Fuzzy Coalition Graphs: A Framework for Understanding Cooperative Dominance in Uncertain Networks

Abstract

In a fuzzy graph $\mathbb{G}$, a fuzzy coalition is formed by two disjoint vertex sets ${\mathbb{V}}_{\mathtt{1}}$ and ${\mathbb{V}}_{\mathtt{2}}$, neither of which is a strongly dominating set, but the union ${\mathbb{V}}_{\mathtt{1}}\cup {\mathbb{V}}_{\mathtt{2}}$ forms a strongly dominating set. A fuzzy coalition partition of $\mathbb{G}$ is defined as $\Pi =\{{\mathbb{V}}_{\mathtt{1}},{\mathbb{V}}_{\mathtt{2}},\cdots ,{\mathbb{V}}_{\mathbb{k}}\}$, where each set ${\mathbb{V}}_{\mathbb{i}}$ either forms a singleton strongly dominating set or is not a strongly dominating set but forms a fuzzy coalition with another non-strongly dominating set in $\Pi$. A fuzzy graph with such a fuzzy coalition partition $\Pi$ is called a fuzzy coalition graph, denoted as $\mathbb{FG}(\mathbb{G},\Pi )$. The vertex set of the fuzzy coalition graph consists of $\{{\mathbb{V}}_{\mathtt{1}},{\mathbb{V}}_{\mathtt{2}},\cdots ,{\mathbb{V}}_{\mathbb{k}}\}$, corresponding one-to-one with the sets of $\Pi$, and the two vertices are adjacent in $\mathbb{FG}(\mathbb{G},\Pi )$ if and only if ${\mathbb{V}}_{\mathbb{i}}$ and ${\mathbb{V}}_{\mathbb{j}}$ are fuzzy coalition partners in $\Pi$. This study demonstrates how fuzzy coalition graphs can model and optimize cybersecurity collaborations across critical infrastructures in smart cities, ensuring comprehensive protection against cyber threats. This study concludes that fuzzy coalition graphs offer a robust framework for optimizing collaboration and decision-making in interconnected systems like smart cities.

1. Introduction

Fuzzy graphs are used in real life to model systems with uncertainty and imprecision, such as social networks, transportation, and decision-making under ambiguity. They provide a flexible framework to represent partial relationships and varying degrees of connectivity in complex environments.

A coalition is formed when two or more political parties choose to collaborate temporarily in pursuit of a common goal. This most commonly happens under parliamentary systems when no political party wins a clear majority in a general election. When this occurs, a parliamentary majority supports the formation of a coalition cabinet by two or more parties. Coalition cabinets are common in many nations. We shall exclusively study scenarios in which coalitions consist of two political parties, despite the fact that coalitions usually involve agreements between more than two parties. This implies the following model of graph theory.

Coalitions in crisp graphs have the advantage of simplicity and clear relationships, making them easy to analyze mathematically and suitable for small, well-defined systems. However, they lack flexibility and fail to represent uncertainty or partial relationships, which limits their applicability to real-world scenarios where relationships are dynamic or imprecise. This oversimplification often overlooks the nuanced interactions and dependencies in complex systems. Fuzzy graphs can overcome these limitations by allowing partial memberships and uncertain connections, providing a realistic framework for modeling coalitions. They capture the dynamic and flexible nature of partnerships, making them ideal for uncertain environments like smart cities, decision-making systems, and interconnected infrastructures, where coalition dynamics evolve with changing factors.

Fuzzy graph theory is ideal for modeling coalition concepts, as it effectively captures the uncertainty and imprecision in real-world systems, where relationships are often ambiguous. By incorporating partial memberships, it provides a nuanced representation of coalition partners, enabling dynamic and flexible partnerships that cannot be modeled using crisp graphs. The use of strong arcs and fuzzy coalitions allows for the analysis of cooperative behavior in uncertain environments. Additionally, it integrates quantitative relationships with qualitative uncertainties, enhancing decision-making in complex coalition-formation scenarios.

