1\right)\times {\eta }_C^{+}\left(w_2\right)$ and ${\eta }_{\overline{D}}^{-}\left(w_1{w}_2\right)$ = $-({\eta }_C^{-}\left(w_1\right)\times {\eta }_C^{-}\left(w_2\right))-{\eta }_D^{-}(w_1,w_2)>-({\eta }_C^{-}\left(w_1\right)\times {\eta }_C^{-}\left(w_2\right))$ for all $w_1,w_2\in W.$ Further, ${\eta }_D^{+}\left(w_1{w}_2\right)>0$, and ${\eta }_D^{-}\left(w_1{w}_2\right)<0.$ Hence, $⋎+\overline{⋎}=2p$ if and only if $0<{\eta }_D^{+}\left(w_1{w}_2\right)<{\eta }_C^{+}\left(w_1\right)\times {\eta }_C^{+}\left(w_2\right)$ and $0>{\eta }_D^{-}\left(w_1{w}_2\right)>-({\eta }_C^{-}\left(w_1\right)\times {\eta }_C^{-}\left(w_2\right))$ for all $w_1,w_2\in W.$    □

Corollary 1. 

Let $\mathcal{H}$ be a ${\mathit{PB}}_f$ graph in such a way that both $\mathcal{H}$ and $\overline{\mathcal{H}}$ have no isolated vertices. Then, $⋎+\overline{⋎}\le p$. Moreover, $⋎+\overline{⋎}=p$ if and only if $⋎=\overline{⋎}=p/2.$

Proof. 

By Theorem 4, we know that $⋎\le p/2$ and $\overline{⋎}\le p/2$

$$\Rightarrow ⋎+\overline{⋎}\le p/2+p/2=p.$$

Hence, $⋎+\overline{⋎}\le p.$ Now, let $⋎=\overline{⋎}=p/2$, and we want to prove that $⋎+\overline{⋎}=p.$ If $⋎=\overline{⋎}=p/2$, then $⋎+\overline{⋎}\le p/2+p/2$ = $p.$ Conversely, suppose that $⋎+\overline{⋎}=p.$ By Theorem 4, $⋎\le p/2$ and $\overline{⋎}\le p/2.$ If either $⋎<p/2$ or $\overline{⋎}<p/2$ or both $⋎<p/2$ and $\overline{⋎}<p/2$, then our supposition will be wrong. However, the only option is that $⋎=\overline{⋎}=p/2.$ Hence, the proof is complete.    □

4.2. Union of Two Product Bipolar Fuzzy Graphs

Definition 20. 

Let ${\mathcal{H}}_1$ = $(C_1,D_1)$ and ${\mathcal{H}}_2$ = $(C_2,D_2)$ be two ${\mathit{PB}}_f$ graphs on $W_1$ and $W_2$, respectively, with $W_1\cap W_2=\varphi$. The union of two ${\mathit{PB}}_f$ graphs ${\mathcal{H}}_1$ and ${\mathcal{H}}_2$ is denoted as $\mathcal{H}$ = $({\mathcal{H}}_1\cup {\mathcal{H}}_2)$ = $(C_1\cup C_2,D_1\cup D_2)$ and defined as follows:

$$\begin{gathered} ({\eta }_{C_1}^{+}\cup {\eta }_{C_2}^{+})\left(w\right)=\left\{\begin{array}{cc}{\eta }_{C_1}^{+}\left(w\right),\hfill & \mathrm{if}w\in W_1,\hfill \\ {\eta }_{C_2}^{+}\left(w\right),\hfill & \mathrm{if}w\in W_2.\hfill \end{array}\right. \\ ({\eta }_{C_1}^{-}\cup {\eta }_{C_2}^{-})\left(w\right)=\left\{\begin{array}{cc}{\eta }_{C_1}^{-}\left(w\right),\hfill & \mathrm{if}w\in W_1,\hfill \\ {\eta }_{C_2}^{-}\left(w\right),\hfill & \mathrm{if}w\in W_2.\hfill \end{array}\right. \end{gathered}$$

for all $w\in W_1\cup W_2$, and

$$\begin{gathered} ({\eta }_{D_1}^{+}\cup {\eta }_{D_2}^{+})\left(w_1{w}_2\right)=\left\{\begin{array}{cc}{\eta }_{D_1}^{+}\left(w_1{w}_2\right),\hfill & \mathrm{if}{w}_1,w_2\in W_1,\hfill \\ {\eta }_{D_2}^{+}\left(w_1{w}_2\right),\hfill & \mathrm{if}{w}_1,w_2\in W_2.\hfill \end{array}\right. \\ ({\eta }_{D_1}^{-}\cup {\eta }_{D_2}^{-})\left(w_1{w}_2\right)=\left\{\begin{array}{cc}{\eta }_{D_1}^{-}\left(w_1{w}_2\right),\hfill & \mathrm{if}{w}_1,w_2\in W_1,\hfill \\ {\eta }_{D_2}^{-}\left(w_1{w}_2\right),\hfill & \mathrm{if}{w}_1,w_2\in W_2.\hfill \end{array}\right. \end{gathered}$$

for all $w\in W_1\cup W_2.$

Remark 4. 

It is not necessary that the union of two complete ${\mathit{PB}}_f$ graphs is a complete ${\mathit{PB}}_f$ graph.

In the following, we give a counterexample that shows that the union of two complete ${\mathrm{PB}}_f$ graphs is not a complete ${\mathrm{PB}}_f$ graph.

Example 7. 

Consider two complete ${\mathit{PB}}_f$ graphs ${\mathcal{H}}_1$ = $(C_1,D_1)$ and ${\mathcal{H}}_2$ = $(C_2,D_2)$ on $W_1$ and $W_2$, respectively, as shown in Figure 4.

The vertex set $C=(C_1\cup C_2)$ and edge set $D=(D_1\cup D_2)$ of $\mathcal{H}$ = $({\mathcal{H}}_1\cup {\mathcal{H}}_2)$ are $\{w_1,w_2,w_3,w_1^{\prime },w_2^{\prime }\}$ and $\{w_1{w}_2,w_1{w}_3,w_2{w}_3,w_1^{\prime }{w}_2^{\prime }\}$. The membership grades of both $C_1\cup C_2$ and $D_1\cup D_2$ of ${\mathcal{H}}_1\cup {\mathcal{H}}_2$ are given in

Table 5. Membership grades of C and $D.$

w	(${\mathit{\eta }}_{\mathit{C}}^{+},{\mathit{\eta }}_{\mathit{C}}^{-}$)	w	(${\mathit{\eta }}_{\mathit{D}}^{+},{\mathit{\eta }}_{\mathit{D}}^{-}$)
$w_1$	$(0.7,-0.4)$	$w_1{w}_2$	$(0.35,-0.24)$
$w_2$	$(0.5,-0.6)$	$w_2{w}_3$	$(0.3,-0.3)$
$w_3$	$(0.6,-0.5)$	$w_1{w}_3$	$(0.42,-0.2)$
$w_1^{\prime }$	$(0.4,-0.3)$	$w_1^{\prime }{w}_2^{\prime }$	$(0.08,-0.06)$
$w_2^{\prime }$	$(0.2,-0.2)$

.

The union of ${\mathcal{H}}_1$ and ${\mathcal{H}}_2$ given in Figure 4 is the same as in Figure 4. Both are complete graphs, but their union is not a complete ${\mathrm{PB}}_f$ graph.

Theorem 6. 

Let ${\mathcal{H}}_1$ and ${\mathcal{H}}_2$ be two ${\mathit{PB}}_f$ graphs on $W_1$ and $W_2$, respectively. Suppose ${\mathcal{D}}_1$ and ${\mathcal{D}}_2$ are two minimum DSs of ${\mathcal{H}}_1$ and ${\mathcal{H}}_2$, respectively. Then, ${\mathcal{D}}_1\cup {\mathcal{D}}_2$ is a DS of ${\mathcal{H}}_1\cup {\mathcal{H}}_2.$

Proof. 

Let ${\mathcal{D}}_1$ be a minimum DS of ${\mathcal{H}}_1$. Then, for every $w^{\prime }\in W_1-{\mathcal{D}}_1$, there exists $w\in {\mathcal{D}}_1$ such that w dominates $w^{\prime }.$ Similarly, let ${\mathcal{D}}_2$ be a minimum DS of ${\mathcal{H}}_2$. Then, for every $w_1^{\prime }\in W_2-{\mathcal{D}}_2$, there exists $w_1\in {\mathcal{D}}_2$ such that $w_1$ dominates $w_1^{\prime }.$ Now, we want to prove that ${\mathcal{D}}_1\cup {\mathcal{D}}_2$ is a DS of ${\mathcal{H}}_1\cup {\mathcal{H}}_2.$ For this, let $w_2\in (W_1-{\mathcal{D}}_1)\cup (W_2-{\mathcal{D}}_2).$ There are two possibilities:

• If $w_2\in W_1-{\mathcal{D}}_1$, then there exists $w_2^{\prime }\in {\mathcal{D}}_1$ such that $w_2^{\prime }$ dominates $w_2.$
• If $w_2\in W_2-{\mathcal{D}}_2$, then there exists $w_3^{\prime }\in {\mathcal{D}}_2$ such that $w_3^{\prime }$ dominates $w_2.$

Thus, $w_2$ was arbitrary; the result holds for every $w_2\in (W_1-{\mathcal{D}}_1)\cup (W_2-{\mathcal{D}}_2).$ Hence, the proof is complete.    □

4.3. Joining Two Product Bipolar Fuzzy Graphs

Definition 21. 

Let ${\mathcal{H}}_1$ = $(C_1,D_1)$ and ${\mathcal{H}}_2$ = $(C_2,D_2)$ be two ${\mathit{PB}}_f$ graphs on $W_1$ and $W_2$, respectively, with $W_1\cap W_2$ = ϕ. The joining $\mathcal{H}$ = $({\mathcal{H}}_1+{\mathcal{H}}_2)$ of two ${\mathit{PB}}_f$ graphs ${\mathcal{H}}_1$ and ${\mathcal{H}}_2$ is a pair $(C_1+C_2,D_1+D_2)$, where

$$\begin{gathered} ({\eta }_{C_1}^{+}+{\eta }_{C_2}^{+})\left(w\right)=\left\{\begin{array}{cc}{\eta }_{C_1}^{+}\left(w\right),\hfill & \mathrm{if}w\in W_1,\hfill \\ {\eta }_{C_2}^{+}\left(w\right),\hfill & \mathrm{if}w\in W_2.\hfill \end{array}\right. \\ ({\eta }_{C_1}^{-}+{\eta }_{C_2}^{-})\left(w\right)=\left\{\begin{array}{cc}{\eta }_{C_1}^{-}\left(w\right),\hfill & \mathrm{if}w\in W_1,\hfill \\ {\eta }_{C_2}^{-}\left(w\right),\hfill & \mathrm{if}w\in W_2,\hfill \end{array}\right. \end{gathered}$$

for all $w\in W_1\cup W_2$, and

$$\begin{array}{ccc}\hfill ({\eta }_{D_1}^{+}+{\eta }_{D_2}^{+})\left(w_1{w}_2\right)& =& \left\{\begin{array}{cc}{\eta }_{D_1}^{+}\left(w_1{w}_2\right),\hfill & \mathrm{if}{w}_1,w_2\in W_1,\hfill \\ {\eta }_{D_2}^{+}\left(w_1{w}_2\right),\hfill & \mathrm{if}{w}_1,w_2\in W_2,\hfill \\ {\eta }_{C_1}^{+}\left(w_1\right)\times {\eta }_{C_2}^{+}\left(w_2\right),\hfill & \mathrm{if}{w}_1\in W_{1}and{w}_2\in W_2.\hfill \end{array}\right.\hfill \\ \hfill ({\eta }_{D_1}^{-}+{\eta }_{D_2}^{-})\left(w_1{w}_2\right)& =& \left\{\begin{array}{cc}{\eta }_{D_1}^{-}\left(w_1{w}_2\right),\hfill & \mathrm{if}{w}_1,w_2\in W_1,\hfill \\ {\eta }_{D_2}^{-}\left(w_1{w}_2\right),\hfill & \mathrm{if}{w}_1,w_2\in W_2,\hfill \\ -({\eta }_{C_1}^{-}\left(w_1\right)\times {\eta }_{C_2}^{-}\left(w_2\right)),\hfill & \mathrm{if}{w}_1\in W_{1}and{w}_2\in W_2,\hfill \end{array}\right.\hfill \end{array}$$

for all $w_1,w_2\in W_1\cup W_2.$

Remark 5. 

From the definition of the union and joining of two ${\mathit{PB}}_f$ graphs, we can see that

$$\begin{array}{ccc}\hfill ({\eta }_{C_1}^{+}+{\eta }_{C_2}^{+})\left(w\right)& =& ({\eta }_{C_1}^{+}\cup {\eta }_{C_2}^{+})\left(w\right),\hfill \\ \hfill ({\eta }_{C_1}^{-}+{\eta }_{C_2}^{-})\left(w\right)& =& ({\eta }_{C_1}^{-}\cup {\eta }_{C_2}^{-})\left(w\right),forallw\in W_1\cup W_2.\hfill \\ \hfill ({\eta }_{D_1}^{+}+{\eta }_{D_2}^{+})\left(w_1{w}_2\right)& =& ({\eta }_{D_1}^{+}\cup {\eta }_{D_2}^{+})\left(w_1{w}_2\right),\hfill \\ \hfill ({\eta }_{D_1}^{-}+{\eta }_{D_2}^{-})\left(w_1{w}_2\right)& =& ({\eta }_{D_1}^{-}\cup {\eta }_{D_2}^{-})\left(w_1{w}_2\right),if{w}_1,w_2\in W_1\cup W_2.\hfill \\ \hfill ({\eta }_{D_1}^{+}+{\eta }_{D_2}^{+})\left(w_1{w}_2\right)& =& {\eta }_{C_1}^{+}\left(w_1\right)\times {\eta }_{C_2}^{+}\left(w_2\right),\hfill \\ \hfill ({\eta }_{D_1}^{-}+{\eta }_{D_2}^{-})\left(w_1{w}_2\right)& =& -({\eta }_{C_1}^{-}\left(w_1\right)\times {\eta }_{C_2}^{-}\left(w_2\right)),if{w}_1\in W_1,w_2\in W_2.\hfill \end{array}$$

Example 8. 

