Exploring Symmetry Structures in Integrity-Based Vulnerability Analysis Using Bipolar Fuzzy Graph Theory

Abstract

The integrity parameter in vulnerability refers to a set of removed vertices and the maximum number of connected components that remain functional. A bipolar fuzzy graph (BFG) assigns membership values to both positive and negative attributes. A new parameter, integrity, is defined and discussed using an example of a BFG. The integrity value of a special type of graph is determined, and the node strength sequence (NSS) for BFG is introduced. Specific NSS values are used to discuss the integrity values of paths and cycles. The integrity of the union, join, and Cartesian product of two BFGs is presented. This parameter is then applied to a road network with both positive and negative attributes, and the findings are discussed with a conclusion.

1. Introduction

A graph can model any network, with crisp graphs assigning membership values of 1 to all vertices and edges. However, in real-world road networks, connectivity often exhibits uncertainty, where membership values range between 0 and 1. If connectivity is absent, the membership value is zero; if fully connected, it is one. A fuzzy graph accommodates such variations by accepting membership values in $\left[0,1\right]$. Incorporating symmetry and symmetric structures into fuzzy graphs provides a novel perspective for road network analysis, enabling the identification of balanced patterns, optimizing traffic flow, and enhancing resilience against disruptions through structural harmony.

If vertex and edge membership values fall within the range $\left[-1,1\right]$, the graph is a BFG (bipolar fuzzy graph). Without diminishing the significance of any linguistic factors, fuzzy graph theory can resolve practical issues. When considering linguistic terms, it often occurs that not all nodes possess the property; some nodes are even against it. All networks have pros and cons. A transport network helps improve a country’s economy but also creates environmental issues. When modeling this type of transport network as a BFG, membership values reflecting economic growth take the form $\left[0,1\right]$, while those indicating negative impact on the environment take the form $\left[-1,0\right]$.

L.A. Zadeh [1] proposed the mathematical concept of fuzzy sets to model uncertainty in real-life situations. Rosenfeld [2] extended this framework by introducing the theory of fuzzy graphs through fuzzy relations on fuzzy sets. Complete and strong fuzzy graphs were studied by Bhutani et al. [3,4], while Sunitha et al. [5] defined fundamental operations on fuzzy graphs such as union, join, composition, and complement. The concept of node strength sequence was introduced by Mathew et al. [6], and a comprehensive overview of fuzzy graph theory was later provided by Sunitha et al. [7]. More recently, Jamil [8] analyzed distance-based topological indices in double graphs, and Ali et al. [9] developed the notion of double resolving sets with the exchange property, emphasizing their applications in network optimization and cybersecurity, thereby extending the practical utility of graph-theoretic tools.

Akram [10] proposed BFGs and developed the concept of regular BFGs [11]. Rashmanlou et al. [12] discussed direct, semi-strong, and strong BFG; Broumi et al. [13] focused on bipolar neutrosophic graphs; and Pramanika et al. [14] introduced the concept of a bipolar fuzzy planar graph. In [11,15,16,17], regular, totally regular, irregular, highly irregular, neighborly irregular, neighborly totally irregular, and highly totally irregular BFGs are defined. Ref. [18] defines strong edge, dominating set, independent set, total dominating number, and independent number of BFGs. The concepts of ordinary order, size, and weak isomorphism of BFGs are discussed in [19,20], and refs. [21,22,23,24] covers self-complement, self-weak complement, strong isomorphism, and other properties of BFGs. Different types of BFGs, such as direct, regular, semi-strong, completely regular, and strong ones, are discussed in [12,16], while normal product and tensor product BFGs are covered in [25]. Various applications of BFGs are presented in [26,27,28,29,30].

Connectivity is a major concern in networking, and vulnerabilities that affect connectivity need to be measured using specific parameters. When disconnecting a network, each component has a certain impact. This disconnection process involves removing certain vertices or edges. The integrity parameter is linked to the set of vertices used for disconnection and the highest-order connected component that remains.

The concept of graph integrity was introduced by Goddard et al. [31], with a survey provided by Bagga et al. [32] covering weak integrity. Kirlangic et al. [33] discussed weak hub integrity, and Mahde et al. [34,35] explored hub integrity and weak hub integrity of graphs. Kilicc et al. [36] proposed edge integrity domination in graphs, and Besirik [37] presented total dominance integrity of graphs.

Saravanan et al. [38] introduced the concept of integrity in fuzzy graphs. Subsequent studies by the same authors [39,40,41] explored integrity across different contexts, including union, join, Cartesian product, domination, and efficient fuzzy graphs. Sujatha et al. [42] focused on span integrity within fuzzy graphs, while Sankar et al. [43,44,45] examined edge integrity, vertex integrity, and domination integrity specifically in signed fuzzy graphs.

1.1. Originality and Significance of Integrity in Bipolar Fuzzy Graphs

A fuzzy graph takes membership values only within the range [0, 1]. While converting profit and loss problems into a fuzzy graph, the membership value assigned to zero for non-profit does not imply a loss. It fails to indicate a loss in the fuzzy sense. Similarly, in the construction of roadways, there can be significant economic profit alongside substantial environmental issues. When converting this into a fuzzy model, traditional fuzzy models are insufficient. There is a need to take values in the range [−1, 1] and discuss determinacy and indeterminacy simultaneously. Bipolar fuzzy graphs provide these advancements.

The integrity parameter in fuzzy graphs deals only with the positive side. Integrity in bipolar fuzzy graphs addresses both positive and negative aspects. This feature allows for the discussion of not only economic growth but also environmental issues. For a single crisp graph, we can draw an infinite number of fuzzy graphs. The bounds for each parameter represent a significant improvement in the field of bipolar fuzzy graphs. The integrity parameter bounds are derived for standard bipolar fuzzy graphs.

1.2. Motivation

Integrity reveals the significance of a set of vertices when they are removed from the graph. In BFGs, it reflects both positive and negative attributes. Similar to the profit and loss network model, the integrity parameter specifies both the minimum gain, even if some nodes are not working, and the maximum loss with a set of vertices that do not contribute. Thus, integrity in BFGs is represented as a 2-tuple value, indicating both positive and negative integrity.

Integrity is the first vulnerability parameter in bipolar fuzzy graphs. Comparison with other parameters is not possible.

1.3. Organization of the Paper

This article begins with Section 1, which outlines the necessity of the parameter and offers a brief literature survey covering fuzzy graph theory, bipolar fuzzy graphs, and integrity parameters. Section 2 provides all the fundamental definitions necessary for understanding both fuzzy graphs and BFGs. Section 3.1 introduces and elucidates the vulnerability parameter, integrity, with an accompanying example. This section also presents the integrity values for some basic and standard BFGs. Section 3.2 delves into the operations of BFGs, discussing the integrity values for union, join, and Cartesian BFGs. Section 3.3 scrutinizes the integrity of BFG values and their significance, using a transport network model as an illustration. The article concludes with a discussion on potential avenues for future research.

2. Preliminaries

A crisp graph ${\Sigma }^{*}=(\Im ,\mathsf{\xi })$ consists of a nonempty set of nodes ℑ and a set of edges $\mathsf{\xi }$. Rosenfeld extended this concept to fuzzy graphs. A fuzzy graph $\Sigma =(\mathsf{\eta },\phi )$ is defined as a pair of functions, where $\mathsf{\eta }:\Im \to [0,1]$ and $\phi :\Im \times \Im \to [0,1]$. For all ${\gimel }_1,{\gimel }_2\in \Im$, we require $\phi ({\gimel }_1,{\gimel }_2)\le \mathsf{\eta }\left({\gimel }_1\right)\wedge \mathsf{\eta }\left({\gimel }_2\right).$

A subgraph of $\Sigma =(\mathsf{\eta },\phi )$ is a fuzzy graph $\eta =(\tau ,\nu )$ if $\tau \left(u\right)\le \mathsf{\eta }\left(u\right)$ and $\nu (u,v)\le \phi (u,v)$ for all $u,v\in \Im$. Furthermore, $\Sigma$ has a spanning subgraph $\Gamma$ if both fuzzy graphs share the same vertex set.

A sequence of distinct nodes ${\gimel }_0,{\gimel }_1,\dots ,{\gimel }_n$ such that $\phi ({\gimel }_i,{\gimel }_{i+1})>0$ for all i is called a path of length n in $\Sigma$. The minimum membership value of the edges along a path determines its strength. A fuzzy graph $\Sigma$ is connected if every pair of vertices is connected by a path. The maximum strength among all paths connecting ${\gimel }_1$ and ${\gimel }_2$ defines their connectedness, denoted by $CON{N}_{\Sigma }({\gimel }_1,{\gimel }_2).$

If, in a fuzzy graph, $\phi ({\gimel }_1,{\gimel }_2)=\mathsf{\eta }\left({\gimel }_1\right)\wedge \mathsf{\eta }\left({\gimel }_2\right)$ for all $({\gimel }_1,{\gimel }_2)\in \mathsf{\xi }$, then $\Sigma$ is called a strong fuzzy graph. A fuzzy graph is complete if for all ${\gimel }_1,{\gimel }_2\in \Im$, $\phi ({\gimel }_1,{\gimel }_2)=\mathsf{\eta }\left({\gimel }_1\right)\wedge \mathsf{\eta }\left({\gimel }_2\right).$

The order and size of a fuzzy graph are defined as $\mathrm{Order}={\displaystyle {\sum }_{{\gimel }_1\in \Im }}\mathsf{\eta }\left({\gimel }_1\right),$$\mathrm{Size}={\sum }_{({\gimel }_1,{\gimel }_2)\in \mathsf{\xi }}\phi ({\gimel }_1,{\gimel }_2).$

The complement of a fuzzy graph $\Sigma =(\mathsf{\eta },\phi )$ is given by $\overline{\Sigma }=(\overline{\mathsf{\eta }},\overline{\phi })$, where $\overline{\mathsf{\eta }}=\mathsf{\eta }$ and $\overline{\phi }({\gimel }_1,{\gimel }_2)=\mathsf{\eta }\left({\gimel }_1\right)\wedge \mathsf{\eta }\left({\gimel }_2\right)-\phi ({\gimel }_1,{\gimel }_2),\forall {\gimel }_1,{\gimel }_2\in \Im .$

Finally, the integrity of a fuzzy graph is defined as ${\displaystyle I(\Sigma )=\underset{S\subseteq \Im }{min}\left\{|S|+m(\Sigma -S)\right\},}$ where $m(\Sigma -S)$ denotes the maximum order of $\Sigma -S$ with respect to the underlying crisp graph ${\Sigma }^{*}$.

To establish the framework for our analysis, we first outline the notation used in this paper. The fundamental concepts of bipolar fuzzy graphs (BFG), vertex and edge membership functions, and integrity measures are consistently represented using the symbols summarized in

Table 1. Summary of key symbols and notations used in this paper.

Symbol	Description
$BFG$	Bipolar Fuzzy Graph
ℑ	Vertex set of the underlying crisp graph
$\mathsf{\xi }$	Edge set of the underlying crisp graph
$\Sigma =(\mathsf{\eta },\phi )$	Fuzzy graph with vertex and edge membership functions
${\Sigma }^{*}$	Underlying crisp graph of $\Sigma$
$\tilde{I}(\Sigma )$	Integrity of a bipolar fuzzy graph
${\tilde{I}}^P$	Positive integrity (based on PVMV)
${\tilde{I}}^N$	Negative integrity (based on NVMV)
NSS	Node strength seguence
PVMV	Positive vertex membership value
NVMV	Negative vertex membership value
$m(\Sigma -S)$	Maximum order of $\Sigma -S$ in PVMV
$M(\Sigma -S)$	Maximum order of $\Sigma -S$ in NVMV
$CON{N}_{\Sigma }({\gimel }_1,{\gimel }_2)$	Connectedness between vertices ${\gimel }_1$ and ${\gimel }_2$

.

Definition 1

([46,47]). Consider the vertex set S to be non-empty. A bipolar fuzzy set $\mathbb{A}$ on S is defined as $\mathbb{A}=\left\{r,{\phi }^P\left(r\right),{\phi }^N\left(r\right)/r\in S\right\}$, where ${\phi }^P\left(r\right):S\to [0,1]$ and ${\phi }^N\left(r\right):S\to [-1,0]$.

Definition 2

([47]). Consider the vertex set S to be non-empty. A bipolar fuzzy relation $\mathbb{A}$ on S is defined as $\mathbb{A}=\left({\phi }_{\mathbb{A}}^P,{\phi }_{\mathbb{A}}^N\right):S\times S\to \left[-1,1\right]\times \left[-1,1\right]$ such that ${\phi }_{\mathbb{A}}^P\left({\gimel }_1,{\gimel }_2\right)\in \left[0,1\right]$ and ${\phi }_{\mathbb{A}}^N\left({\gimel }_1,{\gimel }_2\right)\in \left[-1,0\right]$.

Definition 3

([10]). A BFG with an underlying crisp set ℑ is defined to be a pair $\Sigma =\left(\mathbb{A},\mathbb{B}\right)$, where $\mathbb{A}=\left({\phi }_{\mathbb{A}}^P,{\phi }_{\mathbb{A}}^N\right)$ is a bipolar fuzzy set in ℑ and $\mathbb{B}=\left({\phi }_{\mathbb{B}}^P,{\phi }_{\mathbb{B}}^N\right)$ is a bipolar fuzzy set in $\mathsf{\xi }\subseteq \Im \times \Im$ such that ${\phi }_{\mathbb{B}}^P\left(r,s\right)\le min\left({\phi }_{\mathbb{A}}^P\left(r\right),{\phi }_{\mathbb{A}}^P\left(s\right)\right)$ and ${\phi }_{\mathbb{B}}^N\left(r,s\right)\ge max\left({\phi }_{\mathbb{A}}^N\left(r\right),{\phi }_{\mathbb{A}}^N\left(s\right)\right)$ for all $(r,s)\in \mathsf{\xi }$, where $\mathbb{A}$ is the bipolar fuzzy vertex set of ℑ, and $\mathbb{B}$ is the bipolar fuzzy edge set of $\mathsf{\xi }$. Thus $\Sigma =\left(\mathbb{A},\mathbb{B}\right)$ is a BFG of ${\Sigma }^{*}=\left(\Im ,\Sigma \right)$ if ${\phi }_{\mathbb{B}}^P\left(r,s\right)\le min\left({\phi }_{\mathbb{A}}^P\left(r\right),{\phi }_{\mathbb{A}}^P\left(s\right)\right)$ and ${\phi }_{\mathbb{B}}^N\left(r,s\right)\ge max\left({\phi }_{\mathbb{A}}^N\left(r\right),{\phi }_{\mathbb{A}}^N\left(s\right)\right)$ for all $(r,s)\in \mathsf{\xi }$.

Definition 4

([11]). A BFG $\Sigma =(\mathbb{A},\mathbb{B})$ is complete if ${\phi }_{\mathbb{B}}^P({\gimel }_1,{\gimel }_2)={\phi }_{\mathbb{A}}^P\left({\gimel }_1\right)\wedge {\phi }_{\mathbb{A}}^P\left({\gimel }_2\right)$, ${\phi }_{\mathbb{B}}^N({\gimel }_1,{\gimel }_2)={\phi }_{\mathbb{A}}^N\left({\gimel }_1\right)\vee {\phi }_{\mathbb{A}}^N\left({\gimel }_2\right),\forall {\gimel }_1,{\gimel }_2\in \Im$.

Definition 5

([10]). A BFG $\Sigma =(\mathbb{A},\mathbb{B})$ is strong if ${\phi }_{\mathbb{B}}^P({\gimel }_1,{\gimel }_2)={\phi }_{\mathbb{A}}^P\left({\gimel }_1\right)\wedge {\phi }_{\mathbb{A}}^P\left({\gimel }_2\right)$, ${\phi }_{\mathbb{B}}^N({\gimel }_1,{\gimel }_2)={\phi }_{\mathbb{A}}^N\left({\gimel }_1\right)\vee {\phi }_{\mathbb{A}}^N\left({\gimel }_2\right),\forall ({\gimel }_1,{\gimel }_2)\in \Sigma$.

Definition 6

([48]). Let $\Sigma =(\mathbb{A},\mathbb{B})$ be a BFG then the complement is defined as $\overline{\Sigma }=(\overline{\mathbb{A}},\overline{\mathbb{B}})$. Where $\overline{\mathbb{A}}=\left[\overline{{\phi }_{\mathbb{A}}^P},\overline{{\phi }_{\mathbb{A}}^N}\right]$ and $\overline{\mathbb{B}}=\left[\overline{{\phi }_{\mathbb{B}}^P},\overline{{\phi }_{\mathbb{B}}^N}\right]$ are defined as $\overline{{\phi }_{\mathbb{B}}^P}({\gimel }_1,{\gimel }_2)={\phi }_{\mathbb{A}}^P\left({\gimel }_1\right)\wedge {\phi }_{\mathbb{A}}^P\left({\gimel }_2\right)-{\phi }_{\mathbb{B}}^P({\gimel }_1,{\gimel }_2)$ and $\overline{{\phi }_{\mathbb{B}}^N}({\gimel }_1,{\gimel }_2)={\phi }_{\mathbb{A}}^N\left({\gimel }_1\right)\vee {\phi }_{\mathbb{A}}^N\left({\gimel }_2\right)-{\phi }_{\mathbb{B}}^N({\gimel }_1,{\gimel }_2)$.

Definition 7

([48]). Let $\Sigma =\left(\mathbb{A},\mathbb{B}\right)$ be a strong BFG, then the complement is defined as $\overline{\Sigma }=(\overline{\mathbb{A}},\overline{\mathbb{B}})$, where $\overline{\mathbb{A}}=\left[\overline{{\phi }_{\mathbb{A}}^P},\overline{{\phi }_{\mathbb{A}}^N}\right]$ and $\overline{\mathbb{B}}=\left[\overline{{\phi }_{\mathbb{B}}^P},\overline{{\phi }_{\mathbb{B}}^N}\right]$ are defined as

$\overline{{\phi }_{\mathbb{B}}^P}({\gimel }_1,{\gimel }_2)=\left\{\begin{array}{cc}0\hfill & {\phi }_{\mathbb{B}}^P({\gimel }_1,{\gimel }_2)>0\hfill \\ {\phi }_{\mathbb{A}}^P\left({\gimel }_1\right)\wedge {\phi }_{\mathbb{B}}^P\left({\gimel }_2\right)\hfill & {\phi }_{\mathbb{B}}^P({\gimel }_1,{\gimel }_2)=0\hfill \end{array}\right.$ and

$\overline{{\phi }_{\mathbb{B}}^N}({\gimel }_1,{\gimel }_2)=\left\{\begin{array}{cc}0\hfill & {\phi }_{\mathbb{B}}^N({\gimel }_1,{\gimel }_2)>0\hfill \\ {\phi }_{\mathbb{A}}^N\left({\gimel }_1\right)\vee {\phi }_{\mathbb{B}}^N\left({\gimel }_2\right)\hfill & {\phi }_{\mathbb{B}}^N({\gimel }_1,{\gimel }_2)=0\hfill \end{array}\right.$.

Definition 8

([10]). Let ${\mathbb{A}}_1=\left({\phi }_{{\mathbb{A}}_1}^P,{\phi }_{{\mathbb{A}}_1}^N\right)$ and ${\mathbb{A}}_2=\left({\phi }_{{\mathbb{A}}_2}^P,{\phi }_{{\mathbb{A}}_2}^N\right)$ be bipolar fuzzy subsets on the vertex sets ${\Im }_1$ and ${\Im }_2$, respectively. Let ${\mathbb{B}}_1=\left({\phi }_{{\mathbb{B}}_1}^P,{\phi }_{{\mathbb{B}}_1}^N\right)$ and ${\mathbb{B}}_2=\left({\phi }_{{\mathbb{B}}_2}^P,{\phi }_{{\mathbb{B}}_2}^N\right)$ be bipolar fuzzy subsets on the edge sets ${\mathsf{\xi }}_1$ and ${\mathsf{\xi }}_2$, respectively. Then the union of ${\Sigma }_1=({\mathbb{A}}_1,{\mathbb{B}}_1)$ and ${\Sigma }_2=({\mathbb{A}}_2,{\mathbb{B}}_2)$ is defined as ${\Sigma }_1\cup {\Sigma }_2=\left({\mathbb{A}}_1\cup {\mathbb{A}}_2,{\mathbb{B}}_1\cup {\mathbb{B}}_2\right),$ where

(i) 

$\left({\phi }_{{\mathbb{A}}_1}^P\cup {\phi }_{{\mathbb{A}}_2}^P\right)(\gimel )=\left\{\begin{array}{cc}{\phi }_{{\mathbb{A}}_1}^P(\gimel ),\hfill & \gimel \in {\Im }_1-{\Im }_2,\hfill \\ {\phi }_{{\mathbb{A}}_2}^P(\gimel ),\hfill & \gimel \in {\Im }_2-{\Im }_1,\hfill \\ {\phi }_{{\mathbb{A}}_1}^P(\gimel )\vee {\phi }_{{\mathbb{A}}_2}^P(\gimel ),\hfill & \gimel \in {\Im }_1\cap {\Im }_2,\hfill \end{array}\right.$