Let ${\mathcal{H}}_1$ = $(C_1,D_1)$ and ${\mathcal{H}}_2$ = $(C_2,D_2)$ be two ${\mathit{PB}}_f$ graphs on $W_1$ and $W_2$, respectively, as shown in Figure 5. The vertex set $C=C_1+C_2$ and edge set $D=D_1+D_2$ of $\mathcal{H}$ = ${\mathcal{H}}_1+{\mathcal{H}}_2$ are $\{w_1,w_2,w_3,w_1^{\prime },w_2^{\prime }\}$ and $\{w_1{w}_2,w_1{w}_1^{\prime },w_1{w}_2^{\prime },w_2{w}_3,w_2{w}_1^{\prime },w_2{w}_2^{\prime },w_3{w}_1^{\prime },w_3{w}_2^{\prime },w_1^{\prime }{w}_2^{\prime }\}$, respectively. The membership grades of $C=C_1+C_2$ and $D=D_1+D_2$ of ${\mathcal{H}}_1+{\mathcal{H}}_2$ are given in

Table 6. $C_1+C_2$.

w	(${\mathit{\eta }}_{\mathit{C}}^{+},{\mathit{\eta }}_{\mathit{C}}^{-}$)
$w_1$	$(0.2,-0.2)$
$w_2$	$(0.1,-0.1)$
$w_3$	$(0.3,-0.3)$
$w_1^{\prime }$	$(0.4,-0.4)$
$w_2^{\prime }$	$(0.3,-0.2)$

and

Table 7. $D_1+D_2.$

w	(${\mathit{\eta }}_{\mathit{D}}^{+},{\mathit{\eta }}_{\mathit{D}}^{-}$)	w	(${\mathit{\eta }}_{\mathit{D}}^{+},{\mathit{\eta }}_{\mathit{D}}^{-}$)
$w_1{w}_2$	$(0.02,-0.02)$	$w_1^{\prime }{w}_2^{\prime }$	$(0.12,-0.08)$
$w_1{w}_1^{\prime }$	$(0.08,-0.08)$	$w_1{w}_2^{\prime }$	$(0.06,-0.04)$
$w_2{w}_3$	$(0.03,-0.03)$	$w_2{w}_1^{\prime }$	$(0.04,-0.04)$
$w_2{w}_2^{\prime }$	$(0.03,-0.02)$	$w_3{w}_1^{\prime }$	$(0.12,-0.12)$
$w_3{w}_2^{\prime }$	$(0.09,-0.06)$

, respectively. The joining of ${\mathcal{H}}_1$ and ${\mathcal{H}}_2$ is given in Figure 6.

Proposition 1. 

Let ${\mathcal{H}}_1$ = $(C_1,D_1)$ and ${\mathcal{H}}_2$ = $(C_2,D_2)$ be two complete ${\mathit{PB}}_f$ graphs on $W_1$ and $W_2$, respectively, with $W_1\cap W_2$ = ϕ. Then, ${\mathcal{H}}_1+{\mathcal{H}}_2$ is a complete ${\mathit{PB}}_f$ graph.

Proof. 

Let $w_1\in W_1$ and $w_2\in W_2$. Then,

$$\begin{array}{ccc}\hfill ({\eta }_{D_1}^{+}+{\eta }_{D_2}^{+})\left(w_1{w}_2\right)& =& ({\eta }_{C_1}^{+}\left(w_1\right)\times {\eta }_{C_2}^{+}\left(w_2\right)),\hfill \\ \hfill ({\eta }_{D_1}^{-}+{\eta }_{D_2}^{-})\left(w_1{w}_2\right)& =& -({\eta }_{C_1}^{-}\left(w_1\right)\times {\eta }_{C_2}^{-}\left(w_2\right)),\hfill \\ & =& ({\eta }_{C_1}^{+}\cup {\eta }_{C_2}^{+})\left(w_1\right)\times ({\eta }_{C_1}^{+}\cup {\eta }_{C_2}^{+})\left(w_2\right),\hfill \\ & =& -(({\eta }_{C_1}^{-}\cup {\eta }_{C_2}^{-})\left(w_1\right)\times ({\eta }_{C_1}^{-}\cup {\eta }_{C_2}^{-})\left(w_2\right)),\hfill \\ \hfill ({\eta }_{D_1}^{+}+{\eta }_{D_2}^{+})\left(w_1{w}_2\right)& =& ({\eta }_{C_1}^{+}+{\eta }_{C_2}^{+})\left(w_1\right)\times ({\eta }_{C_1}^{+}+{\eta }_{C_2}^{+})\left(w_2\right),\hfill \\ \hfill ({\eta }_{D_1}^{-}+{\eta }_{D_2}^{-})\left(w_1{w}_2\right)& =& -(({\eta }_{C_1}^{-}+{\eta }_{C_2}^{-})\left(w_1\right)\times ({\eta }_{C_1}^{-}+{\eta }_{C_2}^{-})\left(w_2\right)).\hfill \end{array}$$

Hence, the result holds for every $w_1\in W_1$ and $w_2\in W_2$.    □

Theorem 7. 

Let ${\mathcal{H}}_1$ and ${\mathcal{H}}_2$ be two ${\mathit{PB}}_f$ graphs on $W_1$ and $W_2$, respectively. Suppose ${\mathcal{D}}_1$ and ${\mathcal{D}}_2$ are two minimum DSs of ${\mathcal{H}}_1$ and ${\mathcal{H}}_2$, respectively. Then, ${\mathcal{D}}_1$ and ${\mathcal{D}}_2$ are DSs of ${\mathcal{H}}_1+{\mathcal{H}}_2.$

Proof. 

Let ${\mathcal{D}}_1$ be a minimum DS of ${\mathcal{H}}_1$. Then, for every $w^{\prime }\in W_1-{\mathcal{D}}_1$, there exists $w\in {\mathcal{D}}_1$ such that w dominates $w^{\prime }.$ By the definition of ${\mathcal{H}}_1+{\mathcal{H}}_2$, we know that every vertex of ${\mathcal{H}}_2$ is dominated by all the vertices of ${\mathcal{H}}_1.$ Thus, every element in $W_1-{\mathcal{D}}_1$ and in $W_2$ is dominated by the elements in ${\mathcal{D}}_1.$ Similarly, ${\mathcal{D}}_2$ is also a DS of ${\mathcal{H}}_1+{\mathcal{H}}_2.$ Hence, the proof is complete.    □

Theorem 8. 

Let ${\mathcal{H}}_1$ = $(C_1,D_1)$ and ${\mathcal{H}}_2$ = $(C_2,D_2)$ be two ${\mathit{PB}}_f$ graphs on $W_1$ and $W_2$, respectively. Then,

$$⋎({\mathcal{H}}_1+{\mathcal{H}}_2)=min\{⋎\left({\mathcal{H}}_1\right),⋎\left({\mathcal{H}}_2\right),{\displaystyle \frac{1+{\eta }_{C_1}^{+}\left(w_1\right)+{\eta }_{C_1}^{-}\left(w_1\right)}{2}}+{\displaystyle \frac{1+{\eta }_{C_2}^{+}\left(w_1^{\prime }\right)+{\eta }_{C_2}^{-}\left(w_1^{\prime }\right)}{2}}\},$$

where $w_1\in W_1$ and $w_1^{\prime }\in W_2.$

Proof. 

By the definition of ${\mathcal{H}}_1+{\mathcal{H}}_2$, we know that if we take any edge $w_1{w}_1^{\prime }$ from ${\mathcal{H}}_1+{\mathcal{H}}_2$, where $w_1\in W_1$ and $w_1^{\prime }\in W_2$, then $w_1{w}_1^{\prime }$ is an effective edge. However, any vertex of $W_1$ dominates all the vertices of $W_2.$ Now, suppose $\mathcal{D}$ is any MDS of ${\mathcal{D}}_1+{\mathcal{D}}_2.$ Then, $\mathcal{D}$ is one of the following forms:

• $\mathcal{D}$ = ${\mathcal{D}}_1$, where ${\mathcal{D}}_1$ is an MDS of ${\mathcal{H}}_1.$
• $\mathcal{D}$ = ${\mathcal{D}}_2$, where ${\mathcal{D}}_2$ is an MDS of ${\mathcal{H}}_2.$
• $\mathcal{D}$ = $\{w_1,w_1^{\prime }\}$, where $w_1\in W_1$, $w_1^{\prime }\in W_2$, and both $\left\{w_1\right\}$ and $\left\{w_1^{\prime }\right\}$ sets are not DSs of ${\mathcal{H}}_1$ and ${\mathcal{H}}_2$, respectively.

Hence,

$$⋎({\mathcal{H}}_1+{\mathcal{H}}_2)=min\{⋎\left({\mathcal{H}}_1\right),⋎\left({\mathcal{H}}_2\right),{\displaystyle \frac{1+{\eta }_{C_1}^{+}\left(w_1\right)+{\eta }_{C_1}^{-}\left(w_1\right)}{2}}+{\displaystyle \frac{1+{\eta }_{C_2}^{+}\left(w_1^{\prime }\right)+{\eta }_{C_2}^{-}\left(w_1^{\prime }\right)}{2}}\},$$

where $w_1\in W_1$ and $w_1^{\prime }\in W_2.$    □

4.4. Intersection of Two Product Bipolar Fuzzy Graphs

Definition 22. 

Let ${\mathcal{H}}_1$ = $(C_1,D_1)$ and ${\mathcal{H}}_2$ = $(C_2,D_2)$ be two ${\mathit{PB}}_f$ graphs on $W_1$ and $W_2$, respectively, with $W_1\cap W_2\ne \varphi$. The intersection $\mathcal{H}$ = ${\mathcal{H}}_1\cap {\mathcal{H}}_2$ of two ${\mathit{PB}}_f$ graphs ${\mathcal{H}}_1$ and ${\mathcal{H}}_2$ is a pair $(C_1\cap C_2,D_1\cap D_2)$, where

$$\begin{gathered} ({\eta }_{C_1}^{+}\cap {\eta }_{C_2}^{+})\left(w\right)={\eta }_{C_1}^{+}\left(w\right)\times {\eta }_{C_2}^{+}\left(w\right), \\ ({\eta }_{C_1}^{-}\cap {\eta }_{C_2}^{-})\left(w\right)=-({\eta }_{C_1}^{-}\left(w\right)\times {\eta }_{C_2}^{-}\left(w\right)), \end{gathered}$$

for all $w\in W_1\cap W_2$, and

$$\begin{gathered} ({\eta }_{D_1}^{+}\cap {\eta }_{D_2}^{+})\left(w_1{w}_2\right)={\eta }_{D_1}^{+}\left(w_1{w}_2\right)\times {\eta }_{D_2}^{+}\left(w_1{w}_2\right), \\ ({\eta }_{D_1}^{-}\cap {\eta }_{D_2}^{-})\left(w_1{w}_2\right)=-({\eta }_{D_1}^{+}\left(w_1{w}_2\right)\times {\eta }_{D_2}^{+}\left(w_1{w}_2\right)), \end{gathered}$$

for all $w_1,w_2\in W_1\cap W_2.$

Example 9. 

Consider two ${\mathit{PB}}_f$ graphs ${\mathcal{H}}_1$ = $(C_1,D_1)$ and ${\mathcal{H}}_2$ = $(C_2,D_2)$ on $W_1$ and $W_2$, respectively, as shown in Figure 7.

The vertex set $C=(C_1\cap C_2)$ and edge set $D=(D_1\cap D_2)$ of $\mathcal{H}$ = $({\mathcal{H}}_1\cap {\mathcal{H}}_2)$ are $\{w_1,w_2,w_3\}$ and $\{w_1{w}_2,w_1{w}_3,w_2{w}_3\}.$ The membership grades of both $C_1\cap C_2$ and $D_1\cap D_2$ of ${\mathcal{H}}_1\cap {\mathcal{H}}_2$ are given in

Table 8. Membership grades of $C\cap C$ and $D\cap D.$

w	(${\mathit{\eta }}_{\mathit{C}}^{+},{\mathit{\eta }}_{\mathit{C}}^{-}$)	w	(${\mathit{\eta }}_{\mathit{D}}^{+},{\mathit{\eta }}_{\mathit{D}}^{-}$)
$w_1$	$(0.21,-0.16)$	$w_1{w}_2$	$(0.042,-0.0288)$
$w_2$	$(0.2,-0.18)$	$w_1{w}_3$	$(0.1008,-0.04)$
$w_3$	$(0.48,-0.25)$	$w_2{w}_3$	$(0.096,-0.045)$

.

The intersection of ${\mathcal{H}}_1$ and ${\mathcal{H}}_2$ is given in Figure 8.

Both ${\mathcal{H}}_1$ and ${\mathcal{H}}_2$ are complete ${\mathit{PB}}_f$ graphs, and ${\mathcal{H}}_1\cup {\mathcal{H}}_2$ is also a complete ${\mathit{PB}}_f$ graph.

Proposition 2. 