(ii) 

$\left({\phi }_{{\mathbb{A}}_1}^N\cap {\phi }_{{\mathbb{A}}_2}^N\right)(\gimel )=\left\{\begin{array}{cc}{\phi }_{{\mathbb{A}}_1}^N(\gimel ),\hfill & \gimel \in {\Im }_1-{\Im }_2,\hfill \\ {\phi }_{{\mathbb{A}}_2}^N(\gimel ),\hfill & \gimel \in {\Im }_2-{\Im }_1,\hfill \\ {\phi }_{{\mathbb{A}}_1}^N(\gimel )\wedge {\phi }_{{\mathbb{A}}_2}^N(\gimel ),\hfill & \gimel \in {\Im }_1\cap {\Im }_2,\hfill \end{array}\right.$

(iii) 

$\left({\phi }_{{\mathbb{B}}_1}^P\cup {\phi }_{{\mathbb{B}}_2}^P\right)({\gimel }_1,{\gimel }_2)=\left\{\begin{array}{cc}{\phi }_{{\mathbb{B}}_1}^P({\gimel }_1,{\gimel }_2),\hfill & ({\gimel }_1,{\gimel }_2)\in {\mathsf{\xi }}_1-{\mathsf{\xi }}_2,\hfill \\ {\phi }_{{\mathbb{B}}_2}^P({\gimel }_1,{\gimel }_2),\hfill & ({\gimel }_1,{\gimel }_2)\in {\mathsf{\xi }}_2-{\mathsf{\xi }}_1,\hfill \\ {\phi }_{{\mathbb{B}}_1}^P({\gimel }_1,{\gimel }_2)\vee {\phi }_{{\mathbb{B}}_2}^P({\gimel }_1,{\gimel }_2),\hfill & ({\gimel }_1,{\gimel }_2)\in {\mathsf{\xi }}_1\cap {\mathsf{\xi }}_2,\hfill \end{array}\right.$

(iv) 

$\left({\phi }_{{\mathbb{B}}_1}^N\cap {\phi }_{{\mathbb{B}}_2}^N\right)({\gimel }_1,{\gimel }_2)=\left\{\begin{array}{cc}{\phi }_{{\mathbb{B}}_1}^N({\gimel }_1,{\gimel }_2),\hfill & ({\gimel }_1,{\gimel }_2)\in {\mathsf{\xi }}_1-{\mathsf{\xi }}_2,\hfill \\ {\phi }_{{\mathbb{B}}_2}^N({\gimel }_1,{\gimel }_2),\hfill & ({\gimel }_1,{\gimel }_2)\in {\mathsf{\xi }}_2-{\mathsf{\xi }}_1,\hfill \\ {\phi }_{{\mathbb{B}}_1}^N({\gimel }_1,{\gimel }_2)\wedge {\phi }_{{\mathbb{B}}_2}^N({\gimel }_1,{\gimel }_2),\hfill & ({\gimel }_1,{\gimel }_2)\in {\mathsf{\xi }}_1\cap {\mathsf{\xi }}_2.\hfill \end{array}\right.$

Definition 9

([10]). Let ${\mathbb{A}}_1=({\phi }_{{\mathbb{A}}_1}^P,{\phi }_{{\mathbb{A}}_1}^N)$ and ${\mathbb{A}}_2=({\phi }_{{\mathbb{A}}_2}^P,{\phi }_{{\mathbb{A}}_2}^N)$ be bipolar fuzzy subsets on the vertex sets ${\Im }_1$ and ${\Im }_2$, respectively. Let ${\mathbb{B}}_1=({\phi }_{{\mathbb{B}}_1}^P,{\phi }_{{\mathbb{B}}_1}^N)$ and ${\mathbb{B}}_2=({\phi }_{{\mathbb{B}}_2}^P,{\phi }_{{\mathbb{B}}_2}^N)$ be bipolar fuzzy subsets on the edge sets ${\mathsf{\xi }}_1$ and ${\mathsf{\xi }}_2$, respectively. Then the join of ${\Sigma }_1=({\mathbb{A}}_1,{\mathbb{B}}_1)$ and ${\Sigma }_2=({\mathbb{A}}_2,{\mathbb{B}}_2)$ is ${\Sigma }_1+{\Sigma }_2=({\mathbb{A}}_1+{\mathbb{A}}_2,{\mathbb{B}}_1+{\mathbb{B}}_2),$ where

(i) 

For each $\gimel \in {\Im }_1\cup {\Im }_2$,

$$({\phi }_{{\mathbb{A}}_1}^P+{\phi }_{{\mathbb{A}}_2}^P)(\gimel )=({\phi }_{{\mathbb{A}}_1}^P\cup {\phi }_{{\mathbb{A}}_2}^P)(\gimel ),({\phi }_{{\mathbb{A}}_1}^N+{\phi }_{{\mathbb{A}}_2}^N)(\gimel )=({\phi }_{{\mathbb{A}}_1}^N\cap {\phi }_{{\mathbb{A}}_2}^N)(\gimel ).$$

(ii) 

For each edge $({\gimel }_1,{\gimel }_2)\in {\mathsf{\xi }}_1\cap {\mathsf{\xi }}_2$,

$$({\phi }_{{\mathbb{B}}_1}^P+{\phi }_{{\mathbb{B}}_2}^P)({\gimel }_1,{\gimel }_2)=({\phi }_{{\mathbb{B}}_1}^P\cup {\phi }_{{\mathbb{B}}_2}^P)({\gimel }_1,{\gimel }_2),({\phi }_{{\mathbb{B}}_1}^N+{\phi }_{{\mathbb{B}}_2}^N)({\gimel }_1,{\gimel }_2)=({\phi }_{{\mathbb{B}}_1}^N\cap {\phi }_{{\mathbb{B}}_2}^N)({\gimel }_1,{\gimel }_2).$$

(iii) 

For each new edge $uv\in {\mathsf{\xi }}^{\prime }$, where ${\mathsf{\xi }}^{\prime }$ is the set of edges joining a vertex of ${\Im }_1$ with a vertex of ${\Im }_2$,

$$({\phi }_{{\mathbb{B}}_1}^P+{\phi }_{{\mathbb{B}}_2}^P)({\gimel }_1,{\gimel }_2)={\phi }_{{\mathbb{A}}_1}^P\left({\gimel }_1\right)\vee {\phi }_{{\mathbb{A}}_2}^P\left({\gimel }_2\right),({\phi }_{{\mathbb{B}}_1}^N+{\phi }_{{\mathbb{B}}_2}^N)({\gimel }_1,{\gimel }_2)={\phi }_{{\mathbb{A}}_1}^N\left({\gimel }_1\right)\wedge {\phi }_{{\mathbb{A}}_2}^N\left({\gimel }_2\right).$$

Definition 10

([10]). Let ${\mathbb{A}}_1=({\phi }_{{\mathbb{A}}_1}^P,{\phi }_{{\mathbb{A}}_1}^N)$ and ${\mathbb{A}}_2=({\phi }_{{\mathbb{A}}_2}^P,{\phi }_{{\mathbb{A}}_2}^N)$ be a bipolar fuzzy subsets of ${\Im }_1$ and ${\Im }_2$ and let ${\mathbb{B}}_1=({\phi }_{{\mathbb{B}}_1}^P,{\phi }_{{\mathbb{B}}_1}^N)$ and ${\mathbb{B}}_2=({\phi }_{{\mathbb{B}}_2}^P,{\phi }_{{\mathbb{B}}_2}^N)$ be bipolar fuzzy subsets of ${\mathsf{\xi }}_1$ and ${\mathsf{\xi }}_2$, respectively. Then the cartesian BFG of ${\Sigma }_1$ and ${\Sigma }_2$ is ${\Sigma }_1\times {\Sigma }_2=({\mathbb{A}}_1\times {\mathbb{A}}_2,{\mathbb{B}}_1\times {\mathbb{B}}_2)$ is defined as follows.

(i) 

$\left({\phi }_{{\mathbb{A}}_1}^P\times {\phi }_{{\mathbb{A}}_2}^P\right)(r_1,r_2)={\phi }_{{\mathbb{A}}_1}^P\left(r_1\right)\wedge {\phi }_{{\mathbb{A}}_2}^P\left(r_2\right)$

$\left({\phi }_{{\mathbb{A}}_1}^N\times {\phi }_{{\mathbb{A}}_2}^N\right)(r_1,r_2)={\phi }_{A_1}^N\left(r_1\right)\vee {\phi }_{{\mathbb{A}}_2}^N\left(r_2\right),\forall r_1,r_2\in \Im$,

(ii) 

$\left({\phi }_{{\mathbb{B}}_1}^P\times {\phi }_{{\mathbb{B}}_2}^P\right)\left((r,r_2)(r,s_2)\right)={\phi }_{{\mathbb{A}}_1}^P\left(r\right)\wedge {\phi }_{{\mathbb{B}}_2}^P(r_2,s_2)$

$\left({\phi }_{{\mathbb{B}}_1}^N\times {\phi }_{{\mathbb{B}}_2}^N\right)\left((r,r_2)(r,s_2)\right)={\phi }_{{\mathbb{A}}_1}^N\left(r\right)\vee {\phi }_{{\mathbb{B}}_2}^N(r_2,t_2),\forall r\in {\Im }_1,\forall (r_2,s_2)\in {\mathsf{\xi }}_2$

(iii) 

$\left({\phi }_{{\mathbb{B}}_1}^P\times {\phi }_{{\mathbb{B}}_2}^P\right)\left((r_1,t)(s_1,t)\right)={\phi }_{{\mathbb{B}}_1}^P(r_1,s_1)\wedge {\phi }_{{\mathbb{A}}_2}^P\left(t\right)$

$\left({\phi }_{{\mathbb{B}}_1}^N\times {\phi }_{{\mathbb{B}}_2}^N\right)\left((r_1,t)(s_1,t)\right)={\phi }_{{\mathbb{B}}_1}^N(r_1,s_1)\vee {\phi }_{{\mathbb{A}}_2}^N\left(t\right),\forall t\in {\Im }_2,\forall (r_1,s_1)\in {\mathsf{\xi }}_1$.