If ${\mathcal{H}}_1$ and ${\mathcal{H}}_2$ are complete ${\mathit{PB}}_f$ graphs on $W_1$ and $W_2$, respectively, then ${\mathcal{H}}_1\cap {\mathcal{H}}_2$ is also a complete ${\mathit{PB}}_f$ graph.

Proof. 

Let ${\mathcal{H}}_1$ and ${\mathcal{H}}_2$ be complete ${\mathrm{PB}}_f$ graphs. Suppose $w_1,w_2\in W_1\cap W_2.$ Then, by definition,

$$\begin{array}{ccc}\hfill ({\eta }_{D_1}^{+}\cap {\eta }_{D_2}^{+})\left(w_1{w}_2\right)& =& {\eta }_{D_1}^{+}\left(w_1{w}_2\right)\times {\eta }_{D_2}^{+}\left(w_1{w}_2\right),\hfill \\ \hfill ({\eta }_{D_1}^{-}\cap {\eta }_{D_2}^{-})\left(w_1{w}_2\right)& =& -({\eta }_{D_1}^{+}\left(w_1{w}_2\right)\times {\eta }_{D_2}^{+}\left(w_1{w}_2\right)),\hfill \\ & =& {\eta }_{C_1}^{+}\left(w_1\right)\times {\eta }_{C_1}^{+}\left(w_2\right)\times {\eta }_{C_2}^{+}\left(w_1\right)\times {\eta }_{C_2}^{+}\left(w_2\right),\hfill \\ & =& -(-({\eta }_{C_1}^{-}\left(w_1\right)\times {\eta }_{C_1}^{-}\left(w_2\right))\times (-({\eta }_{C_2}^{-}\left(w_1\right)\times {\eta }_{C_2}^{-}\left(w_2\right)))),\hfill \\ & =& {\eta }_{C_1}^{+}\left(w_1\right)\times {\eta }_{C_2}^{+}\left(w_1\right)\times {\eta }_{C_1}^{+}\left(w_2\right)\times {\eta }_{C_2}^{+}\left(w_2\right),\hfill \\ & =& -({\eta }_{C_1}^{-}\left(w_1\right)\times {\eta }_{C_2}^{-}\left(w_1\right)\times {\eta }_{C_1}^{-}\left(w_2\right)\times {\eta }_{C_2}^{-}\left(w_2\right)),\hfill \\ \hfill ({\eta }_{D_1}^{+}\cap {\eta }_{D_2}^{+})\left(w_1{w}_2\right)& =& ({\eta }_{C_1}^{+}\cap {\eta }_{C_2}^{+})\left(w_1\right)\times ({\eta }_{C_1}^{+}\cap {\eta }_{C_2}^{+})\left(w_2\right),\hfill \\ \hfill ({\eta }_{D_1}^{-}\cap {\eta }_{D_2}^{-})\left(w_1{w}_2\right)& =& -(({\eta }_{C_1}^{-}\cap {\eta }_{C_2}^{-})\left(w_1\right)\times ({\eta }_{C_1}^{-}\cap {\eta }_{C_2}^{-})\left(w_2\right)).\hfill \end{array}$$

Hence, the proof is complete.    □

Theorem 9. 

Let ${\mathcal{H}}_1$ and ${\mathcal{H}}_2$ be two disjoint ${\mathit{PB}}_f$ graphs. Then,

$$⋎({\mathcal{H}}_1\cap {\mathcal{H}}_2)=0.$$

Proof. 

Let ${\mathcal{D}}_1$ and ${\mathcal{D}}_2$ be two minimum DSs or ⋎-sets of ${\mathit{PB}}_f$ graphs ${\mathcal{H}}_1$ and ${\mathcal{H}}_2$, respectively. Since ${\mathcal{H}}_1$ and ${\mathcal{H}}_2$ are two disjoint ${\mathit{PB}}_f$ graphs,

$${\mathcal{D}}_1\cap {\mathcal{D}}_2=\varphi .$$

Therefore,

$$⋎({\mathcal{H}}_1\cap {\mathcal{H}}_2)=|{\mathcal{D}}_1\cap {\mathcal{D}}_2|=|\varphi |=0.$$

   □

4.5. Cartesian Product of Two Product Bipolar Fuzzy Graphs

Definition 23. 

Let ${\mathcal{H}}_1$ = $(C_1,D_1)$ and ${\mathcal{H}}_2$ = $(C_2,D_2)$ be two ${\mathit{PB}}_f$ graphs on $W_1$ and $W_2$, respectively. The Cartesian product $\mathcal{H}$ = ${\mathcal{H}}_1\times {\mathcal{H}}_2$ of two ${\mathit{PB}}_f$ graphs ${\mathcal{H}}_1$ and ${\mathcal{H}}_2$ is a pair $(C_1\times C_2,D_1\times D_2)$, where

$$\begin{array}{ccc}\hfill ({\eta }_{C_1}^{+}\times {\eta }_{C_2}^{+})(w_1,w_1^{\prime })& =& {\eta }_{C_1}^{+}\left(w_1\right)\times {\eta }_{C_2}^{+}\left(w_1^{\prime }\right),\hfill \\ \hfill ({\eta }_{C_1}^{-}\times {\eta }_{C_2}^{-})(w_1,w_1^{\prime })& =& -({\eta }_{C_1}^{-}\left(w_1\right)\times {\eta }_{C_2}^{-}\left(w_1^{\prime }\right)),\hfill \end{array}$$

for all $(w_1,w_1^{\prime })\in W_1\times W_2$, and

$$({\eta }_{D_1}^{+}\times {\eta }_{D_2}^{+})\left((w_1,w_1^{\prime })(w_2,w_2^{\prime })\right)=\left\{\begin{array}{cc}{\left({\eta }_{C_1}^{+}\left(w_1\right)\right)}^2\times {\eta }_{D_2}^{+}\left(w_1^{\prime }{w}_2^{\prime }\right),\hfill & \mathrm{if}{w}_1=w_2,\hfill \\ {\eta }_{D_1}^{+}\left(w_1{w}_2\right)\times {\left({\eta }_{C_2}^{+}\left(w_1^{\prime }\right)\right)}^2,\hfill & \mathrm{if}{w}_1^{\prime }=w_2^{\prime }.\hfill \end{array}\right.$$

$$({\eta }_{D_1}^{-}\times {\eta }_{D_2}^{-})\left((w_1,w_1^{\prime })(w_2,w_2^{\prime })\right)=\left\{\begin{array}{cc}{\left({\eta }_{C_1}^{-}\left(w_1\right)\right)}^2\times {\eta }_{D_2}^{-}\left(w_1^{\prime }{w}_2^{\prime }\right),\hfill & \mathrm{if}{w}_1=w_2,\hfill \\ {\eta }_{D_1}^{-}\left(w_1{w}_2\right)\times {\left({\eta }_{C_2}^{-}\left(w_1^{\prime }\right)\right)}^2,\hfill & \mathrm{if}{w}_1^{\prime }=w_2^{\prime },\hfill \end{array}\right.$$

for all $(w_1,w_1^{\prime }),(w_2,w_2^{\prime })\in W_1\times W_2.$

Example 10. 

Let ${\mathcal{H}}_1$ = $(C_1,D_1)$ and ${\mathcal{H}}_2$ = $(C_2,D_2)$ be two ${\mathit{PB}}_f$ graphs on $W_1=\{w_1,w_2\}$ and $W_2=\{w_1^{\prime },w_2^{\prime }\}$, respectively, as shown in Figure 9.

The vertex set $C=C_1\times C_2$ and edge set $D=D_1\times D_2$ of $\mathcal{H}={\mathcal{H}}_1\times {\mathcal{H}}_2$ are $\{(w_1,w_1^{\prime }),(w_1,w_2^{\prime }),(w_2,w_1^{\prime }),(w_2,w_2^{\prime })\}$ and $\{\left((w_1,w_1^{\prime })(w_1,w_2^{\prime })\right),\left((w_1,w_1^{\prime })(w_2,w_1^{\prime })\right),$

$\left((w_2,w_1^{\prime })(w_2,w_2^{\prime })\right),\left((w_1,w_2^{\prime })(w_2,w_2^{\prime })\right)\}$. The membership grades of $C\times C$ and $D\times D$ of ${\mathcal{H}}_1\times {\mathcal{H}}_2$ are given in

Table 9. Membership grades of $C\times C$ and $D\times D.$

($\mathit{w}\times \mathit{w}$)	(${\mathit{\eta }}_{\mathit{C}}^{+},{\mathit{\eta }}_{\mathit{C}}^{-}$)	($\mathit{w}\times \mathit{w}$)	(${\mathit{\eta }}_{\mathit{D}}^{+},{\mathit{\eta }}_{\mathit{D}}^{-}$)
$(w_1,w_1^{\prime })$	$(0.01,-0.02)$	$\left((w_1,w_1^{\prime })(w_1,w_2^{\prime })\right)$	$(0.0003,-0.0008)$
$(w_1,w_2^{\prime })$	$(0.03,-0.04)$	$\left((w_1,w_1^{\prime })(w_2,w_1^{\prime })\right)$	$(0.0002,-0.0006)$
$(w_2,w_1^{\prime })$	$(0.02,-0.03)$	$\left((w_2,w_1^{\prime })(w_2,w_2^{\prime })\right)$	$(0.0012,-0.0018)$
$(w_2,w_2^{\prime })$	$(0.06,-0.06)$	$\left((w_1,w_2^{\prime })(w_2,w_2^{\prime })\right)$	$(0.0018,-0.0024)$

.

The Cartesian product of ${\mathcal{H}}_1\times {\mathcal{H}}_2$ is given in Figure 10.

Theorem 10. 

Let ${\mathcal{H}}_1$ = $(C_1,D_1)$ and ${\mathcal{H}}_2$ = $(C_2,D_2)$ be two ${\mathit{PB}}_f$ graphs on $W_1$ and $W_2$, respectively. Suppose ${\mathcal{D}}_1$ and ${\mathcal{D}}_2$ are two minimum DSs of ${\mathcal{H}}_1$ and ${\mathcal{H}}_2$, respectively. Then,

$$⋎({\mathcal{H}}_1\times {\mathcal{H}}_2)\le min\left\{\right|{\mathcal{D}}_1\times W_2|,|W_1\times {\mathcal{D}}_2\left|\right\}.$$

Proof. 

We first prove that ${\mathcal{D}}_1\times W_2$ is a DS of ${\mathcal{H}}_1\times {\mathcal{H}}_2.$ Let $(w_1,w_2)\notin {\mathcal{D}}_1\times W_2$; therefore, $w_1\notin {\mathcal{D}}_1.$ Since ${\mathcal{D}}_1$ is a DS of ${\mathcal{H}}_1$, there exists $w_1^{\prime }\in {\mathcal{D}}_1$ such that

$$\begin{gathered} {\eta }_{D_1}^{+}\left(w_1{w}_1^{\prime }\right)={\eta }_{C_1}^{+}\left(w_1\right)\times {\eta }_{C_1}^{+}\left(w_1^{\prime }\right), \\ {\eta }_{D_1}^{-}\left(w_1{w}_1^{\prime }\right)=-({\eta }_{C_1}^{-}\left(w_1\right)\times {\eta }_{C_1}^{-}\left(w_1^{\prime }\right)). \end{gathered}$$

Now, $(w_1^{\prime },w_2)\in {\mathcal{D}}_1\times W_2$. So,

$$\begin{array}{ccc}\hfill ({\eta }_{D_1}^{+}\times {\eta }_{D_2}^{+})\left((w_1,w_2)(w_1^{\prime },w_2)\right)& =& {\eta }_{D_1}^{+}\left(w_1{w}_1^{\prime }\right)\times {\left({\eta }_{C_2}^{+}\left(w_2\right)\right)}^2,\hfill \\ \hfill ({\eta }_{D_1}^{-}\times {\eta }_{D_2}^{-})\left((w_1,w_2)(w_1^{\prime },w_2)\right)& =& {\eta }_{D_1}^{-}\left(w_1{w}_1^{\prime }\right)\times {\left({\eta }_{C_2}^{-}\left(w_2\right)\right)}^2,\hfill \\ & =& {\eta }_{C_1}^{+}\left(w_1\right)\times {\eta }_{C_1}^{+}\left(w_1^{\prime }\right)\times {\eta }_{C_2}^{+}\left(w_2\right)\times {\eta }_{C_2}^{+}\left(w_2\right),\hfill \\ & =& -({\eta }_{C_1}^{-}\left(w_1\right)\times {\eta }_{C_1}^{-}\left(w_1^{\prime }\right))\times {\eta }_{C_2}^{-}\left(w_2\right)\times {\eta }_{C_2}^{-}\left(w_2\right),\hfill \\ & =& {\eta }_{C_1}^{+}\left(w_1\right)\times {\eta }_{C_2}^{+}\left(w_2\right)\times {\eta }_{C_1}^{+}\left(w_1^{\prime }\right)\times {\eta }_{C_2}^{+}\left(w_2\right),\hfill \\ & =& -({\eta }_{C_1}^{-}\left(w_1\right)\times {\eta }_{C_2}^{-}\left(w_2\right)\times {\eta }_{C_1}^{-}\left(w_1^{\prime }\right)\times {\eta }_{C_2}^{-}\left(w_2\right)),\hfill \\ \hfill ({\eta }_{D_1}^{+}\times {\eta }_{D_2}^{+})\left((w_1,w_2)(w_1^{\prime },w_2)\right)& =& ({\eta }_{C_1}^{+}\times {\eta }_{C_2}^{+})(w_1,w_2)\times ({\eta }_{C_1}^{+}\times {\eta }_{C_2}^{+})(w_1^{\prime },w_2),\hfill \\ \hfill ({\eta }_{D_1}^{-}\times {\eta }_{D_2}^{-})\left((w_1,w_2)(w_1^{\prime },w_2)\right)& =& -(({\eta }_{C_1}^{-}\times {\eta }_{C_2}^{-})(w_1,w_2)\times ({\eta }_{C_1}^{-}\times {\eta }_{C_2}^{-})(w_1^{\prime },w_2)),\hfill \end{array}$$

Thus, $(w_1^{\prime },w_2)$ dominates $(w_1,w_2)$ in ${\mathcal{H}}_1\times {\mathcal{H}}_2$ so that ${\mathcal{D}}_1\times W_2$ is a DS of ${\mathcal{H}}_1\times {\mathcal{H}}_2$. Similarly, $W_1\times {\mathcal{D}}_2$ is also a DS of ${\mathcal{H}}_1\times {\mathcal{H}}_2$. Hence, it follows that

$$⋎({\mathcal{H}}_1\times {\mathcal{H}}_2)\le min\left\{\right|{\mathcal{D}}_1\times W_2|,|W_1\times {\mathcal{D}}_2\left|\right\}.$$

   □

The Cartesian product of two ${\mathrm{PB}}_f$ graphs ${\mathcal{H}}_1$ and ${\mathcal{H}}_2$ is not a complete ${\mathrm{PB}}_f$ graph (see Example 10).