Definition 11

([27]). Let $\Sigma =(\mathbb{A},\mathbb{B})$ be a Bipolar Fuzzy Graph (BFG) with an underlying crisp graph ${\Sigma }^{*}=(\Im ,\mathsf{\xi })$, where ℑ is the vertex set and $\mathsf{\xi }$ is the edge set. For each vertex $\gimel \in \Im$, the positive and negative membership functions of $\mathbb{A}$ are denoted by ${\phi }_{\mathbb{A}}^P(\gimel )$ and ${\phi }_{\mathbb{A}}^N(\gimel )$, respectively. For each edge $({\gimel }_1,{\gimel }_2)\in \mathsf{\xi }$, the positive and negative membership functions of $\mathbb{B}$ are denoted by ${\phi }_{\mathbb{B}}^P({\gimel }_1,{\gimel }_2)$ and ${\phi }_{\mathbb{B}}^N({\gimel }_1,{\gimel }_2)$, respectively. Then:

1. 

The order of Σ is defined as

$$O(\Sigma )=\left(\sum _{\gimel \in \Im }{\phi }_{\mathbb{A}}^P(\gimel ),\sum _{\gimel \in \Im }{\phi }_{\mathbb{A}}^N(\gimel )\right).$$

2. 

The size of Σ is defined as

$$S(\Sigma )=\left(\sum _{\begin{array}{c}{\gimel }_1,{\gimel }_2\in \Im \\ {\gimel }_1\ne {\gimel }_2\end{array}}{\phi }_{\mathbb{B}}^P({\gimel }_1,{\gimel }_2),\sum _{\begin{array}{c}{\gimel }_1,{\gimel }_2\in \Im \\ {\gimel }_1\ne {\gimel }_2\end{array}}{\phi }_{\mathbb{B}}^N({\gimel }_1,{\gimel }_2)\right).$$

3. Results

3.1. Integrity of Bipolar Fuzzy Graph

In this section, the integrity parameter is defined and explained with an example.

Definition 12.

Let Σ be a BFG with an underlying crisp graph ${\Sigma }^{*}$ having a vertex set ℑ and an edge set $\mathsf{\xi }\subseteq \Im \times \Im$. The integrity of Σ is represented as a tuple value, denoted by $\tilde{I}(\Sigma )=({\tilde{I}}^P,{\tilde{I}}^N)$, where ${\tilde{I}}^P$ denotes the positive integrity, and ${\tilde{I}}^N$ denotes the negative integrity of Σ.

The positive integrity of a BFG is defined as ${\tilde{I}}^P=min\left\{|S_1| + m(\Sigma -S_1)/S_1\subset \Im \right\}$, and the negative integrity is defined as ${\tilde{I}}^N=max\left\{|S_2| + M(\Sigma -S_2)/S_2\subset \Im \right\}$, where $|S_1|, |S_2|$ represent the vertex cardinality of vertex subsets $S_1$ and $S_2$, respectively. Furthermore, $m(\Sigma -S_1)$ denotes the maximum order of the connected component after removing $S_1$, and $M(\Sigma -S_2)$ denotes the maximum order of the connected component after removing $S_2$.

Definition 13.

The sets $S_1$ and $S_2$ are referred to as the positive and negative integrity sets of Σ if ${\tilde{I}}^P=|S_1|+m(\Sigma -S_1)$ and ${\tilde{I}}^N=|S_2|+M(\Sigma -S_2)$.

Example 1.

Consider a BFG $\Sigma =\left(\Im ,\mathsf{\xi }\right)$ such that $\Im =\left\{{\gimel }_1,{\gimel }_2,{\gimel }_3,{\gimel }_4\right\}$, $\mathsf{\xi }=\{{\gimel }_1{\gimel }_2,{\gimel }_2{\gimel }_3,{\gimel }_3{\gimel }_4,{\gimel }_4{\gimel }_1,$ ${\gimel }_2{\gimel }_4\}$. Let $\mathbb{A}=\left({\phi }_{\mathbb{A}}^P,{\Sigma }_{\mathbb{A}}^N\right)$ be a bipolar fuzzy subset of $\mathsf{\xi }\subseteq \Im \times \Im$ as shown in Figure 1. The integrity value is calculated as shown in

Table 2. Integrity of BFGs.

S	${|\mathit{S}|}^{\mathit{P}}$	$\mathit{m}{\left(\mathbf{\Sigma }-\mathit{S}\right)}^{\mathit{P}}$	$\mathit{\alpha }$	${|\mathit{S}|}^{\mathit{N}}$	$\mathit{M}{\left(\mathbf{\Sigma }-\mathit{S}\right)}^{\mathit{N}}$	$\mathit{\beta }$
$\left\{\varphi \right\}$	0	2.4	2.4	0	−1.3	−1.3
$\left\{{\gimel }_1\right\}$	0.7	1.7	2.4	−0.8	−0.5	−1.3
$\left\{{\gimel }_2\right\}$	0.6	1.8	2.4	−0.1	−1.2	−1.3
$\left\{{\gimel }_3\right\}$	0.9	1.5	2.4	−0.3	−1	−1.3
$\left\{{\gimel }_4\right\}$	0.2	2.2	2.4	−0.1	−1.2	−1.3
$\left\{{\gimel }_1,{\gimel }_2\right\}$	1.3	1.1	2.4	−0.9	−0.4	−1.3
$\left\{{\gimel }_1,{\gimel }_3\right\}$	1.6	0.8	2.4	−1.1	−0.2	−1.3
$\left\{{\gimel }_1,{\gimel }_4\right\}$	0.9	1.5	2.4	−0.9	−0.4	−1.3
$\left\{{\gimel }_2,{\gimel }_3\right\}$	1.5	0.9	2.4	−0.4	−0.9	−1.3
$\left\{{\gimel }_2,{\gimel }_4\right\}$	0.8	0.9	1.7	−0.2	−0.8	−1
$\left\{{\gimel }_1,{\gimel }_2,{\gimel }_3\right\}$	2.2	0.2	2.4	−1.2	−0.1	−1.3
$\left\{{\gimel }_1,{\gimel }_2,{\gimel }_4\right\}$	1.5	0.9	2.4	−1	−0.3	−1.3
$\left\{{\gimel }_2,{\gimel }_3,{\gimel }_4\right\}$	1.7	0.7	2.4	−0.5	−0.8	−1.3
$\left\{{\gimel }_1,{\gimel }_2,{\gimel }_3,{\gimel }_4\right\}$	2.4	0	2.4	−1.3	0	−1.3

Where $\alpha ={|S|}^P+m{\left(\Sigma -S\right)}^P$ and $\beta ={|S|}^N+M{\left(\Sigma -S\right)}^N$. From the table, the integrity is $\left(1.7,-1\right)$.

.

The computation of integrity in a bipolar fuzzy graph involves systematically removing subsets of vertices and evaluating the resulting connected components. The step-by-step procedure used in this paper is outlined in Algorithm 1.

Algorithm 1 Computation of Integrity of a BFG

Require: A BFG $\Sigma$ with underlying crisp graph ${\Sigma }^{*}=(\Im ,\mathsf{\xi })$

Ensure: Positive integrity ${\tilde{I}}^P$, Negative integrity ${\tilde{I}}^N$

1: Convert $\Sigma$ into its underlying crisp graph ${\Sigma }^{*}$ 2: Initialize ${\tilde{I}}^P\leftarrow +\infty$, ${\tilde{I}}^N\leftarrow -\infty$ 3: for all vertex subset S ⊆ ℑ do 4:  Remove vertices in S from ${\Sigma }^{*}$ to obtain ${\Sigma }^{*}-S$ 5:  Compute $C\leftarrow$ sizes of connected components of ${\Sigma }^{*}-S$ 6:  $m\leftarrow max\left(C\right)$ {maximum component size after removal} 7:  $value\leftarrow |S|+m$ 8:  if $value<{\tilde{I}}^P$then 9:       ${\tilde{I}}^P\leftarrow value$ 10:     $S_1^{*}\leftarrow S$ {store subset attaining positive integrity} 11:  end if 12:  if $value>{\tilde{I}}^N$then 13:     ${\tilde{I}}^N\leftarrow value$ 14:     $S_2^{*}\leftarrow S$ {store subset attaining negative integrity} 15:  end if 16: end for 17: return $\tilde{I}(\Sigma )=({\tilde{I}}^P,{\tilde{I}}^N)$

The evaluation of connected components, which can be completed in $O(n+m)$ time for a graph ${\Sigma }^{*}=(\Im ,\mathsf{\xi })$ with $n=|\Im |$ and $m=|\mathsf{\xi }|$, is the main source of computational complexity in the suggested framework. All $2^n$ vertex subsets are enumerated by the brute-force (exact) method, which results in a total time complexity of $O\left(2^n(n+m)\right)$ and space complexity of $O(n+m)$.

Theorem 1.

Let Σ be a BFG with the underlying crisp graph ${\Sigma }^{*}$ with n vertices. If the PVMV of Σ is ${\gimel }_1$ and NVMV of Σ is ${\gimel }_2$ then the integrity of Σ is

1. 

Null BFG is $\tilde{I}(\Sigma )=({\gimel }_1,{\gimel }_2)$.

2. 

Complete BFG $K_n$ is $\tilde{I}(\Sigma )=(n{\gimel }_1,n{\gimel }_2)$.

3. 

Star BFG $K_{1,n}$ is $\tilde{I}(\Sigma )=(2{\gimel }_1,2{\gimel }_2)$.

4. 

Path BFG of length n is

$\tilde{I}(\Sigma )=\left(\left(\lceil 2\sqrt{n+1}\rceil -2\right){\gimel }_1,\left(\lceil 2\sqrt{n+1}\rceil -2\right){\gimel }_2\right)$.

5. 

Cycle BFG $C_n$ is

$\tilde{I}(\Sigma )=\left(\left(\lceil 2\sqrt{n}\rceil -1\right){\gimel }_1,\left(\lceil 2\sqrt{n}\rceil -1\right){\gimel }_2\right)$.

6. 

Complete bipartite BFG $K_{m,n}$ is

$\tilde{I}(\Sigma )=\left(\left(1+min\{m,n\}\right){\gimel }_1,\left(1+min\{m,n\}\right){\gimel }_2\right)$.

Proof. 

1.

Consider a null BFG with constant membership values. Choosing the integrity set as empty implies $m(\Sigma -S_1)={\gimel }_1$ and $M(\Sigma -S_2)={\gimel }_2$. Therefore the integrity of $\Sigma$ becomes $\tilde{I}(\Sigma )=({\gimel }_1,{\gimel }_2)$.

2.

In a complete BFG, every vertex is adjacent to all other remaining vertices. Removing a vertex set with r number of vertices leaves a single connected component with remaining $n-r$ vertices. Then $m(\Sigma -S_1)=(n-r){\gimel }_1$ and $M(\Sigma -S_2)=(n-r){\gimel }_2$ and hence $\tilde{I}(\Sigma )=(n{\gimel }_1,n{\gimel }_2)$.