4.6. Composition of Two Product Bipolar Fuzzy Graphs

Definition 24. 

Let ${\mathcal{H}}_1$ = $(C_1,D_1)$ and ${\mathcal{H}}_2$ = $(C_2,D_2)$ be two ${\mathit{PB}}_f$ graphs on $W_1$ and $W_2$, respectively. The composition $\mathcal{H}$ = ${\mathcal{H}}_1\circ {\mathcal{H}}_2$ of two ${\mathit{PB}}_f$ graphs ${\mathcal{H}}_1$ and ${\mathcal{H}}_2$ is a pair $(C_1\circ C_2,D_1\circ D_2)$, where

$$\begin{array}{ccc}\hfill ({\eta }_{C_1}^{+}\circ {\eta }_{C_2}^{+})(w_1,w_1^{\prime })& =& {\eta }_{C_1}^{+}\left(w_1\right)\times {\eta }_{C_2}^{+}\left(w_1^{\prime }\right),\hfill \\ \hfill ({\eta }_{C_1}^{-}\circ {\eta }_{C_2}^{-})(w_1,w_1^{\prime })& =& -({\eta }_{C_1}^{-}\left(w_1\right)\times {\eta }_{C_2}^{-}\left(w_1^{\prime }\right)),\hfill \end{array}$$

for all $(w_1,w_1^{\prime })\in W_1\times W_2$, and

$$({\eta }_{D_1}^{+}\circ {\eta }_{D_2}^{+})\left((w_1,w_1^{\prime })(w_2,w_2^{\prime })\right)=\left\{\begin{array}{cc}{\left({\eta }_{C_1}^{+}\left(w_1\right)\right)}^2\times {\eta }_{D_2}^{+}\left(w_1^{\prime }{w}_2^{\prime }\right),\hfill & \mathrm{if}{w}_1=w_2,\hfill \\ {\eta }_{D_1}^{+}\left(w_1{w}_2\right)\times {\left({\eta }_{C_2}^{+}\left(w_1^{\prime }\right)\right)}^2,\hfill & \mathrm{if}{w}_1^{\prime }=w_2^{\prime },\hfill \\ {\eta }_{D_1}^{+}\left(w_1{w}_2\right)\times {\eta }_{C_2}^{+}\left(w_1^{\prime }\right)\times {\eta }_{C_2}^{+}\left(w_2^{\prime }\right),\hfill & otherwise.\hfill \end{array}\right.$$

$$({\eta }_{D_1}^{-}\circ {\eta }_{D_2}^{-})\left((w_1,w_1^{\prime })(w_2,w_2^{\prime })\right)=\left\{\begin{array}{cc}{\eta }_{C_1}^{-}\left(w_1\right)\times {\eta }_{D_2}^{-}\left(w_1^{\prime }{w}_2^{\prime }\right),\hfill & \mathrm{if}{w}_1=w_2,\hfill \\ {\eta }_{D_1}^{-}\left(w_1{w}_2\right)\times {\eta }_{C_2}^{-}\left(w_1^{\prime }\right),\hfill & \mathrm{if}{w}_1^{\prime }=w_2^{\prime },\hfill \\ {\eta }_{D_1}^{-}\left(w_1{w}_2\right)\times {\eta }_{C_2}^{-}\left(w_1^{\prime }\right)\times {\eta }_{C_2}^{-}\left(w_2^{\prime }\right),\hfill & otherwise.\hfill \end{array}\right.$$

for all $(w_1,w_1^{\prime }),(w_2,w_2^{\prime })\in W_1\times W_2.$

The composition of ${\mathcal{H}}_1$ and ${\mathcal{H}}_2$ of Figure 9 is given in Figure 11.

Proposition 3. 

If ${\mathcal{H}}_1$ and ${\mathcal{H}}_2$ are complete ${\mathit{PB}}_f$ graphs on $W_1$ and $W_2$, respectively, then ${\mathcal{H}}_1\circ {\mathcal{H}}_2$ is also a complete ${\mathit{PB}}_f$ graph.

Proof. 

Let ${\mathcal{H}}_1$ and ${\mathcal{H}}_2$ be complete ${\mathrm{PB}}_f$ graphs. Suppose $(w_1,w_1^{\prime }),(w_2,w_2^{\prime })\in W_1\times W_2.$ There are three cases:

• Case (i) Suppose $w_1=w_2$. Then,

  $$\begin{array}{ccc}\hfill ({\eta }_{D_1}^{+}\circ {\eta }_{D_2}^{+})\left((w_1,w_1^{\prime })(w_1,w_2^{\prime })\right)& =& {\left({\eta }_{C_1}^{+}\left(w_1\right)\right)}^2\times {\eta }_{D_2}^{+}\left(w_1^{\prime }{w}_2^{\prime }\right),\hfill \\ \hfill ({\eta }_{D_1}^{-}\circ {\eta }_{D_2}^{-})\left((w_1,w_1^{\prime })(w_1,w_2^{\prime })\right)& =& {\left({\eta }_{C_1}^{-}\left(w_1\right)\right)}^2\times {\eta }_{D_2}^{-}\left(w_1^{\prime }{w}_2^{\prime }\right)0,\hfill \\ & =& {\eta }_{C_1}^{+}\left(w_1\right)\times {\eta }_{C_1}^{+}\left(w_1\right)\times {\eta }_{C_2}^{+}\left(w_1^{\prime }\right)\times {\eta }_{C_2}^{+}\left(w_2^{\prime }\right),\hfill \\ & =& -({\eta }_{C_1}^{-}\left(w_1\right)\times {\eta }_{C_1}^{-}\left(w_1\right)\times {\eta }_{C_2}^{-}\left(w_1^{\prime }\right)\times {\eta }_{C_2}^{-}\left(w_2^{\prime }\right)),\hfill \\ & =& {\eta }_{C_1}^{+}\left(w_1\right)\times {\eta }_{C_2}^{+}\left(w_1^{\prime }\right)\times {\eta }_{C_1}^{+}\left(w_1\right)\times {\eta }_{C_2}^{+}\left(w_2^{\prime }\right),\hfill \\ & =& -({\eta }_{C_1}^{-}\left(w_1\right)\times {\eta }_{C_2}^{-}\left(w_1^{\prime }\right)\times {\eta }_{C_1}^{-}\left(w_1\right)\times {\eta }_{C_2}^{-}\left(w_2^{\prime }\right)),\hfill \\ \hfill ({\eta }_{D_1}^{+}\circ {\eta }_{D_2}^{+})\left((w_1,w_1^{\prime })(w_1,w_2^{\prime })\right)& =& ({\eta }_{C_1}^{+}\circ {\eta }_{C_2}^{+})(w_1,w_1^{\prime })\times ({\eta }_{C_1}^{+}\circ {\eta }_{C_2}^{+})(w_1,w_2^{\prime }),\hfill \\ \hfill ({\eta }_{D_1}^{-}\circ {\eta }_{D_2}^{-})\left((w_1,w_1^{\prime })(w_1,w_2^{\prime })\right)& =& -(({\eta }_{C_1}^{-}\circ {\eta }_{C_2}^{-})(w_1,w_1^{\prime })\times ({\eta }_{C_1}^{-}\circ {\eta }_{C_2}^{-})(w_1,w_2^{\prime })).\hfill \end{array}$$
• Case (ii) Suppose $w_1^{\prime }=w_2^{\prime }$. Then,

  $$\begin{array}{ccc}\hfill ({\eta }_{D_1}^{+}\circ {\eta }_{D_2}^{+})\left((w_1,w_1^{\prime })(w_2,w_1^{\prime })\right)& =& {\eta }_{D_1}^{+}\left(w_1{w}_2\right)\times {\left({\eta }_{C_2}^{+}\left(w_1^{\prime }\right)\right)}^2,\hfill \\ \hfill ({\eta }_{D_1}^{-}\circ {\eta }_{D_2}^{-})\left((w_1,w_1^{\prime })(w_2,w_1^{\prime })\right)& =& {\eta }_{D_1}^{-}\left(w_1{w}_2\right)\times {\left({\eta }_{C_2}^{-}\left(w_1^{\prime }\right)\right)}^2,\hfill \\ & =& {\eta }_{C_1}^{+}\left(w_1\right)\times {\eta }_{C_1}^{+}\left(w_2\right)\times {\eta }_{C_2}^{+}\left(w_1^{\prime }\right)\times {\eta }_{C_2}^{+}\left(w_1^{\prime }\right),\hfill \\ & =& -({\eta }_{C_1}^{-}\left(w_1\right)\times {\eta }_{C_1}^{-}\left(w_2\right)\times {\eta }_{C_2}^{-}\left(w_1^{\prime }\right)\times {\eta }_{C_2}^{-}\left(w_1^{\prime }\right)),\hfill \\ & =& {\eta }_{C_1}^{+}\left(w_1\right)\times {\eta }_{C_2}^{+}\left(w_1^{\prime }\right)\times {\eta }_{C_1}^{+}\left(w_2\right)\times {\eta }_{C_2}^{+}\left(w_1^{\prime }\right),\hfill \\ & =& -({\eta }_{C_1}^{-}\left(w_1\right)\times {\eta }_{C_2}^{-}\left(w_1^{\prime }\right)\times {\eta }_{C_1}^{-}\left(w_2\right)\times {\eta }_{C_2}^{-}\left(w_1^{\prime }\right)),\hfill \\ \hfill ({\eta }_{D_1}^{+}\circ {\eta }_{D_2}^{+})\left((w_1,w_1^{\prime })(w_2,w_1^{\prime })\right)& =& ({\eta }_{C_1}^{+}\circ {\eta }_{C_2}^{+})(w_1,w_1^{\prime })\times ({\eta }_{C_1}^{+}\circ {\eta }_{C_2}^{+})(w_2,w_1^{\prime }),\hfill \\ \hfill ({\eta }_{D_1}^{-}\circ {\eta }_{D_2}^{-})\left((w_1,w_1^{\prime })(w_2,w_1^{\prime })\right)& =& -(({\eta }_{C_1}^{-}\circ {\eta }_{C_2}^{-})(w_1,w_1^{\prime })\times ({\eta }_{C_1}^{-}\circ {\eta }_{C_2}^{-})(w_2,w_1^{\prime })).\hfill \end{array}$$
• Case (iii) Suppose $(w_1,w_1^{\prime }),(w_2,w_2^{\prime })\in W_1\times W_2$. Then,

  $$\begin{array}{ccc}\hfill ({\eta }_{D_1}^{+}\circ {\eta }_{D_2}^{+})\left((w_1,w_1^{\prime })(w_2,w_2^{\prime })\right)& =& {\eta }_{D_1}^{+}\left(w_1{w}_2\right)\times {\eta }_{C_2}^{+}\left(w_1^{\prime }\right)\times {\eta }_{C_2}^{+}\left(w_2^{\prime }\right),\hfill \\ \hfill ({\eta }_{D_1}^{-}\circ {\eta }_{D_2}^{-})\left((w_1,w_1^{\prime })(w_2,w_1^{\prime })\right)& =& {\eta }_{D_1}^{-}\left(w_1{w}_2\right)\times {\eta }_{C_2}^{-}\left(w_1^{\prime }\right)\times {\eta }_{C_2}^{-}\left(w_2^{\prime }\right),\hfill \\ & =& {\eta }_{C_1}^{+}\left(w_1\right)\times {\eta }_{C_1}^{+}\left(w_2\right)\times {\eta }_{C_2}^{+}\left(w_1^{\prime }\right)\times {\eta }_{C_2}^{+}\left(w_2^{\prime }\right),\hfill \\ & =& -({\eta }_{C_1}^{-}\left(w_1\right)\times {\eta }_{C_1}^{-}\left(w_2\right)\times {\eta }_{C_2}^{-}\left(w_1^{\prime }\right)\times {\eta }_{C_2}^{-}\left(w_2^{\prime }\right)),\hfill \\ & =& {\eta }_{C_1}^{+}\left(w_1\right)\times {\eta }_{C_2}^{+}\left(w_1^{\prime }\right)\times {\eta }_{C_1}^{+}\left(w_2\right)\times {\eta }_{C_2}^{+}\left(w_2^{\prime }\right),\hfill \\ & =& -({\eta }_{C_1}^{-}\left(w_1\right)\times {\eta }_{C_2}^{-}\left(w_1^{\prime }\right)\times {\eta }_{C_1}^{-}\left(w_2\right)\times {\eta }_{C_2}^{-}\left(w_2^{\prime }\right)),\hfill \\ \hfill ({\eta }_{D_1}^{+}\circ {\eta }_{D_2}^{+})\left((w_1,w_1^{\prime })(w_2,w_2^{\prime })\right)& =& ({\eta }_{C_1}^{+}\circ {\eta }_{C_2}^{+})(w_1,w_1^{\prime })\times ({\eta }_{C_1}^{+}\circ {\eta }_{C_2}^{+})(w_2,w_2^{\prime }),\hfill \\ \hfill ({\eta }_{D_1}^{-}\circ {\eta }_{D_2}^{-})\left((w_1,w_1^{\prime })(w_2,w_2^{\prime })\right)& =& -(({\eta }_{C_1}^{-}\circ {\eta }_{C_2}^{-})(w_1,w_1^{\prime })\times ({\eta }_{C_1}^{-}\circ {\eta }_{C_2}^{-})(w_2,w_2^{\prime })).\hfill \end{array}$$

Hence, the proof is complete.    □

Theorem 11. 

Let ${\mathcal{D}}_1$ and ${\mathcal{D}}_2$ be two DSs of ${\mathit{PB}}_f$ graphs ${\mathcal{H}}_1$ = $(C_1,D_1)$ and ${\mathcal{H}}_2$ = $(C_2,D_2)$, respectively. Then, ${\mathcal{D}}_1\times {\mathcal{D}}_2$ is a DS of ${\mathcal{H}}_1\circ {\mathcal{H}}_2.$

Proof. 

Let $(w,w^{\prime })\notin {\mathcal{D}}_1\times {\mathcal{D}}_2.$

• Case (i) Let $w\notin {\mathcal{D}}_1$ and $w^{\prime }\in {\mathcal{D}}_2$. Consider $w_1\in {\mathcal{D}}_1$ such that $w_1$ dominates w. Then,

  $$\begin{gathered} {\eta }_{D_1}^{+}\left(w{w}_1\right)={\eta }_{C_1}^{+}\left(w\right)\times {\eta }_{C_1}^{+}\left(w_1\right), \\ {\eta }_{D_1}^{-}\left(w{w}_1\right)=-({\eta }_{C_1}^{-}\left(w\right)\times {\eta }_{C_1}^{-}\left(w_1\right)). \end{gathered}$$

Now, $(w_1,w^{\prime })\in {\mathcal{D}}_1\times {\mathcal{D}}_2.$ So,

$$\begin{array}{ccc}\hfill ({\eta }_{D_1}^{+}\circ {\eta }_{D_2}^{+})\left((w,w^{\prime })(w_1,w^{\prime })\right)& =& {\eta }_{D_1}^{+}\left(w{w}_1\right)\times {\left({\eta }_{C_2}^{+}\left(w^{\prime }\right)\right)}^2,\hfill \\ \hfill ({\eta }_{D_1}^{-}\circ {\eta }_{D_2}^{-})\left((w,w^{\prime })(w_1,w^{\prime })\right)& =& {\eta }_{D_1}^{-}\left(w{w}_1\right)\times {\left({\eta }_{C_2}^{-}\left(w^{\prime }\right)\right)}^2,\hfill \\ & =& {\eta }_{C_1}^{+}\left(w\right)\times {\eta }_{C_1}^{+}\left(w_1\right)\times {\eta }_{C_2}^{+}\left(w^{\prime }\right)\times {\eta }_{C_2}^{+}\left(w^{\prime }\right),\hfill \\ & =& -({\eta }_{C_1}^{-}\left(w\right)\times {\eta }_{C_1}^{-}\left(w_1\right))\times {\eta }_{C_2}^{-}\left(w^{\prime }\right)\times {\eta }_{C_2}^{-}\left(w^{\prime }\right),\hfill \\ & =& {\eta }_{C_1}^{+}\left(w\right)\times {\eta }_{C_2}^{+}\left(w^{\prime }\right)\times {\eta }_{C_1}^{+}\left(w_1\right)\times {\eta }_{C_2}^{+}\left(w^{\prime }\right),\hfill \\ & =& -({\eta }_{C_1}^{-}\left(w\right)\times {\eta }_{C_2}^{-}\left(w^{\prime }\right)\times {\eta }_{C_1}^{-}\left(w_1\right)\times {\eta }_{C_2}^{-}\left(w^{\prime }\right)),\hfill \\ \hfill ({\eta }_{D_1}^{+}\circ {\eta }_{D_2}^{+})\left((w,w^{\prime })(w_1,w^{\prime })\right)& =& ({\eta }_{C_1}^{+}\circ {\eta }_{C_2}^{+})(w,w^{\prime })\times ({\eta }_{C_1}^{+}\circ {\eta }_{C_2}^{+})(w_1,w^{\prime }),\hfill \\ \hfill ({\eta }_{D_1}^{-}\circ {\eta }_{D_2}^{-})\left((w,w^{\prime })(w_1,w^{\prime })\right)& =& -(({\eta }_{C_1}^{-}\circ {\eta }_{C_2}^{-})(w,w^{\prime })\times ({\eta }_{C_1}^{-}\circ {\eta }_{C_2}^{-})(w_1,w^{\prime })).\hfill \end{array}$$

Hence, $(w_1,w^{\prime })$ dominates $(w,w^{\prime })$ in ${\mathcal{H}}_1\circ {\mathcal{H}}_2.$

• Case (ii) Let $w\in {\mathcal{D}}_1$ and $w^{\prime }\notin {\mathcal{D}}_2$. Consider $w_1^{\prime }\in {\mathcal{D}}_2$ such that $w_1^{\prime }$ dominates $w^{\prime }$. Then,

  $$\begin{gathered} {\eta }_{D_2}^{+}\left(w^{\prime }{w}_1^{\prime }\right)={\eta }_{C_2}^{+}\left(w^{\prime }\right)\times {\eta }_{C_2}^{+}\left(w_1^{\prime }\right), \\ {\eta }_{D_2}^{-}\left(w^{\prime }{w}_1^{\prime }\right)=-({\eta }_{C_2}^{-}\left(w^{\prime }\right)\times {\eta }_{C_2}^{-}\left(w_1^{\prime }\right)). \end{gathered}$$

Now, $(w,w_1^{\prime })\in {\mathcal{D}}_1\times {\mathcal{D}}_2.$ So,

$$\begin{array}{ccc}\hfill ({\eta }_{D_1}^{+}\circ {\eta }_{D_2}^{+})\left((w,w^{\prime })(w,w_1^{\prime })\right)& =& {\left({\eta }_{C_1}^{+}\left(w\right)\right)}^2\times {\eta }_{D_2}^{+}\left(w^{\prime }{w}_1^{\prime }\right),\hfill \\ \hfill ({\eta }_{D_1}^{-}\circ {\eta }_{D_2}^{-})\left((w,w^{\prime })(w,w_1^{\prime })\right)& =& {\left({\eta }_{C_1}^{-}\left(w\right)\right)}^2\times {\eta }_{D_2}^{-}\left(w^{\prime }{w}_1^{\prime }\right),\hfill \\ & =& {\eta }_{C_1}^{+}\left(w\right)\times {\eta }_{C_1}^{+}\left(w\right)\times {\eta }_{C_2}^{+}\left(w^{\prime }\right)\times {\eta }_{C_2}^{+}\left(w_1^{\prime }\right),\hfill \\ & =& {\eta }_{C_1}^{-}\left(w\right)\times {\eta }_{C_1}^{-}\left(w\right)\times (-({\eta }_{C_2}^{-}\left(w^{\prime }\right)\times {\eta }_{C_2}^{-}\left(w_1^{\prime }\right))),\hfill \\ & =& {\eta }_{C_1}^{+}\left(w\right)\times {\eta }_{C_2}^{+}\left(w^{\prime }\right)\times {\eta }_{C_1}^{+}\left(w\right)\times {\eta }_{C_2}^{+}\left(w_1^{\prime }\right),\hfill \\ & =& -({\eta }_{C_1}^{-}\left(w\right)\times {\eta }_{C_2}^{-}\left(w^{\prime }\right)\times {\eta }_{C_1}^{-}\left(w\right)\times {\eta }_{C_2}^{-}\left(w_1^{\prime }\right)),\hfill \\ \hfill ({\eta }_{D_1}^{+}\circ {\eta }_{D_2}^{+})\left((w,w^{\prime })(w,w_1^{\prime })\right)& =& ({\eta }_{C_1}^{+}\circ {\eta }_{C_2}^{+})(w,w^{\prime })\times ({\eta }_{C_1}^{+}\circ {\eta }_{C_2}^{+})(w,w_1^{\prime }),\hfill \\ \hfill ({\eta }_{D_1}^{-}\circ {\eta }_{D_2}^{-})\left((w,w^{\prime })(w,w_1^{\prime })\right)& =& ({\eta }_{C_1}^{-}\circ {\eta }_{C_2}^{-})(w,w^{\prime })\times ({\eta }_{C_1}^{-}\circ {\eta }_{C_2}^{-})(w,w_1^{\prime }).\hfill \end{array}$$

Hence, $(w,w_1^{\prime })$ dominates $(w,w^{\prime })$ in ${\mathcal{H}}_1\circ {\mathcal{H}}_2.$

• Case (iii) Let $w\notin {\mathcal{D}}_1$ and $w^{\prime }\notin {\mathcal{D}}_2$. Consider $w_1\in {\mathcal{D}}_1$ and $w_1^{\prime }\in {\mathcal{D}}_2$ such that $w_1$ dominates w in ${\mathcal{H}}_1$ and $w_1^{\prime }$ dominates $w^{\prime }$ in ${\mathcal{H}}_2.$ Then,

  $$\begin{gathered} {\eta }_{D_1}^{+}\left(w{w}_1\right)={\eta }_{C_1}^{+}\left(w\right)\times {\eta }_{C_1}^{+}\left(w_1\right), \\ {\eta }_{D_1}^{-}\left(w{w}_1\right)=-({\eta }_{C_1}^{-}\left(w\right)\times {\eta }_{C_1}^{-}\left(w_1\right)), \\ {\eta }_{D_2}^{+}\left(w^{\prime }{w}_1^{\prime }\right)={\eta }_{C_2}^{+}\left(w^{\prime }\right)\times {\eta }_{C_2}^{+}\left(w_1^{\prime }\right), \\ {\eta }_{D_2}^{-}\left(w^{\prime }{w}_1^{\prime }\right)=-({\eta }_{C_2}^{-}\left(w^{\prime }\right)\times {\eta }_{C_2}^{-}\left(w_1^{\prime }\right)). \end{gathered}$$

Now, $(w_1,w_1^{\prime })\in {\mathcal{D}}_1\times {\mathcal{D}}_2.$ So,

$$\begin{array}{ccc}\hfill ({\eta }_{D_1}^{+}\circ {\eta }_{D_2}^{+})\left((w,w^{\prime })(w_1,w_1^{\prime })\right)& =& {\eta }_{D_1}^{+}\left(w{w}_1\right)\times {\eta }_{C_2}^{+}\left(w^{\prime }\right)\times {\eta }_{C_2}^{+}\left(w_1^{\prime }\right),\hfill \\ \hfill ({\eta }_{D_1}^{-}\circ {\eta }_{D_2}^{-})\left((w,w^{\prime })(w_1,w_1^{\prime })\right)& =& {\eta }_{D_1}^{-}\left(w{w}_1\right)\times {\eta }_{C_2}^{-}\left(w^{\prime }\right)\times {\eta }_{C_2}^{+}\left(w_1^{\prime }\right),\hfill \\ & =& {\eta }_{C_1}^{+}\left(w\right)\times {\eta }_{C_1}^{+}\left(w_1\right)\times {\eta }_{C_2}^{+}\left(w^{\prime }\right)\times {\eta }_{C_2}^{+}\left(w_1^{\prime }\right),\hfill \\ & =& -({\eta }_{C_1}^{-}\left(w\right)\times {\eta }_{C_1}^{-}\left(w_1\right))\times {\eta }_{C_2}^{-}\left(w^{\prime }\right)\times {\eta }_{C_2}^{-}\left(w_1^{\prime }\right),\hfill \\ & =& {\eta }_{C_1}^{+}\left(w\right)\times {\eta }_{C_2}^{+}\left(w^{\prime }\right)\times {\eta }_{C_1}^{+}\left(w_1\right)\times {\eta }_{C_2}^{+}\left(w_1^{\prime }\right),\hfill \\ & =& -({\eta }_{C_1}^{-}\left(w\right)\times {\eta }_{C_2}^{-}\left(w^{\prime }\right)\times {\eta }_{C_1}^{-}\left(w_1\right)\times {\eta }_{C_2}^{-}\left(w_1^{\prime }\right)),\hfill \\ \hfill ({\eta }_{D_1}^{+}\circ {\eta }_{D_2}^{+})\left((w,w^{\prime })(w_1,w_1^{\prime })\right)& =& ({\eta }_{C_1}^{+}\circ {\eta }_{C_2}^{+})(w,w^{\prime })\times ({\eta }_{C_1}^{+}\circ {\eta }_{C_2}^{+})(w_1,w_1^{\prime }),\hfill \\ \hfill ({\eta }_{D_1}^{-}\circ {\eta }_{D_2}^{-})\left((w,w^{\prime })(w_1,w_1^{\prime })\right)& =& ({\eta }_{C_1}^{-}\circ {\eta }_{C_2}^{-})(w,w^{\prime })\times ({\eta }_{C_1}^{-}\circ {\eta }_{C_2}^{-})(w_1,w_1^{\prime }).\hfill \end{array}$$

Hence, $(w_1,w_1^{\prime })$ dominates $(w,w^{\prime })$ in ${\mathcal{H}}_1\circ {\mathcal{H}}_2.$ Thus, ${\mathcal{D}}_1\times {\mathcal{D}}_2$ is a DS of ${\mathcal{H}}_1\circ {\mathcal{H}}_2.$    □

5. Relations Between Product Bipolar Fuzzy Graphs

Definition 25. 