3.

In a star BFG, $K_{1,n}$, a single vertex is adjacent to all the remaining other n vertices. Choosing that center vertex as integrity set S and removing implies all the other n vertices are isolated with constant vertex membership values. Then $m(\Sigma -S_1)={\gimel }_1$ and $M(\Sigma -S_2)={\gimel }_2$ implies $\tilde{I}(\Sigma )=(2{\gimel }_1,2{\gimel }_2)$.

4.

By Theorem 2.1 in [32], if the membership values are constant, then the integrity set for both the crisp graph and fuzzy graph are same. From the definition of a BFG, it also has the same integrity set while considering constant vertex membership values. Removing a vertex set S with k vertices, at least one component will have $\left({\displaystyle \frac{n-k}{k+1}}\right)$ number of vertices. The function $f\left(k\right)=k{\gimel }_1+\left({\displaystyle \frac{n-k}{k+1}}\right){\gimel }_1$ attains its minimum at $\left(\lceil 2\sqrt{n+1}\rceil -2\right){\gimel }_1$ and its maximum at $\left(\lceil 2\sqrt{n+1}\rceil +2\right){\gimel }_2$. Therefore $\tilde{I}(\Sigma ) = (\left(\lceil 2\sqrt{n+1}\rceil -2\right){\gimel }_1,\left(\lceil 2\sqrt{n+1}\rceil +2\right){\gimel }_2)$.

5.

Removing a vertex from a cycle with n vertices turns into a path of $n-1$ vertices. Including a vertex membership value with the integrity value of a path with $n-1$, the integrity value of the cycle becomes $(\left(\lceil 2\sqrt{n}\rceil -1\right){\gimel }_1,\left(\lceil 2\sqrt{n}\rceil -1\right){\gimel }_2)$.

6.

Without loss of generality, assume $m<n$. Let $u_1,u_2,\dots ,u_m$ and $v_1,v_2,\dots ,v_n$ be the $mn$ vertices in $K_{m,n}$. Then removing m vertices from $K_{m,n}$, there are n isolated vertices each with constant positive membership value ${\gimel }_1$ and negative membership value ${\gimel }_2$. Choosing the integrity set S with $u_1,u_2,\dots ,u_m$ leaves a set of isolated vertices with each membership value $({\gimel }_1,{\gimel }_2)$, which implies $m(\Sigma -S_1)={\gimel }_1$ and $M(\Sigma -S_2)={\gimel }_2$. Thus the integrity value becomes $\left(1+min\{m,n\}\right){\gimel }_1,\left(1+min\{m,n\}\right){\gimel }_2$.

□

Remark 1.

For the above Theorem 1, if the membership values are the same then the integrity for the BFG, fuzzy graph, and crisp graph have same integrity set.

Remark 2.

In a BFG, positive and negative integrity sets need not be same. For example, consider a BFG $\Sigma =\left(\Im ,\mathsf{\xi }\right)$ such that $\Im =\left\{{\gimel }_1,{\gimel }_2,{\gimel }_3,{\gimel }_4\right\}$, $\mathsf{\xi }=\left\{{\gimel }_1{\gimel }_2,{\gimel }_2{\gimel }_3,{\gimel }_3{\gimel }_4,{\gimel }_4{\gimel }_1,{\gimel }_2{\gimel }_4\right\}$. Let $\mathbb{A}=\left({\phi }_{\mathbb{A}}^P,{\phi }_{\mathbb{A}}^N\right)$ be a bipolar fuzzy subset of $\mathsf{\xi }\subseteq \Im \times \Im$ as shown in Figure 2. Then integrity is $\tilde{I}(\Sigma )=(1.4,-0.7)$, as shown in

Table 3. Integrity of BFG.

S	${|\mathit{S}|}^{\mathit{P}}$	$\mathit{m}{\left(\mathbf{\Sigma }-\mathit{S}\right)}^{\mathit{P}}$	$\mathit{\alpha }$	${|\mathit{S}|}^{\mathit{N}}$	$\mathit{M}{\left(\mathbf{\Sigma }-\mathit{S}\right)}^{\mathit{N}}$	$\mathit{\beta }$
$\left\{{\gimel }_1,{\gimel }_3\right\}$	1.4	0.4	1.8	−0.4	−0.3	−0.7
$\left\{{\gimel }_2,{\gimel }_4\right\}$	0.6	0.8	1.4	−0.6	−0.3	−0.9

Where $\alpha ={|S|}^P+m{\left(\Sigma -S\right)}^P$ and $\beta ={|S|}^N+M{\left(\Sigma -S\right)}^N$. From the table, the integrity is $\left(1.4,-0.7\right)$.

.

Definition 14.

Let Σ be a bipolar fuzzy graph (BFG) with n vertices. The NSS of Σ is defined as $\{p_1,p_2,\dots ,p_n\},$ where each $p_i=(q_i,r_i)$, with $q_i\in [0,1]$ (positive membership value) and $r_i\in [-1,0]$ (negative membership value). Here, $p_1=(q_1,r_1)$ denotes the vertex with the minimum membership values in the NSS, and $p_n=(q_n,r_n)$ denotes the vertex with the maximum membership values in the NSS.

Theorem 2.

Let Σ be a bipolar fuzzy path graph whose underlying crisp graph ${\Sigma }^{*}$ is the path $P_l$ on $l\ge 3$ vertices. Suppose the NSS of Σ is $\{p_1^{l-2},p_2,p_3\},p_1^{l-2}=(q_1^{l-2},r_1^{l-2}),p_2=(q_2,r_2),p_3=(q_3,r_3),$ i.e., there are $l-2$ vertices with positive/negative strengths $q_1-r_1$, and two exceptional vertices with strengths $(q_2,r_2)$ and $(q_3,r_3)$. Let $S_1$ and $S_2$ be the positive and negative integrity sets of Σ. Then the integrity $\tilde{I}(\Sigma )=({\tilde{I}}^P,{\tilde{I}}^N)$ is obtained as follows. Then the positive part ${\tilde{I}}^P$ is equal to one of the four quantities (depending on whether $q_2,q_3$ lie in $S_1$):

$${\tilde{I}}^P=\left\{\begin{array}{c}{q}_1\left(\lceil 2\sqrt{l+1}\rceil -2\right)+q_2+q_3-2q_1,\mathit{if}{q}_2,q_3\in S_1\mathit{and}{q}_1\in m\hfill \\ q_1\left(\lceil 2\sqrt{l+1}\rceil -2\right)+q_2-q_1,\mathit{if}{q}_2\in S_1\mathit{and}{q}_1\in m\hfill \\ q_1\left(\lceil 2\sqrt{l+1}\rceil -2\right)+q_3-q_1,\mathit{if}{q}_3\in S_1\mathit{and}{q}_1\in m\hfill \\ q_1\left(\lceil 2\sqrt{l+1}\rceil -2\right),\mathit{if}{q}_2,q_3\notin S_1,m\hfill \end{array}\right.$$

Similarly, the negative part ${\tilde{I}}^N$ equals one of the corresponding four quantities with $q_i$ replaced by $r_i$ and $S_1$ replaced by $S_2$:

$${\tilde{I}}^N=\left\{\begin{array}{c}{r}_1\left(\lceil 2\sqrt{l+1}\rceil -2\right)+r_2+r_3-2r_1,\mathit{if}{r}_2,r_3\in S_2\mathit{and}{r}_1\in M\hfill \\ r_1\left(\lceil 2\sqrt{l+1}\rceil -2\right)+r_2-r_1,\mathit{if}{r}_2\in S_2\mathit{and}{r}_1\in M\hfill \\ r_1\left(\lceil 2\sqrt{l+1}\rceil -2\right)+r_3-r_1,\mathit{if}{r}_3\in S_2\mathit{and}{r}_1\in M\hfill \\ r_1\left(\lceil 2\sqrt{l+1}\rceil -2\right),\mathit{if}{r}_2,r_3\notin S_2,M.\hfill \end{array}\right.$$

Proof. 

Let Σ be a BFG with underlying crisp graph ${\Sigma }^{*}$ being a path $P_l$ with NSS $\left\{p_1^{l-2},p_2,p_3\right\}$ where $p_1^{l-2}=\left(q_1^{l-2},r_1^{l-2}\right),p_2=\left(q_2,r_2\right),p_3=\left(q_3,r_3\right)$. The proof consists of two cases, where the first case deals with positive integrity and the second case discusses negative integrity.

• Case 1 Let $S_1$ be the positive integrity set with x vertices. Then removing $S_1$ from $\Sigma$ leaves $x+1$ or less components and at least one of them must contain ${\displaystyle \frac{l-x}{x+1}}$ vertices. Thus for a positive integrity value, the function $\{|S_1|+{\displaystyle \frac{l-x}{x+1}}\}$ should get a minimum value and for negative integrity, it should get a maximum value. Let ℷ be the highest order component of $\Sigma -S_1$. The cases arise while $q_2$ and $q_3$ lies either in $S_1$ or ℷ or any other component. Here the vertices with membership values $q_2$ and $q_3$ are the deciding factor in finding the integrity value.
  • Case 1.1 Then $x-2$ vertices have node strength $q_1$, one vertex with node strength $q_2$ and another vertex with strength $q_3$ and ℷ have $\left({\displaystyle \frac{l-x}{x+1}}\right)$ node strength $q_1$. Or, if $q_1,q_2\in S_1$, then $x-1$ vertices have node strength $q_1$, and one vertex with node strength $q_2$ and ℷ have $\left({\displaystyle \frac{l-x}{x+1}}\right)-1$ node strength $q_1$ and one node with $q_3$. Or, if $q_1,q_3\in S_1$, then $x-1$ vertices have node strength $q_1$, and one vertex with node strength $q_3$ and ℷ have $\left({\displaystyle \frac{l-x}{x+1}}\right)-1$ node strength $q_1$ and one node with $q_2$. Or, if $q_1\in S_1$, then x vertices have node strength $q_1$ and ℷ have $\left({\displaystyle \frac{l-x}{x+1}}\right)-2$ node strength $q_1$, one node with $q_2$ and one vertex with node strength $q_3$.
    Therefore $|S_1| = \left(x-2\right)q_1+q_2+q_3$ and $m\left(\Sigma -S_1\right)=\left({\displaystyle \frac{l-x}{x+1}}\right)q_1$

    $$\begin{array}{cc}\hfill I^P& =min\left\{\left(x-2\right)q_1+q_2+p_3+\left({\displaystyle \frac{l-x}{x+1}}\right)q_1\right\}\hfill \\ & =q_1\left(\lceil 2\sqrt{l+1}\rceil -2\right)+q_2+q_3-2q_1\hfill \end{array}$$
  • Case 1.2 In $q_1,q_2\in S_1$, then $x-1$ vertices have node strength $q_1$ and one vertex with strength $q_2$ and ℷ have ${\displaystyle \frac{l-x}{x+1}}$ node strength $q_1$. Therefore $|S_1| = \left(x-1\right)q_1+q_2$ and $m\left(\Sigma -S_1\right)=\left({\displaystyle \frac{l-x}{x+1}}\right)q_1$

    $$\begin{array}{cc}\hfill I^P& =min\left\{\left(x-1\right)q_1+q_2+\left({\displaystyle \frac{l-x}{x+1}}\right)q_1\right\}\hfill \\ & =q_1\left(\lceil 2\sqrt{l+1}\rceil -2\right)+q_2-q_1\hfill \end{array}$$
  • Case 1.3 If $q_1,q_3\in S_1$, then $x-1$ vertices have node strengths of $q_1$ and one vertex with strength $q_3$ and ℷ have ${\displaystyle \frac{l-x}{x+1}}$ node strength $q_1$. Therefore $|S_1| = \left(x-1\right)q_1+q_3$ and $m\left(\Sigma -S_1\right)=\left({\displaystyle \frac{l-x}{x+1}}\right)q_1$

    $$\begin{array}{cc}\hfill I^P& =min\left\{\left(x-1\right)q_1+q_3+\left({\displaystyle \frac{l-x}{x+1}}\right)q_1\right\}\hfill \\ & =q_1\left(\lceil 2\sqrt{l+1}\rceil -2\right)+q_3-q_1\hfill \end{array}$$
  • Case 1.4 If $q_2,q_3\notin S_1,q_1\in S_1$, then x vertices each have node strength $q_1$ and ℷ have $\left({\displaystyle \frac{l-x}{x+1}}\right)$ node strength $q_1$. Therefore $|S_1| = x{p}_1$ and $m\left(\Sigma -S_1\right)=\left({\displaystyle \frac{l-x}{x+1}}\right)q_1$

    $$\begin{array}{cc}\hfill I^P& =min\left\{x{q}_1+\left({\displaystyle \frac{l-x}{x+1}}\right)q_1\right\}\hfill \\ & =q_1\left(\lceil 2\sqrt{l+1}\rceil -2\right)\hfill \end{array}$$
    The proof for ${\tilde{I}}^N$ is identical after replacing $q_i$ by $r_i$ and $S_1$ by $S_2$.

□

Example 2.

Consider a BFG $\Sigma =\left(\Im ,\mathsf{\xi }\right)$ such that $\Im =\left\{{\gimel }_1,{\gimel }_2,{\gimel }_3,{\gimel }_4,{\gimel }_5\right\}$, $\mathsf{\xi }=\{{\gimel }_1{\gimel }_2,{\gimel }_2{\gimel }_3,{\gimel }_3{\gimel }_4,{\gimel }_4{\gimel }_5\}$. Let $\mathbb{A}=\left({\phi }_{\mathbb{A}}^P,{\phi }_{\mathbb{A}}^N\right)$ be a bipolar fuzzy subset of $\mathsf{\xi }\subseteq \Im \times \Im$ as shown in Figure 3. The integrity of the path BFG is $(0.5,-0.6)$.

Theorem 3.

A bipolar fuzzy null graph integrity is defined as $\tilde{I}(\Sigma )=$ (maximum of the PVMV, minimum of NVMV).

Proof. 

Every vertex in the given graph is an isolated vertex because it is a null bipolar graph. Let S be the empty set, then $\Sigma -S$ remains the same null BFG. For PVMV, $m\left(\Sigma -S\right)$ is maximum. Therefore the integrity is $\tilde{I}(\Sigma )=({\tilde{I}}^P,{\tilde{I}}^N)$, where ${\tilde{I}}^P=$ maximum of PVMV and for NVMV, $M\left(\Sigma -S\right)$ is minimum. Therefore the integrity is ${\tilde{I}}^N=$ minimum of NVMV.  □

Theorem 4.

In the integrity of a complete BFG is equal to $\tilde{I}(\Sigma )=$ (Order of the PVMV, Order of the NVMV).

Proof. 

G is a complete BFG, therefore between every pair of vertex there is an edge. If any vertex is removed from $\Sigma$ then the remaining vertex will be a connected graph. For PVMV $m\left(\Sigma -S\right)$ is sum of the PVMV, and for NVMV $m\left(\Sigma -S\right)$ is sum of NVMV. Integrity of a complete BFG is $\tilde{I}(\Sigma )=({\tilde{I}}^P,{\tilde{I}}^N)$, where ${\tilde{I}}^P = |S|+m\left(\Sigma -S\right)$ = Order of the PVMV, ${\tilde{I}}^N=|S|+M\left(\Sigma -S\right)$ = Order of the NVMV.  □

Corollary 1.

Assume Σ is a complete BFG with n vertices and an NSS $\left\{p_1^{n-2},p_2,p_3\right\}$; $n\ge 2$, where $p_1^{n-2}=(q_1^{n-2},r_1^{n-2}),p_2=(q_2,r_2),p_3=(q_3,r_3)$. Then integrity is $\tilde{I}(\Sigma )=\left((n-2)q_1+q_2+q_3,(n-2)r_1+r_2+r_3\right)$.

Theorem 5.

The integrity of a bipolar star graph $K_{1,n}$ is $\tilde{I}\left(K_{1,n}\right)=(max\left(\mathsf{\eta }\left({\gimel }_1\right)+\mathsf{\eta }\left({\gimel }_2\right)\right),min\left(\mathsf{\eta }\left({\gimel }_1\right)+\mathsf{\eta }\left({\gimel }_2\right)\right)$, where ${\gimel }_1$ is the centre vertex of the star graph and vertex ${\gimel }_2$ is adjacent to ${\gimel }_1$.

Proof. 

The centre vertex of the star bipolar graph is connected to every other vertex in $K_{1,n}$. Let S be the centre vertex. When it is removed then every other vertex gets disconnected. For PVMV $m\left(K_{1,n}-S\right)$ is the maximum value of the PVMV and for NVMV $M\left(\Sigma -S\right)$ is the minimum of NVMV. Therefore integrity of a bipolar star graph $K_{1,n}$ is equal to $\tilde{I}\left(K_{1,n}\right)=({\tilde{I}}^P,{\tilde{I}}^N)$, where ${\tilde{I}}^P=max\left(\mathsf{\eta }\left({\gimel }_1\right)+\mathsf{\eta }\left({\gimel }_2\right)\right),{\tilde{I}}^N=min\left(\mathsf{\eta }\left({\gimel }_1\right)+\mathsf{\eta }\left({\gimel }_2\right)\right)$.  □

Example 3.

Consider a BFG $K_{1,n}=\left(\Im ,\mathsf{\xi }\right)$ such that $\Im =\left\{{\gimel }_1,{\gimel }_2,{\gimel }_3,{\gimel }_4,{\gimel }_5,{\gimel }_6\right\}$, $\mathsf{\xi }=\left\{{\gimel }_1{\gimel }_2,{\gimel }_1{\gimel }_3,{\gimel }_1{\gimel }_4,{\gimel }_1{\gimel }_5,{\gimel }_1{\gimel }_6\right\}$. Let $\mathbb{A}=\left({\phi }_{\mathbb{A}}^P,{\phi }_{\mathbb{A}}^N\right)$ be a bipolar fuzzy subset of $\mathsf{\xi }\subseteq \Im \times \Im$ as shown in Figure 4. The integrity of the star BFG is $(1.1,-0.9)$.

To highlight the distinction between classical and fuzzy models, we compare the integrity measures of crisp graphs, fuzzy graphs, and bipolar fuzzy graphs. The summary of these comparisons for standard graphs such as null, complete, and star graphs is presented in

Table 4. Comparison of integrity of crisp graph, fuzzy graph, and BFG.

Standard Graph	Integrity of Crisp Graph	Integrity of Fuzzy Graph	Integrity of Bipolar Fuzzy Graph
Null Graph $K_n$	1	Maximum vertex membership value	$\tilde{I}\left(G\right)=$ (maximum of the PVMV, minimum of NVMV).
Complete Graph $K_n$	n	Order of Fuzzy Graph	$\tilde{I}(\Gamma )=$ (Order of the PVMV, Order of the NVMV).
Star Graph $K_{1,n}$	2	Maximum of $\left\{\mathsf{\eta }\left({\gimel }_1\right)+\mathsf{\eta }\left({\gimel }_2\right)\right\}$ for all $({\gimel }_1,{\gimel }_2)$ in $K_{1,n}$	$\tilde{I}\left(K_{1,n}\right)=(max\left(\mathsf{\eta }\left({\gimel }_1\right)+\mathsf{\eta }\left({\gimel }_2\right)\right),$ min(η(ℷ1) + η(ℷ2)), where ℷ1 is the centre vertex of the star graph and vertex ℷ2 is adjacent to ℷ1

.

3.2. Union, Join and Cartesian BFG

Integrity values for the union of BFGs, the join of BFGs, and the Cartesian product of BFGs are discussed with examples, and integrity values are found.

Theorem 6.

Let Σ be a strong BFG and $\overline{\Sigma }$ be the complement of Σ. Then the integrity of BFG $\Sigma \cup \overline{\Sigma }$ is $\tilde{I}(\Sigma \cup \overline{\Sigma })=(\mathit{Order}\mathit{of}\mathit{PVMV},\mathit{Order}\mathit{of}\mathit{NVMV})$.

Proof. 

Let $\Sigma$ be a strong BFG and $\overline{\Sigma }$ be the complement of $\Sigma$. Then the union of $\Sigma$ and $\overline{\Sigma }$ is a complete BFG. Therefore by Theorem 4, the integrity is $\tilde{I}(\Sigma \cup \overline{\Sigma })=$ (Order of PVMV, Order of NVMV).  □

Theorem 7.

Let Σ be the union of two disjoint BFGs, with $|{\Sigma }_1| \ge |{\Sigma }_2|$. Then $\tilde{I}(\Sigma )=({\tilde{I}}^P,{\tilde{I}}^N)$, where ${\tilde{I}}^P(\Sigma )=min\{|{\Sigma }_1|,I\left({\Sigma }_1\right),|S| + |{\Sigma }_2|,|S|+max\left\{m({\Sigma }_1-S),m({\Sigma }_2-S)\right\}\}$ and ${\tilde{I}}^N(\Sigma )=max\{|{\Sigma }_1|,I\left({\Sigma }_1\right),|S| + |{\Sigma }_2|,|S|+min\left\{m({\Sigma }_1-S),m({\Sigma }_2-S)\right\}\}$.