Let ${\mathcal{H}}_1$ = $(C_1,D_1)$ and ${\mathcal{H}}_2$ = $(C_2,D_2)$ be two ${\mathit{PB}}_f$ graphs on $W_1$ and $W_2$, respectively. An isomorphism of ${\mathit{PB}}_f$ graphs $f:{\mathcal{H}}_1⟶{\mathcal{H}}_2$ is a bijective map $f:W_1⟶W_2$ that fulfills the following properties:

1. 

${\eta }_{C_1}^{+}\left(w\right)={\eta }_{C_2}^{+}\left(f\left(w\right)\right)$ and ${\eta }_{C_1}^{-}\left(w\right)={\eta }_{C_2}^{-}\left(f\left(w\right)\right)$ for all $w\in W_1.$

2. 

${\eta }_{D_1}^{+}\left(w_1{w}_2\right)={\eta }_{D_2}^{+}\left(f\left(w_1\right)f\left(w_2\right)\right)$ and ${\eta }_{D_1}^{-}\left(w_1{w}_2\right)={\eta }_{D_2}^{-}\left(f\left(w_1\right)f\left(w_2\right)\right)$ for all $w_1,w_2\in W_1.$

If there is an isomorphism from ${\mathcal{H}}_1$ to ${\mathcal{H}}_2$, then we denote ${\mathcal{H}}_1\cong {\mathcal{H}}_2.$

Theorem 12. 

Let ${\mathcal{H}}_1$ = $(C_1,D_1)$ and ${\mathcal{H}}_2$ = $(C_2,D_2)$ be two ${\mathit{PB}}_f$ graphs on $W_1$ and $W_2$, respectively. Then,

$$\left(\overline{{\mathcal{H}}_1+{\mathcal{H}}_2}\right)=\overline{{\mathcal{H}}_1}\cup \overline{{\mathcal{H}}_2}.$$

Proof. 

Consider the identity map $I:W_1\cup W_2⟶W_1\cup W_2.$ It is enough to prove the following:

(i)

$\left(\overline{{\eta }_{C_1}^{+}+{\eta }_{C_2}^{+}}\right)\left(w\right)$ = $({\eta }_{\overline{C_1}}^{+}\cup {\eta }_{\overline{C_2}}^{+})\left(w\right)$ and $\left(\overline{{\eta }_{C_1}^{-}+{\eta }_{C_2}^{-}}\right)\left(w\right)$ = $({\eta }_{\overline{C_1}}^{-}\cup {\eta }_{\overline{C_2}}^{-})\left(w\right)$ for all $w\in W_1\cup W_2.$

(ii)

$\left(\overline{{\eta }_{D_1}^{+}+{\eta }_{D_2}^{+}}\right)\left(w_1{w}_2\right)$ = $({\eta }_{\overline{D_1}}^{+}\cup {\eta }_{\overline{D_2}}^{+})\left(w_1{w}_2\right)$ and $\left(\overline{{\eta }_{D_1}^{-}+{\eta }_{D_2}^{-}}\right)\left(w_1{w}_2\right)$ = $({\eta }_{\overline{D_1}}^{-}\cup {\eta }_{\overline{D_2}}^{-})\left(w_1{w}_2\right)$ for all $w_1,w_2\in W_1\cup W_2.$

By the definition of a complement, we know that

$$\left(\overline{{\eta }_{C_1}^{+}+{\eta }_{C_2}^{+}}\right)\left(w\right)=\left({\eta }_{C_1}^{+}+{\eta }_{C_2}^{+}\right)\left(w\right)and\left(\overline{{\eta }_{C_1}^{-}+{\eta }_{C_2}^{-}}\right)\left(w\right)=\left({\eta }_{C_1}^{-}+{\eta }_{C_2}^{-}\right)\left(w\right),$$

for all $w\in W_1\cup W_2.$ Then,

$$\begin{gathered} ({\eta }_{C_1}^{+}+{\eta }_{C_2}^{+})\left(w\right)=\left\{\begin{array}{cc}{\eta }_{C_1}^{+}\left(w\right),\hfill & \mathrm{if}w\in W_1,\hfill \\ {\eta }_{C_2}^{+}\left(w\right),\hfill & \mathrm{if}w\in W_2.\hfill \end{array}\right. \\ ({\eta }_{C_1}^{-}+{\eta }_{C_2}^{-})\left(w\right)=\left\{\begin{array}{cc}{\eta }_{C_1}^{-}\left(w\right),\hfill & \mathrm{if}{w}_1\in W_1,\hfill \\ {\eta }_{C_2}^{-}\left(w\right),\hfill & \mathrm{if}{w}_1\in W_2,\hfill \end{array}\right. \end{gathered}$$

$$\begin{gathered} ({\eta }_{C_1}^{+}+{\eta }_{C_2}^{+})\left(w\right)=\left\{\begin{array}{cc}{\eta }_{\overline{C_1}}^{+}\left(w\right),\hfill & \mathrm{if}w\in W_1,\hfill \\ {\eta }_{\overline{C_2}}^{+}\left(w\right),\hfill & \mathrm{if}w\in W_2.\hfill \end{array}\right. \\ ({\eta }_{C_1}^{-}+{\eta }_{C_2}^{-})\left(w\right)=\left\{\begin{array}{cc}{\eta }_{\overline{C_1}}^{-}\left(w\right),\hfill & \mathrm{if}{w}_1\in W_1,\hfill \\ {\eta }_{\overline{C_2}}^{-}\left(w\right),\hfill & \mathrm{if}{w}_1\in W_2,\hfill \end{array}\right. \end{gathered}$$

$$\begin{array}{ccc}& =& \left({\eta }_{\overline{C_1}}^{+}\cup {\eta }_{\overline{C_2}}^{+}\right)\left(w\right),\hfill \\ & =& \left({\eta }_{\overline{C_1}}^{-}\cup {\eta }_{\overline{C_2}}^{-}\right)\left(w\right),\hfill \end{array}$$

Hence,

$$\left(\overline{{\eta }_{C_1}^{+}+{\eta }_{C_2}^{+}}\right)\left(w\right)=\left({\eta }_{\overline{C_1}}^{+}\cup {\eta }_{\overline{C_2}}^{+}\right)\left(w\right)\mathrm{and}\left(\overline{{\eta }_{C_1}^{-}+{\eta }_{C_2}^{-}}\right)\left(w\right)=\left({\eta }_{\overline{C_1}}^{-}\cup {\eta }_{\overline{C_2}}^{-}\right)\left(w\right),$$

for all $w\in W_1\cup W_2.$

$$\begin{array}{ccc}\hfill \left(\overline{{\eta }_{D_1}^{+}+{\eta }_{D_2}^{+}}\right)\left(w_1{w}_2\right)& =& ({\eta }_{D_1}^{+}+{\eta }_{D_2}^{+})\left(w_1\right)\times ({\eta }_{D_1}^{+}+{\eta }_{D_2}^{+})\left(w_2\right)-({\eta }_{D_1}^{+}+{\eta }_{D_2}^{+})\left(w_1{w}_2\right),\hfill \\ \hfill \left(\overline{{\eta }_{D_1}^{-}+{\eta }_{D_2}^{-}}\right)\left(w_1{w}_2\right)& =& -(({\eta }_{D_1}^{-}+{\eta }_{D_2}^{-})\left(w_1\right)\times ({\eta }_{D_1}^{-}+{\eta }_{D_2}^{-})\left(w_2\right))-({\eta }_{D_1}^{-}+{\eta }_{D_2}^{-})\left(w_1{w}_2\right).\hfill \end{array}$$

$$\begin{array}{ccc}\hfill \left(\overline{{\eta }_{D_1}^{+}+{\eta }_{D_2}^{+}}\right)\left(w_1{w}_2\right)& =& \left\{\begin{array}{cc}{\eta }_{C_1}^{+}\left(w_1\right)\times {\eta }_{C_1}^{+}\left(w_2\right)-{\eta }_{D_1}^{+}\left(w_1{w}_2\right),\hfill & \mathrm{if}{w}_1,w_2\in W_1,\hfill \\ {\eta }_{C_2}^{+}\left(w_1\right)\times {\eta }_{C_2}^{+}\left(w_2\right)-{\eta }_{D_2}^{+}\left(w_1{w}_2\right),\hfill & \mathrm{if}{w}_1,w_2\in W_2,\hfill \\ {\eta }_{C_1}^{+}\left(w_1\right)\times {\eta }_{C_2}^{+}\left(w_2\right)-{\eta }_{C_1}^{+}\left(w_1\right)\times {\eta }_{C_2}^{+}\left(w_2\right),\hfill & \mathrm{if}{w}_1\in W_{1}and{w}_2\in W_2.\hfill \end{array}\right.\hfill \\ \hfill \left(\overline{{\eta }_{D_1}^{-}+{\eta }_{D_2}^{-}}\right)\left(w_1{w}_2\right)& =& \left\{\begin{array}{cc}-({\eta }_{C_1}^{-}\left(w_1\right)\times {\eta }_{C_1}^{-}\left(w_2\right))-{\eta }_{D_1}^{-}\left(w_1{w}_2\right),\hfill & \mathrm{if}{w}_1,w_2\in W_1,\hfill \\ -({\eta }_{C_2}^{-}\left(w_1\right)\times {\eta }_{C_2}^{-}\left(w_2\right))-{\eta }_{D_2}^{-}\left(w_1{w}_2\right),\hfill & \mathrm{if}{w}_1,w_2\in W_2,\hfill \\ -({\eta }_{C_1}^{-}\left(w_1\right)\times {\eta }_{C_2}^{-}\left(w_2\right))-(-({\eta }_{C_1}^{-}\left(w_1\right)\times {\eta }_{C_2}^{-}\left(w_2\right))),\hfill & \mathrm{if}{w}_1\in W_{1}and{w}_2\in W_2.\hfill \end{array}\right.\hfill \end{array}$$

$$=\left\{\begin{array}{cc}{\eta }_{\overline{D_1}}^{+}\left(w_1{w}_2\right),\hfill & \mathrm{if}{w}_1,w_2\in W_1,\hfill \\ {\eta }_{\overline{D_2}}^{+}\left(w_1{w}_2\right),\hfill & \mathrm{if}{w}_1,w_2\in W_2,\hfill \\ 0,\hfill & \mathrm{if}{w}_1\in W_{1}and{w}_2\in W_2.\hfill \end{array}\right.$$

$$=\left\{\begin{array}{cc}{\eta }_{\overline{D_1}}^{-}\left(w_1{w}_2\right),\hfill & \mathrm{if}{w}_1,w_2\in W_1,\hfill \\ {\eta }_{\overline{D_2}}^{-}\left(w_1{w}_2\right),\hfill & \mathrm{if}{w}_1,w_2\in W_2,\hfill \\ 0,\hfill & \mathrm{if}{w}_1\in W_{1}and{w}_2\in W_2.\hfill \end{array}\right.$$

Hence,

$$\begin{array}{ccc}\hfill \left(\overline{{\eta }_{D_1}^{+}+{\eta }_{D_2}^{+}}\right)\left(w_1{w}_2\right)& =& ({\eta }_{\overline{D_1}}^{+}\cup {\eta }_{\overline{D_2}}^{+})\left(w_1{w}_2\right),\hfill \\ \hfill \left(\overline{{\eta }_{D_1}^{-}+{\eta }_{D_2}^{-}}\right)\left(w_1{w}_2\right)& =& ({\eta }_{\overline{D_1}}^{-}\cup {\eta }_{\overline{D_2}}^{-})\left(w_1{w}_2\right).\hfill \end{array}$$

for all $w_1,w_2\in W_1\cup W_2.$    □

Theorem 13. 

Let ${\mathcal{H}}_1$ = $(C_1,D_1)$ and ${\mathcal{H}}_2$ = $(C_2,D_2)$ be two ${\mathit{PB}}_f$ graphs on $W_1$ and $W_2$, respectively. Then,

$$\left(\overline{{\mathcal{H}}_1\cup {\mathcal{H}}_2}\right)=\overline{{\mathcal{H}}_1}+\overline{{\mathcal{H}}_2}.$$

Proof. 