Proof. 

Let Σ be the union of two disjoint BFGs with $|{\Sigma }_1| \ge |{\Sigma }_2|$.

• Case 1: The integrity set S of a BFG $\Sigma$ is empty. Then, the integrity for PVMV of $\Sigma$ is the maximum of ${\Sigma }_1$ and ${\Sigma }_2$ and integrity for NVMV of $\Sigma$ is the minimum of ${\Sigma }_1$ and ${\Sigma }_2$.
• Case 2: If the integrity set S is from BFG ${\Sigma }_1$. For PVMV $m\left({\Sigma }_1-S\right)\ge |{\Sigma }_1|$, then $\tilde{I}{(\Sigma )}^P=\tilde{I}{\left({\Sigma }_1\right)}^P$. If for any set S, $m\left({\Sigma }_1-S\right)<|{\Sigma }_1|$, then $\tilde{I}{(\Sigma )}^P=|S| + |{\Sigma }_2|$. For NVMV $M\left({\Sigma }_1-S\right)\le |{\Sigma }_1|$, then $\tilde{I}{(\Sigma )}^N=I{\left({\Sigma }_1\right)}^N$. If for any set S, $M\left({\Sigma }_1-S\right)>|{\Sigma }_2|$. Then $\tilde{I}{(\Sigma )}^N=|S| + |{\Sigma }_2|$.
• Case 3: If S is from ${\Sigma }_1$ and ${\Sigma }_2$. If S is a cut vertex from ${\Sigma }_1$ or ${\Sigma }_2$, we get at least three components in $\Sigma -S$. Then the positive integrity is ${\tilde{I}}^P=|S|+max\left\{m({\Sigma }_1-S_1),m({\Sigma }_2-S_2)\right\}$ and negative integrity is ${\tilde{I}}^N=|S|+min\left\{m({\Sigma }_1-S_1),m({\Sigma }_2-S_2)\right\}$, where $S_1=S\cap \mathsf{\eta }\left({\Sigma }_1\right)$ and $S_2=S\cap \mathsf{\eta }\left({\Sigma }_2\right)$. If S does not have any cut vertex of ${\Sigma }_1$ or ${\Sigma }_2$, it cannot be the integrity set for $\Sigma$, since ${\Sigma }_1$ and ${\Sigma }_2$ are disjoint. Combining the above cases, integrity is ${\tilde{I}}^P(\Sigma )=min\{|{\Sigma }_1|,I\left({\Sigma }_1\right),|S| + |{\Sigma }_2|,|S|+max\left\{m({\Sigma }_1-S),m({\Sigma }_2-S)\right\}\}$ and ${\tilde{I}}^N(\Sigma )=max\{|{\Sigma }_1|,\tilde{I}\left({\Sigma }_1\right),|S| + |{\Sigma }_2|,|S|+min\left\{m({\Sigma }_1-S),m({\Sigma }_2-S)\right\}\}$.

□

Theorem 8.

Let Σ be the union of two disjoint connected BFGs, ${\Sigma }_1$ and ${\Sigma }_2$. Then $\tilde{I}(\Sigma )=({\tilde{I}}^P,{\tilde{I}}^N)$, where ${\tilde{I}}^P(\Sigma )$ is $max\left\{{\tilde{I}}^P\left({\Sigma }_1\right),{\tilde{I}}^P\left({\Sigma }_2\right)\right\}\le {\tilde{I}}^P(\Sigma )$ and ${\tilde{I}}^N(\Sigma )$ is $min\left\{{\tilde{I}}^N\left({\Sigma }_1\right),{\tilde{I}}^N\left({\Sigma }_2\right)\right\}\le {\tilde{I}}^N(\Sigma )$.

Theorem 9.

Let ${\Sigma }_1$ and ${\Sigma }_2$ be two connected BFGs, then the join $\Sigma ={\Sigma }_1+{\Sigma }_2$ with ${\Im }_1\cap {\Im }_2=\varphi$. Then $\tilde{I}(\Sigma )=({\tilde{I}}^P,{\tilde{I}}^N)$, where ${\tilde{I}}^P(\Sigma )=min\left\{{\tilde{I}}^P\left({\Sigma }_1\right) + |{\Sigma }_2{|}^P,{\tilde{I}}^P\left({\Sigma }_2\right)+{|{\Sigma }_1|}^P\right\}$ and ${\tilde{I}}^N(\Sigma )=max\left\{{\tilde{I}}^N\left({\Sigma }_1\right) + |{\Sigma }_2{|}^N,{\tilde{I}}^N\left({\Sigma }_2\right)+{|{\Sigma }_1|}^N\right\}$.

Proof. 

According to the definition of the join of two BFGs, one edge connects each vertex from ${\Sigma }_1$ to ${\Sigma }_2$.

If ${\Sigma }_1$ and ${\Sigma }_2$ are complete BFGs, then the join of ${\Sigma }_1$ and ${\Sigma }_2$ is also a complete BFG. Then the integrity for PVMV is ${\tilde{I}}^P(\Sigma )=$ sum of PVMV and for NVMV integrity is ${\tilde{I}}^N(\Sigma )=$ sum of NVMV.

If ${\Sigma }_1$ and ${\Sigma }_2$ are two disjoint connected BFGs, then the join of $\Sigma ={\Sigma }_1+{\Sigma }_2$ is a complete BFG. Then the integrity for positive vertex membership value is ${\tilde{I}}^P(\Sigma )=$ sum of PVMV and the integrity for NVMV is ${\tilde{I}}^N(\Sigma )=$ sum of NVMV.  □

Example 4.

Consider two BFGs ${\Sigma }_1=\left({\Im }_1,{\mathsf{\xi }}_1\right)$ such that ${\Im }_1=\left\{{\gimel }_1,{\gimel }_2\right\}$, ${\mathsf{\xi }}_1=\left\{{\gimel }_1{\gimel }_2\right\}$ and ${\Sigma }_2=\left({\Im }_2,{\mathsf{\xi }}_2\right)$ such that ${\Im }_2=\left\{{\gimel }_3,{\gimel }_4,{\gimel }_5\right\}$, ${\mathsf{\xi }}_2=\left\{{\gimel }_3{\gimel }_4,{\gimel }_4{\gimel }_5\right\}$. Then the join of ${\Sigma }_1$ and ${\Sigma }_2$ is shown in Figure 5.

Theorem 10.

Let $K_{1,n}$ be a star BFG. Then $\tilde{I}\left(\overline{K_{1,n}}\right)=({\tilde{I}}^P,{\tilde{I}}^N)$, where ${\tilde{I}}^P=$ maximum $\left\{|K_1{|}^P,{|K_n|}^P\right\}$ and ${\tilde{I}}^N=$ minimum $\left\{|K_1{|}^N,{|K_n|}^N\right\}$.

Theorem 11.

Let ${\Sigma }_1$ and ${\Sigma }_2$ be two connected BFGs with vertex set ${\Im }_1=\left\{{\gimel }_1,{\gimel }_2\right\}$ and ${\Im }_2=\left\{u_1,u_2,\dots ,u_n\right\}$ and edges ${\mathsf{\xi }}_1=\left\{e_1\right\}$ and ${\mathsf{\xi }}_2=\left\{e_1,e_2,\dots ,e_n\right\}$, respectively. Let Σ be the cartesian BFG of ${\Sigma }_1$ and ${\Sigma }_2$. Let ${\Im }_1$ and ${\Im }_2$ be two partial subgraphs of Σ. Then the integrity of $\Sigma ={\Sigma }_1\times {\Sigma }_2$ is ${\tilde{I}}^P(\Sigma )=min\left\{|{\Im }_1{|}^P + |S_2{|}^P,|{\Im }_2{|}^P + {|S_1|}^P\right\}$ and ${\tilde{I}}^N(\Sigma )=min\left\{|{\Im }_1{|}^N + |S_2{|}^N,|{\Im }_2{|}^N + {|S_1|}^N\right\}$, where $S_i,i=1,2$ is the minimal set of ${\Im }_1$ and ${\Im }_2$.

Proof. 

The BFG $\Sigma$ consists of two partial subgraphs ${\Im }_1$ and ${\Im }_2$, which are isomorphic to ${\Sigma }_2$. $\Sigma$ remains a single connected component after less than n vertices are removed. Therefore, in order to break the graph, we must take away at least n vertices from ${\Im }_1$ and ${\Im }_2$. In ${\Im }_1$ BFG, if n vertices are removed then ${\Im }_2$ with minimal set $S_2$ is the integrity. Similarly in ${\Im }_2$ BFG if n vertices are removed then ${\Im }_1$ with minimal set $S_1$ is the integrity. The integrity value of $\Sigma$ is ${\tilde{I}}^P(\Sigma )=min\left\{|{\Im }_1{|}^P + |S_2{|}^P,|{\Im }_2{|}^P + {|S_1|}^P\right\}$ and ${\tilde{I}}^N(\Sigma )=max\left\{|{\Im }_1{|}^N + |S_2{|}^N,|{\Im }_2{|}^N + {|S_1|}^N\right\}$, where $S_i,i=1,2$ is the minimal set of ${\Im }_1$ and ${\Im }_2$. □

Example 5.

Consider two BFG ${\Sigma }_1=\left({\Im }_1,{\mathsf{\xi }}_1\right)$ such that ${\Im }_1=\left\{{\gimel }_1,{\gimel }_2\right\}$, ${\mathsf{\xi }}_1=\left\{{\gimel }_1{\gimel }_2\right\}$ and ${\Sigma }_2=\left({\Im }_2,{\mathsf{\xi }}_2\right)$ such that ${\Im }_2=\left\{{\gimel }_3,{\gimel }_4,{\gimel }_5\right\}$, ${\mathsf{\xi }}_2=\left\{{\gimel }_3{\gimel }_4,{\gimel }_4{\gimel }_5\right\}$. Then the cartesian of ${\Sigma }_1$ and ${\Sigma }_2$ is shown in Figure 6.

Theorem 12.

Let $K_2$ and $K_{1,n}(n\ge 1)$ be two BFGs, then the cartesian BFG is $\Sigma =K_2\times K_{1,n}$. Then $\tilde{I}(\Sigma )=\left({\tilde{I}}^P,{\tilde{I}}^N\right)$, where ${\tilde{I}}^P(\Sigma )=$ Sum of PVMV $\left\{(v_1,u),(v_2,u)\right\}$+maximum order of the PVMV in n components and ${\tilde{I}}^N(\Sigma )=$ Sum of NVMV $\left\{(v_1,u),(v_2,u)\right\}$+minimum order of the NVMV in n components.