Consider the identity map $I:W_1\cup W_2⟶W_1\cup W_2.$ It is enough to prove the following:

(i)

$\left(\overline{{\eta }_{C_1}^{+}\cup {\eta }_{C_2}^{+}}\right)\left(w\right)$ = $({\eta }_{\overline{C_1}}^{+}+{\eta }_{\overline{C_2}}^{+})\left(w\right)$ and $\left(\overline{{\eta }_{C_1}^{-}\cup {\eta }_{C_2}^{-}}\right)\left(w\right)$ = $({\eta }_{\overline{C_1}}^{-}+{\eta }_{\overline{C_2}}^{-})\left(w\right)$ for all $w\in W_1\cup W_2.$

(ii)

$\left(\overline{{\eta }_{D_1}^{+}\cup {\eta }_{D_2}^{+}}\right)\left(w_1{w}_2\right)$ = $({\eta }_{\overline{D_1}}^{+}+{\eta }_{\overline{D_2}}^{+})\left(w_1{w}_2\right)$ and $\left(\overline{{\eta }_{D_1}^{-}\cup {\eta }_{D_2}^{-}}\right)\left(w_1{w}_2\right)$ = $({\eta }_{\overline{D_1}}^{-}+{\eta }_{\overline{D_2}}^{-})\left(w_1{w}_2\right)$ for all $w_1,w_2\in W_1\cup W_2.$

By the definition of a complement, we know that

$$\left(\overline{{\eta }_{C_1}^{+}\cup {\eta }_{C_2}^{+}}\right)\left(w\right)=\left({\eta }_{C_1}^{+}\cup {\eta }_{C_2}^{+}\right)\left(w\right)\mathrm{and}\left(\overline{{\eta }_{C_1}^{-}\cup {\eta }_{C_2}^{-}}\right)\left(w\right)=\left({\eta }_{C_1}^{-}\cup {\eta }_{C_2}^{-}\right)\left(w\right),$$

for all $w\in W_1\cup W_2.$ Then,

$$\begin{gathered} ({\eta }_{C_1}^{+}\cup {\eta }_{C_2}^{+})\left(w\right)=\left\{\begin{array}{cc}{\eta }_{C_1}^{+}\left(w\right),\hfill & \mathrm{if}w\in W_1,\hfill \\ {\eta }_{C_2}^{+}\left(w\right),\hfill & \mathrm{if}w\in W_2.\hfill \end{array}\right. \\ ({\eta }_{C_1}^{-}\cup {\eta }_{C_2}^{-})\left(w\right)=\left\{\begin{array}{cc}{\eta }_{C_1}^{-}\left(w\right),\hfill & \mathrm{if}{w}_1\in W_1,\hfill \\ {\eta }_{C_2}^{-}\left(w\right),\hfill & \mathrm{if}{w}_1\in W_2,\hfill \end{array}\right. \end{gathered}$$

$$\begin{gathered} ({\eta }_{C_1}^{+}\cup {\eta }_{C_2}^{+})\left(w\right)=\left\{\begin{array}{cc}{\eta }_{\overline{C_1}}^{+}\left(w\right),\hfill & \mathrm{if}w\in W_1,\hfill \\ {\eta }_{\overline{C_2}}^{+}\left(w\right),\hfill & \mathrm{if}w\in W_2.\hfill \end{array}\right. \\ ({\eta }_{C_1}^{-}\cup {\eta }_{C_2}^{-})\left(w\right)=\left\{\begin{array}{cc}{\eta }_{\overline{C_1}}^{-}\left(w\right),\hfill & \mathrm{if}{w}_1\in W_1,\hfill \\ {\eta }_{\overline{C_2}}^{-}\left(w\right),\hfill & \mathrm{if}{w}_1\in W_2,\hfill \end{array}\right. \end{gathered}$$

$$\begin{array}{ccc}\hfill ({\eta }_{C_1}^{+}\cup {\eta }_{C_2}^{+})\left(w\right)& =& \left({\eta }_{\overline{C_1}}^{+}+{\eta }_{\overline{C_2}}^{+}\right)\left(w\right),\hfill \\ \hfill ({\eta }_{C_1}^{-}\cup {\eta }_{C_2}^{-})\left(w\right)& =& \left({\eta }_{\overline{C_1}}^{-}+{\eta }_{\overline{C_2}}^{-}\right)\left(w\right).\hfill \end{array}$$

Hence,

$$\left(\overline{{\eta }_{C_1}^{+}\cup {\eta }_{C_2}^{+}}\right)\left(w\right)=\left({\eta }_{\overline{C_1}}^{+}+{\eta }_{\overline{C_2}}^{+}\right)\left(w\right)\mathrm{and}\left(\overline{{\eta }_{C_1}^{-}\cup {\eta }_{C_2}^{-}}\right)\left(w\right)=\left({\eta }_{\overline{C_1}}^{-}+{\eta }_{\overline{C_2}}^{-}\right)\left(w\right),$$

for all $w\in W_1\cup W_2.$

$$\begin{array}{ccc}\hfill \left(\overline{{\eta }_{D_1}^{+}\cup {\eta }_{D_2}^{+}}\right)\left(w_1{w}_2\right)& =& ({\eta }_{D_1}^{+}\cup {\eta }_{D_2}^{+})\left(w_1\right)\times ({\eta }_{D_1}^{+}\cup {\eta }_{D_2}^{+})\left(w_2\right)-({\eta }_{D_1}^{+}\cup {\eta }_{D_2}^{+})\left(w_1{w}_2\right),\hfill \\ \hfill \left(\overline{{\eta }_{D_1}^{-}\cup {\eta }_{D_2}^{-}}\right)\left(w_1{w}_2\right)& =& -(({\eta }_{D_1}^{-}\cup {\eta }_{D_2}^{-})\left(w_1\right)\times ({\eta }_{D_1}^{-}\cup {\eta }_{D_2}^{-})\left(w_2\right))-({\eta }_{D_1}^{-}\cup {\eta }_{D_2}^{-})\left(w_1{w}_2\right).\hfill \end{array}$$

$$\begin{array}{ccc}\hfill \left(\overline{{\eta }_{D_1}^{+}\cup {\eta }_{D_2}^{+}}\right)\left(w_1{w}_2\right)& =& \left\{\begin{array}{cc}{\eta }_{C_1}^{+}\left(w_1\right)\times {\eta }_{C_1}^{+}\left(w_2\right)-{\eta }_{D_1}^{+}\left(w_1{w}_2\right),\hfill & \mathrm{if}{w}_1,w_2\in W_1,\hfill \\ {\eta }_{C_2}^{+}\left(w_1\right)\times {\eta }_{C_2}^{+}\left(w_2\right)-{\eta }_{D_2}^{+}\left(w_1{w}_2\right),\hfill & \mathrm{if}{w}_1,w_2\in W_2,\hfill \\ {\eta }_{C_1}^{+}\left(w_1\right)\times {\eta }_{C_2}^{+}\left(w_2\right)-{\eta }_{C_1}^{+}\left(w_1\right)\times {\eta }_{C_2}^{+}\left(w_2\right),\hfill & \mathrm{if}{w}_1\in W_{1}and{w}_2\in W_2.\hfill \end{array}\right.\hfill \\ \hfill \left(\overline{{\eta }_{D_1}^{-}\cup {\eta }_{D_2}^{-}}\right)\left(w_1{w}_2\right)& =& \left\{\begin{array}{cc}-({\eta }_{C_1}^{-}\left(w_1\right)\times {\eta }_{C_1}^{-}\left(w_2\right))-{\eta }_{D_1}^{-}\left(w_1{w}_2\right),\hfill & \mathrm{if}{w}_1,w_2\in W_1,\hfill \\ -({\eta }_{C_2}^{-}\left(w_1\right)\times {\eta }_{C_2}^{-}\left(w_2\right))-{\eta }_{D_2}^{-}\left(w_1{w}_2\right),\hfill & \mathrm{if}{w}_1,w_2\in W_2,\hfill \\ -({\eta }_{C_1}^{-}\left(w_1\right)\times {\eta }_{C_2}^{-}\left(w_2\right))-(-({\eta }_{C_1}^{-}\left(w_1\right)\times {\eta }_{C_2}^{-}\left(w_2\right))),\hfill & \mathrm{if}{w}_1\in W_{1}and{w}_2\in W_2.\hfill \end{array}\right.\hfill \end{array}$$

$$=\left\{\begin{array}{cc}{\eta }_{\overline{D_1}}^{+}\left(w_1{w}_2\right),\hfill & \mathrm{if}{w}_1,w_2\in W_1,\hfill \\ {\eta }_{\overline{D_2}}^{+}\left(w_1{w}_2\right),\hfill & \mathrm{if}{w}_1,w_2\in W_2,\hfill \\ 0,\hfill & \mathrm{if}{w}_1\in W_{1}and{w}_2\in W_2.\hfill \end{array}\right.$$

$$=\begin{array}{cc}{\eta }_{\overline{D_1}}^{-}\left(w_1{w}_2\right),\hfill & \mathrm{if}{w}_1,w_2\in W_1,\hfill \\ {\eta }_{\overline{D_2}}^{-}\left(w_1{w}_2\right),\hfill & \mathrm{if}{w}_1,w_2\in W_2,\hfill \\ 0,\hfill & \mathrm{if}{w}_1\in W_{1}and{w}_2\in W_2.\hfill \end{array}$$

Hence,

$$\begin{array}{ccc}\hfill \left(\overline{{\eta }_{D_1}^{+}\cup {\eta }_{D_2}^{+}}\right)\left(w_1{w}_2\right)& =& ({\eta }_{\overline{D_1}}^{+}+{\eta }_{\overline{D_2}}^{+})\left(w_1{w}_2\right),\hfill \\ \hfill \left(\overline{{\eta }_{D_1}^{-}\cup {\eta }_{D_2}^{-}}\right)\left(w_1{w}_2\right)& =& ({\eta }_{\overline{D_1}}^{-}+{\eta }_{\overline{D_2}}^{-})\left(w_1{w}_2\right).\hfill \end{array}$$

for all $w_1,w_2\in W_1\cup W_2.$    □

Theorem 14. 

Let ${\mathcal{H}}_1$ = $(C_1,D_1)$ be a ${\mathit{PB}}_f$ graph of ${\mathcal{G}}_1^{*}=(W_1,E_1)$ such that ${\eta }_{D_1}^{+}\left(w_1{w}_2\right)={\eta }_{C_1}^{+}\left(w_1\right)\times {\eta }_{C_1}^{+}\left(w_2\right)$ and ${\eta }_{D_1}^{-}\left(w_1{w}_2\right)=-({\eta }_{C_1}^{-}\left(w_1\right)\times {\eta }_{C_1}^{-}\left(w_2\right))$ for all $w_1{w}_2\in E_1$. Similarly, let ${\mathcal{H}}_2$ = $(C_2,D_2)$ be a ${\mathit{PB}}_f$ graph of ${\mathcal{G}}_2^{*}=(W_2,E_2)$ such that ${\eta }_{D_2}^{+}\left(w_1^{\prime }{w}_2^{\prime }\right)={\eta }_{C_2}^{+}\left(w_1^{\prime }\right)\times {\eta }_{C_2}^{+}\left(w_2^{\prime }\right)$ and ${\eta }_{D_2}^{-}\left(w_1^{\prime }{w}_2^{\prime }\right)=-({\eta }_{C_2}^{-}\left(w_1^{\prime }\right)\times {\eta }_{C_2}^{-}\left(w_2^{\prime }\right))$ for all $w_1^{\prime }{w}_2^{\prime }\in E_2$. Both ${\mathcal{H}}_1$ and ${\mathcal{H}}_2$ are disjoint ${\mathit{PB}}_f$ graphs. Then,

$$⋎({\mathcal{H}}_1\cup {\mathcal{H}}_2)=⋎\left({\mathcal{H}}_1\right)+⋎\left({\mathcal{H}}_2\right).$$

Proof. 

Let ${\mathcal{D}}_1$ be a ${⋎}_1$-set of a ${\mathit{PB}}_f$ graph ${\mathcal{H}}_1$ and ${\mathcal{D}}_2$ be a ${⋎}_2$-set of a ${\mathit{PB}}_f$ graph ${\mathcal{H}}_2$. Then, by Theorem 14, ${\mathcal{D}}_1\cup {\mathcal{D}}_2$ is a DS of ${\mathcal{H}}_1\cup {\mathcal{H}}_2$. Since both ${\mathcal{H}}_1$ and ${\mathcal{H}}_2$ are disjoint ${\mathit{PB}}_f$ graphs, $|{\mathcal{D}}_1\cup {\mathcal{D}}_2|$ is a minimum DS of ${\mathcal{H}}_1\cup {\mathcal{H}}_2$. Hence,

$$⋎({\mathcal{H}}_1\cup {\mathcal{H}}_2)=|{\mathcal{D}}_1\cup {\mathcal{D}}_2|=⋎\left({\mathcal{H}}_1\right)+⋎\left({\mathcal{H}}_2\right).$$

   □

6. Application of Domination in Product Bipolar Fuzzy Graph

Domination plays a vital role in solving real-life problems. In this section, we utilize the concept of domination to address decision-making problems related to locating metro bus stations. For instance, the Lahore Metro Bus Service in Punjab, Pakistan, operates from Gajju Matah to Shahdara via Ferozepur Road, allowing passengers to reach their destinations on time. However, there are many cities where metro bus services are unavailable, causing inconvenience for travelers.

Consider two cities where people frequently travel, but the metro bus service is not yet available. To save travel time, the government plans to introduce a metro bus service between these cities. Since these cities also contain several rural areas, it is not feasible to establish stations in every locality. Therefore, the concept of domination sets helps us strategically select stations in such a way that every area can benefit from the metro bus service.

6.1. Locating Metro Bus Stations Using Domination

Consider City A as a source point and City B as a destination point. We know that many areas exist between any two cities. In our discussion, we highlight some main areas between City A and City B. Consider a set W of City A, City B, and the main areas that exist between these two cities.

$$W=\{\mathrm{City}\mathrm{A}, \mathrm{Area}1, \mathrm{Area}2, \mathrm{Area}3, \mathrm{Area}4, \mathrm{Area}5, \mathrm{Area}6, \mathrm{Area}7, \mathrm{Area}8, \mathrm{City}\mathrm{B}\}.$$

Now, we want to construct a track that connects these places only. To build a route between these two cities, we convert this problem into a ${\mathrm{PB}}_f$ graph. We consider that the places are vertices and the existing direct roadways between these places are edges. Let C be a ${\mathrm{B}}_f$ set on W as defined in

Table 10. ${\mathrm{B}}_f$ set C of places.

W	Places	Positive Value	Negative Value
$w_1$	City A	$0.9$	$-0.1$
$w_2$	Area 1	$0.8$	$-0.2$
$w_3$	Area 2	$0.4$	$-0.3$
$w_4$	Area 3	$0.8$	$-0.2$
$w_5$	Area 4	$0.5$	$-0.1$
$w_6$	Area 5	$0.9$	$-0.1$
$w_7$	Area 6	$0.9$	$-0.1$
$w_8$	Area 7	$0.4$	$-0.5$
$w_9$	Area 8	$0.2$	$-0.8$
$w_{10}$	City B	$0.9$	$-0.1$

.

The positive membership value of each place indicates the degree of frequent transport usage of the people at this place. The negative membership value of each place indicates the degree of infrequent transport usage of the people at this place. In a set form, these attributes can be represented as

$$\{\mathrm{frequent} \mathrm{usage} \mathrm{of} \mathrm{transport}, \mathrm{infrequent} \mathrm{usage} \mathrm{of} \mathrm{transport}\}.$$

For example, the positive membership value of City A is $0.9$, which indicates that there are $90\%$ people in City A that are frequently using transport services. The negative membership value of City A is $-0.1$, which indicates that there are $10\%$ people who are infrequent transport users. Now, let D be a ${\mathrm{B}}_f$ relation on W as defined in

Table 11. ${\mathrm{B}}_f$ relation D of places.

W	Places	Positive Value	Negative Value
$w_1{w}_2$	(City A, Area 1)	$0.72$	$-0.02$
$w_2{w}_3$	(Area 1, Area 2)	$0.3$	$-0.05$
$w_3{w}_4$	(Area 2, Area 3)	$0.36$	$-0.03$
$w_4{w}_5$	(Area 3, Area 4)	$0.45$	$-0.01$
$w_5{w}_6$	(Area 4, Area 5)	$0.45$	$-0.01$
$w_6{w}_7$	(Area 5, Area 6)	$0.81$	$-0.01$
$w_7{w}_8$	(Area 6, Area 7)	$0.36$	$-0.05$
$w_8{w}_9$	(Area 7, Area 8)	$0.07$	$-0.4$
$w_9{w}_{10}$	(Area 8, City B)	$0.18$	$-0.08$

. In Table 11, we list the membership values between two places, and the membership value of each pair of places is expressed as

$${\eta }_D^{+}\left(w_1{w}_2\right)\le {\eta }_C^{+}\left(w_1\right)\times {\eta }_C^{+}\left(w_2\right)\mathrm{and}{\eta }_D^{-}\left(w_1{w}_2\right)\ge -({\eta }_C^{-}\left(w_1\right)\times {\eta }_C^{-}\left(w_2\right)),$$

for all $w_1,w_2\in W.$

The positive membership value of each pair indicates the degree of comfortable routes while traveling between two areas. The negative membership value of the edge indicates the degree of uncomfortable routes while traveling between two areas. In a set form, these membership characteristics can be stated as

$$\{\mathrm{comfortable} \mathrm{routes} \mathrm{during} \mathrm{traveling}, \mathrm{uncomfortable} \mathrm{routes} \mathrm{during} \mathrm{traveling}\}.$$

For example, the edge between Area 1 and Area 2 has a membership value of $(0.3,-0.05)$, which indicates that $30\%$ of routes are comfortable for people traveling between Area 1 and Area 2 and $0.5\%$ of routes are uncomfortable when people travel between Area 1 and Area 2. Based on the data above,

$$W=\{w_1, w_2, w_3, w_4, w_5, w_6, w_7, w_8, w_9, w_{10}\},$$

is a crisp set.

$$\begin{gathered} C= \\ \{(w_1,0.9,-0.1),(w_2,0.8,-0.2),(w_3,0.4,-0.3),(w_4,0.9,-0.1),(w_5,0.5,-0.1),(w_6,0.9,-0.1), \\ (w_7,0.9,-0.1),(w_8,0.4,-0.5),(w_9,0.2,-0.8),(w_{10},0.9,-0.1)\}, \end{gathered}$$

is a ${\mathrm{B}}_f$ set on $W.$

$$\begin{gathered} D=\{(w_1{w}_2,0.72,-0.02),(w_2{w}_3,0.3,-0.05),(w_3{w}_4,0.36,-0.03),(w_4{w}_5,0.45,-0.01), \\ (w_5{w}_6,0.45,-0.01),(w_6{w}_7,0.81,-0.01),(w_7{w}_8,0.36,-0.05),(w_8{w}_9,0.07,-0.4),(w_9{w}_{10},0.18,-0.08)\}, \end{gathered}$$

is a ${\mathrm{B}}_f$ relation on $W.$ This is a ${\mathrm{PB}}_f$ graph $\mathcal{H}$ = $(C,D)$, which is given in Figure 12.

From Figure 12, we can observe that a track is constructed between the given places. There are eight areas between City A and City B, but it is not feasible to place stations at all of them. The concept of domination plays a crucial role in selecting the optimal locations for stations. We need to find an MDS of a ${\mathrm{PB}}_f$ (Figure 12) that decides which are the places where we need to place stations. So, to find the MDS, first, we need to find all DSs of Figure 12. To check whether a given subset $\mathcal{D}$ of W is a DS or not, we need to check the _j $\in$$W-\mathcal{D}$ places we have, where place _i $\in \mathcal{D}$ must satisfy the following condition:

$${\eta }_D^{+}\left({\mathrm{place}}_i{\mathrm{place}}_j\right)={\eta }_C^{+}\left({\mathrm{place}}_i\right)\times {\eta }_C^{+}\left({\mathrm{place}}_j\right)\mathrm{and}{\eta }_D^{+}\left({\mathrm{place}}_i{\mathrm{place}}_j\right)=-({\eta }_C^{+}\left({\mathrm{place}}_i\right)\times {\eta }_C^{+}\left({\mathrm{place}}_j\right)).$$

And the DS indicates that all the places that are in $W-\mathcal{D}$ must have a nearby station in $\mathcal{D}$.

Let $\mathcal{D}$ = {City A, Area 3, Area 5, Area 6, City B}. So, $W-\mathcal{D}$ = {Area 1, Area 2, Area 4, Area 7, Area 8}. It is easy to check that $\mathcal{D}$ is a DS:

• The people who are living in Area 1 may use the nearby station in City A.
• The people who are living in Area 2 may use the nearby station in Area 3.
• The people who are living in Area 4 may use both nearby stations in Area 3 and Area 5.
• The people who are living in Area 7 may use the nearby station in Area 6.
• The people who are living in Area 8 may use the nearby station in City B.

So, {City A, Area 3, Area 5, Area 6, City B} is a DS. If we select stations in both Areas, Area 5, and Area 6, then the DS is not an MDS. However, we need to find an MDS to place stations with maximum benefits. So, we choose a station in just Area 6. Hence,

$$\{\mathrm{City}A, \mathrm{Area}3, \mathrm{Area}6, \mathrm{City}B\},$$

is an MDS. Therefore, we conclude that these are the optimal locations for the stations. In Figure 13, the circled areas indicate the placement of the stations, and a line between two places shows that the people who are in Area 1 may use the nearby station in City A. Dotted circles indicate those areas that have no station, and a dotted line between two areas indicates that the people who are in Area 1 cannot use Area 2 because there exists no station in Area 2. We added Algorithm 1 for placing metro bus stations.

Algorithm 1: Method for Placing Stations of Metro Bus

Input the set W of cities and areas between the source and destination. Construct a graph $G=(W,E)$, where E represents the possible routes between places. Form a candidate set $\mathcal{D}\subseteq W$. Check whether $\mathcal{D}$ is a dominating set: $$\forall w_j\in W-\mathcal{D},\exists w_i\in \mathcal{D},(w_i,w_j)\in E$$ If the condition holds, check the minimal condition by removing any vertex from $\mathcal{D}$. If no smaller subset satisfies the dominating condition, $\mathcal{D}$ is a minimal dominating set (MDS). Repeat the process to find all MDSs.

6.2. Comparison with Bipolar Fuzzy Graphs

Domination plays a vital role in graph theory to solve the problems that arise in real-world situations including the problem of determining locations for army posts, radio stations, bus route stations, and many others. To find the solution to these problems, the best way is to convert these problems into graphs. The main parts of the problems and the relations between the main parts are converted into vertices and edges, respectively. Depending on the problems, assign membership grades to the vertices and edges, and then, with the help of domination theory, find the solution to the problems. In ${\mathrm{B}}_f$ graphs, the membership grades of edges are less than or equal to and greater than or equal to the minimum and maximum of two ${\mathrm{B}}_f$ vertices, respectively:

$${\eta }_N^{+}\left(w_1{w}_2\right)⩽{\eta }_M^{+}\left(w_1\right)\wedge {\eta }_M^{+}\left(w_2\right)\mathrm{and}{\eta }_N^{-}\left(w_1{w}_2\right)⩾{\eta }_M^{-}\left(w_1\right)\vee {\eta }_M^{-}\left(w_2\right),$$

for all $w_1,w_2\in W$ (see Definition 1). However, sometimes in real-life problems, it is possible that the membership grades of edges are less than or equal to and greater than or equal to the product of two ${\mathrm{B}}_f$ vertices, respectively:

$${\eta }_D^{+}\left(w_1{w}_2\right)⩽{\eta }_C^{+}\left(w_1\right)\times {\eta }_C^{+}\left(w_2\right)\mathrm{and}{\eta }_D^{-}\left(w_1{w}_2\right)⩾-({\eta }_C^{-}\left(w_1\right)\times {\eta }_C^{-}\left(w_2\right)),$$

for all $w_1,w_2\in W$ (see Definition 11). We call this a ${\mathrm{PB}}_f$ graph, which is also called a ${\mathrm{B}}_f$ graph. For example, in locating stations for the bus routing problem, the membership grades between places are less than or equal to and greater than or equal to the product of the membership grades of incident places, respectively; see Figure 12. In this case, we aim to place stations for bus routing. However, solving this problem using domination theory on the ${\mathrm{B}}_f$ graph does not yield better results. For instance, if we take a set $\mathcal{D}$ = {City A, Area 3, Area 5, Area 6, City B}, we can check whether it is a DS or not. So, $W-\mathcal{D}$ = {Area 1, Area 2, Area 4, Area 7, Area 8}. Clearly, this set $\mathcal{D}$ is not a dominating set because we have no place in $\mathcal{D}$ that satisfies the following condition:

$$\begin{gathered} {\eta }_D^{+}\left({\mathrm{place}}_i{\mathrm{place}}_j\right)={\eta }_C^{+}\left({\mathrm{place}}_i\right)\times {\eta }_C^{+}\left({\mathrm{place}}_j\right)\mathrm{and}{\eta }_D^{-}\left({\mathrm{place}}_i{\mathrm{place}}_j\right)= \\ -({\eta }_C^{-}\left({\mathrm{place}}_i\right)\times {\eta }_C^{-}\left({\mathrm{place}}_j\right)), \end{gathered}$$

where ${\mathrm{place}}_i\in \mathcal{D}$ and ${\mathrm{place}}_j\in W-\mathcal{D}.$ Thus, the only DS, in this case, is

$$W=\{\mathrm{City} \mathrm{A}, \mathrm{Area} 1, \mathrm{Area} 2, \mathrm{Area} 3, \mathrm{Area} 4, \mathrm{Area} 5, \mathrm{Area} 6, \mathrm{Area} 7, \mathrm{Area} 8, \mathrm{City} \mathrm{B}\}.$$

This set indicates that these stations should be located at these places. However, if we solve this problem with the help of domination theory on the ${\mathrm{PB}}_f$ graph, we obtain $\mathcal{D}$ = {City A, Area 3, Area 6, City B}, which is an MDS and indicates that these are the best places for the stations. So, we see that in this case, the idea of domination in the ${\mathrm{PB}}_f$ graph gives a better result compared to the theory of domination on the ${\mathrm{B}}_f$ graph. So, from this example, we conclude the following two facts:

•

If the membership grades of edges are less than equal to and greater than equal to the minimum and maximum of two ${\mathrm{B}}_f$ vertices, respectively, we use the idea of domination on ${\mathrm{B}}_f$ graphs.

•

In the case of the membership grades of edges being less than equal to and greater than equal to the product of two ${\mathrm{B}}_f$ vertices, respectively, we use the theory of domination on ${\mathrm{PB}}_f$ graphs because in some cases, the theory of domination on ${\mathrm{B}}_f$ graphs does not yield better results.

7. Conclusions and Future Directions

The concept of domination in the bipolar fuzzy (Bf) model is a highly powerful tool with significant applications across various research domains. It offers greater accuracy and flexibility compared to traditional fuzzy models, making it a robust framework for addressing complex problems. In this research paper, we defined and explored various operations—such as union (∪), join (+), intersection (∩), Cartesian product (×), and composition (◦)—on product bipolar fuzzy (PBf) graphs, as well as the complement of PBf graphs, to generate new PBf graphs. We observed that the standard definitions of Cartesian product and composition for Bf graphs cannot be directly extended to PBf graphs, as the resulting structures do not satisfy the properties of PBf graphs.

We introduced and investigated the notions of dominating sets (DSs), minimal dominating sets (MDSs), and the domination number ($⋎\left(\mathcal{H}\right)$) for PBf graphs, supported by illustrative examples. Our study revealed that many results related to domination on Bf graphs also hold true for PBf graphs, particularly in the context of complete PBf graphs. Furthermore, we demonstrated the practical utility of domination on PBf graphs by applying it to a real-world decision-making problem involving the optimization of bus routing and station placement. This application highlights the critical role of domination sets in PBf graphs for solving complex logistical challenges. The present study opens several avenues for future research, including domination on PBf graphs using strong arcs, edge domination on PBf graphs, and strong and weak domination on PBf graphs. These directions promise to further enrich the theoretical and practical applications of bipolar fuzzy graph theory.