Proof. 

Let the vertices of $K_2$ be $\left\{v_1,v_2\right\}$ and $K_{1,m}$ be $u,u_1,u_2,...,u_m$, where vertex u is the central vertex in $K_{1,n}$ with degree n. Let the minimal set $S=\left\{(v_1,u),(v_2,u)\right\}\subset \Sigma$, then $\Sigma -S$ has m components of $K_2$. The integrity of $\Sigma$ is ${\tilde{I}}^P(\Sigma )=$ Sum of PVMV $\left\{(v_1,u),(v_2,u)\right\}$+maximum order of the PVMV in n components and ${\tilde{I}}^N(\Sigma )=$ Sum of NVMV $\left\{(v_1,u),(v_2,u)\right\}$+minimum order of the NVMV in n components.  □

Theorem 13.

Let $K_2$ and $K_{n,m}(n\le m,n,m\ge 2)$ be two BFGs, then the cross BFG is $\Sigma =K_2\times K_{n,m}$. Then $\tilde{I}(\Sigma )=\left({\tilde{I}}^P,{\tilde{I}}^N\right)$, where ${\tilde{I}}^P(\Sigma )={|S|}^P$+maximum order of the PVMV in m components and ${\tilde{I}}^N(\Sigma )={|S|}^N$+minimum order of the NVMV in m components, where S is the minimal set.

Proof. 

Let the vertices of $K_2$ be $\left\{v_1,v_2\right\}$ and $K_{n,m}$ has vertices $u_1,u_2,\dots ,u_n$ in one partite set and $u_1^{\prime },u_2^{\prime },\dots ,u_n^{\prime }$ are in the other partite set, respectively. Let the minimal set of $\Sigma$ be $S=\left\{(v_1,u_i),(v_2,u_i);i=1,2,\dots ,n\right\}$, then $\Sigma -S$ has m components of $K_2$. The integrity of $\Sigma$ is $I^P(\Sigma )={|S|}^P$+maximum order of the PVMV in m components and $I^N(\Sigma )={|S|}^N$+minimum order of the NVMV in m components, where S is the minimal set.  □

Example 6.

Consider two BFG of $K_2$ and $K_{2,3}$. Then cartesian of $K_2$ and $K_{2,3}$ is shown in Figure 7.

Theorem 14.

Let $K_2$ and $W_{1,m}(m\ge 3)$ be two BFGs, then the cartesian BFG is $\Sigma =K_2\times W_{1,m}$. Then $\tilde{I}(\Sigma )=\left({\tilde{I}}^P,{\tilde{I}}^N\right)$, where ${\tilde{I}}^P(\Sigma )={|S|}^P$ + $I{\left(K_2\times C_{1,m}\right)}^P$ and $I^N(\Sigma )={|S|}^N$ + $\tilde{I}{\left(K_2\times C_{1,m}\right)}^N$.

Proof. 

Let the vertices of $K_2$ be $\left\{v_1,v_2\right\}$ and the vertices of $W_{1,m}$ be $\left\{x,u_1,u_2,\dots ,u_n\right\}$. Let the minimal set of $\Sigma$ be $S=\left\{(v_1,x),(v_2,x)\right\}$, then $\Sigma -S$ has only one component of $K_2\times C_m$. The integrity of BFG $\Sigma$ is ${\tilde{I}}^P(\Sigma )={|S|}^P$ + $\tilde{I}{\left(K_2\times C_m\right)}^P$ and ${\tilde{I}}^N(\Sigma )={|S|}^N$ + $\tilde{I}{\left(K_2\times C_m\right)}^N$. □

3.3. Application

When modeling real-world problems with mathematical models, it is important to recognize that every model has limitations. For new projects, it is essential to consider both advantages and disadvantages. By assigning fuzzy membership values to these pros and cons, the pros receive positive membership values, while the cons are assigned negative membership values.

If these values are considered as membership values of a fuzzy graph, it transforms into a bipolar fuzzy graph (BFG). The road network serves as a fundamental transportation mode for goods and people, significantly impacting a country’s economy. When implementing a project within the road network, choosing the right path optimizes both profit and loss comprehensively. Longer road lengths directly lead to increased gasoline and diesel consumption, contributing to environmental issues. At times, the compensation cost significantly surpasses the actual cost of road construction.

As a result, road construction has both positive and negative effects on social and economic conditions. This network type can be considered as a BFG. The positive impact is represented by positive membership values, while the negative impact is represented by negative membership values on specific nodes.

Consider a road network where nodes in the graph represent specific areas, places, villages, or towns, and the edges denote proposed road routes. Each node’s positive membership value represents economic gain, while the negative membership value represents economic loss due to environmental issues for that node. The integrity parameter is particularly helpful in identifying the optimal transmission path in such problems.

The network depicted in Figure 8 illustrates all possible road routes from node ${\gimel }_1$ to node ${\gimel }_8$. There are seven distinct paths between these two nodes: ${\gimel }_1-{\gimel }_2-{\gimel }_5-{\gimel }_8,{\gimel }_1-{\gimel }_2-{\gimel }_6-{\gimel }_8,{\gimel }_1-{\gimel }_3-{\gimel }_5-{\gimel }_8,{\gimel }_1-{\gimel }_3-{\gimel }_6-{\gimel }_8,$ ${\gimel }_1-{\gimel }_3-{\gimel }_7-{\gimel }_8,{\gimel }_1-{\gimel }_4-{\gimel }_6-{\gimel }_8,{\gimel }_1-{\gimel }_4-{\gimel }_7-{\gimel }_8.$

For each path S, the integrity set is defined as $\alpha ={|S|}^P+m{(\Sigma -S)}^P,\beta ={|S|}^N+M{(\Sigma -S)}^N,$ where ${|S|}^P$ and ${|S|}^N$ are the cumulative positive and negative membership values of nodes in S, $m{(\Sigma -S)}^P$ is the maximum positive membership value in the complement, and $M{(\Sigma -S)}^N$ is the maximum negative membership value in the complement.

From

Table 5. Integrity of BFG.

S	${|\mathit{S}|}^{\mathit{P}}$	$\mathit{m}{\left(\mathbf{\Sigma }-\mathit{S}\right)}^{\mathit{P}}$	$\mathit{\alpha }$	${|\mathit{S}|}^{\mathit{N}}$	$\mathit{M}{\left(\mathbf{\Sigma }-\mathit{S}\right)}^{\mathit{N}}$	$\mathit{\beta }$
$\left\{{\gimel }_1\right\}$	0.6	0.9	1.5	−0.3	−1.3	−1.6
$\left\{{\gimel }_2\right\}$	0.4	0.6	1	−0.5	−1.1	−1.6
$\left\{{\gimel }_6\right\}$	0.3	1	1.3	−0.2	−0.8	−1
$\left\{{\gimel }_8\right\}$	0.2	1.3	1.5	−0.6	−1	−1.6
$\left\{{\gimel }_1,{\gimel }_2\right\}$	1	0.5	1.5	−0.8	−0.8	−1.6
$\left\{{\gimel }_1,{\gimel }_6\right\}$	0.9	0.4	1.3	−0.5	−0.6	−1.1
$\left\{{\gimel }_1,{\gimel }_4\right\}$	0.8	0.7	1.5	−0.9	−0.7	−1.6
$\left\{{\gimel }_2,{\gimel }_6\right\}$	0.7	0.6	1.3	−0.7	−0.6	−1.3
$\left\{{\gimel }_2,{\gimel }_8\right\}$	0.6	0.6	1.2	−1.1	−0.3	−1.4
$\left\{{\gimel }_6,{\gimel }_8\right\}$	0.5	1	1.5	−0.8	−0.8	−1.6
$\left\{{\gimel }_1,{\gimel }_2,{\gimel }_6\right\}$	1.3	0.2	1.5	−1	−0.6	−1.6
$\left\{{\gimel }_1,{\gimel }_2,{\gimel }_8\right\}$	1.2	0.3	1.5	−1.4	−0.2	−1.6
$\left\{{\gimel }_2,{\gimel }_6,{\gimel }_8\right\}$	0.9	0.6	1.5	−1.3	−0.3	−1.6
$\left\{{\gimel }_1,{\gimel }_2,{\gimel }_6,{\gimel }_8\right\}$	1.5	0	1.5	−1.6	0	−1.6

Where $\alpha ={|S|}^P+m{\left(\Sigma -S\right)}^P$ and $\beta ={|S|}^N+M{\left(\Sigma -S\right)}^N$. From the table, the integrity is $\left(1,-1\right)$.

, the path ${\gimel }_1-{\gimel }_2-{\gimel }_6-{\gimel }_8$ achieves the minimal positive integrity, and the path ${\gimel }_1-{\gimel }_4-{\gimel }_6-{\gimel }_8$ achieves the maximal negative integrity. Thus, the overall integrity of the BFG is obtained as $\tilde{I}(\Sigma )=({\tilde{I}}^P,{\tilde{I}}^N)=(1,-1),$ after normalization. This ensures that the transmission from ${\gimel }_1$ to ${\gimel }_8$ is selected based on both economic gain and environmental loss in a balanced manner.

4. Conclusions

Both advantages and disadvantages are simultaneously reflected in the study of integrity in bipolar fuzzy graphs (BFGs), which extend traditional graph theory by incorporating both positive and negative membership values. By considering vertex removal and the resulting components, the integrity parameter defined as a tuple of positive and negative integrities effectively evaluates the robustness and vulnerability of BFGs. Interestingly, as the examples demonstrate, the integrity parameter can assume both positive and negative values, reflecting the complex trade-offs observed in real-world systems modelled by BFGs.

A comprehensive understanding of the structural resilience of BFGs is obtained by investigating integrity across specific graph classes, including null, complete, star, path, cycle, and bipartite graphs, as well as under operations such as union, join, and Cartesian product. Modeling a road network as a BFG further demonstrates the practical value of the integrity measure in a real-world application. In this case, positive membership values represent financial gains, while negative values capture environmental losses, showing how the network’s integrity can guide decisions that balance ecological impact with economic profit.

Overall, the integrity parameter of bipolar fuzzy graphs serves as a valuable tool for evaluating and improving complex systems with competing criteria, thereby enabling better decision-making in real-world applications. Moreover, this concept can be extended to intuitionistic fuzzy graphs and generalized fuzzy graphs, offering broader opportunities for research and application.