Ref. [1] A subset $\mathbb{D}\subseteq \mathbb{V}$ is defined as a dominating set of a graph $\mathbb{G}$ if every vertex not in $\mathbb{D}$ has at least one neighbor within $\mathbb{D}.$ Ref. [1] The open neighborhood of $\mathbb{y}\in \mathbb{V}\left(\mathbb{G}\right)$ is $\mathbb{N}\left(\mathbb{y}\right)=\left\{\mathbb{x}\right|\mathbb{xy}\in \mathbb{V}\left(\mathbb{G}\right)\}.$ Every vertex $\mathbb{x}\in \mathbb{N}\left(\mathbb{y}\right)$ is referred to as a neighbor of $\mathbb{y}$ and $\left|\mathbb{N}\right(\mathbb{y}\left)\right|,$ the $\mathbf{degree}$ of $\mathbb{y},$ denoted $\mathbb{deg}\left(\mathbb{y}\right).$ Ref. [2] A fuzzy graph $\mathbb{G}:(\mathbb{V},\mu ,\sigma )$ consists of a finite and non-empty set $\mathbb{V}\left(\mathbb{G}\right),$ where $\mu :\mathbb{V}\to [\mathtt{0},\mathtt{1}]$ represents the true membership and $\sigma :\mathbb{V}\times \mathbb{V}\to [\mathtt{0},\mathtt{1}]$ represents the false membership value of $\mathbb{G}.$ The condition $\sigma (\mathbb{x},\mathbb{y})\le \mu \left(\mathbb{x}\right)\wedge \mu \left(\mathbb{y}\right)$ must hold for all pairs of vertices $\mathbb{x},\mathbb{y}\in \mathbb{V}.$ Ref. [3] The strength of connectedness between two nodes $\mathbb{x}$ and $\mathbb{y}$ is defined as the maximum of the strength of all paths between $\mathbb{x}$ and $\mathbb{y}$ and is denoted by ${\mathbb{CONN}}_{\mathbb{G}}(\mathbb{x},\mathbb{y})$. Ref. [3] An arc of a fuzzy graph is called strong if its weight is at least as great as the strength of connectedness of its end nodes when it is deleted. Ref. [3] Let $\mathbb{G}$ be a fuzzy graph. Let $\mathbb{u}$ and $\mathbb{v}$ be two nodes of $\mathbb{G}$. We say that $\mathbb{u}$strongly dominates $\mathbb{v}$ if $(\mathbb{u},\mathbb{v})$ is a strong arc. Ref. [3] A subset $\mathbb{D}$ of $\mathbb{V}\left(\mathbb{G}\right)$ is considered a strong domination set if every vertex $\mathbb{y}\in \mathbb{V}\left(\mathbb{G}\right)-\mathbb{D}$ is strongly dominated by at least one neighbor in $\mathbb{D}.$ The strong domination number of $\mathbb{G},$ denoted as ${\gamma }_{\mathbb{s}}\left(\mathbb{G}\right),$ is the smallest size of a strong domination set in $\mathbb{G}.$ Ref. [1] In a graph $\mathbb{G},$ with $\mathbb{n}$ vertices, a vertex that has a degree of $\mathbb{n}-\mathtt{1}$ is called a full vertex. Ref. [1] If $|{\mathbb{V}}_{\mathbb{i}}|=\mathtt{1},$ then it is said to be a singleton set. Ref. [1] An induced subgraph $\mathbb{G}\left[\mathbb{D}\right]$ is a graph whose vertex set is $\mathbb{D}$ and whose edge set consists of all of the edges in $E\left(\mathbb{G}\right)$ that have both endpoints in $\mathbb{D}.$ Ref. [1] A coalition in a graph G consists of two disjoint sets of vertices ${\mathbb{V}}_{\mathtt{1}}$ and ${\mathbb{V}}_{\mathtt{2}}$, neither of which is a dominating set but whose union ${\mathbb{V}}_{\mathtt{1}}\cup {\mathbb{V}}_{\mathtt{2}}$ is a dominating set. We say that the set ${\mathbb{V}}_{\mathtt{1}}$ and ${\mathbb{V}}_{\mathtt{2}}$ form a coalition and are coalition partners. Ref. [1] A coalition partition, henceforth called a $\mathbb{c}$-partition, in a graph $\mathbb{G}$ is a vertex partition $\Pi =\{{\mathbb{V}}_{\mathtt{1}},{\mathbb{V}}_{\mathtt{2}},·,{\mathbb{V}}_{\mathbb{k}}\}$, such that every set ${\mathbb{V}}_{\mathbb{i}}$ of $\Pi$ is either a singleton dominating set, or is not a dominating set but forms a coalition with another set ${\mathbb{V}}_{\mathbb{j}}$ in $\Pi$. The coalition number $\mathbb{C}\left(\mathbb{G}\right)$ equals the maximum order $\mathbb{k}$ of a $\mathbb{c}$-partition of $\mathbb{G}$ having order $\mathbb{C}\left(\mathbb{G}\right)$, called a $\mathbb{C}\left(\mathbb{G}\right)$- partition. Ref. [4] Let $\mathbb{G}$ be a graph with a $\mathbb{c}$- partition $\pi =\{{\mathbb{V}}_{\mathtt{1}},{\mathbb{V}}_{\mathtt{2}},\cdots ,{\mathbb{V}}_{\mathbb{k}}\}$. The coalition graph $\mathbb{CG}(\mathbb{G},\pi )$ of $\mathbb{G}$ is the graph with vertex set ${\mathbb{V}}_{\mathtt{1}},{\mathbb{V}}_{\mathtt{2}},\cdots ,{\mathbb{V}}_{\mathbb{k}}$, corresponding one-to-one with the sets of $\pi$, and the two vertices ${\mathbb{V}}_{\mathbb{i}}$ and ${\mathbb{V}}_{\mathbb{j}}$ are adjacent in $\mathbb{CG}(\mathbb{G},\pi )$ if and only if the sets ${\mathbb{V}}_{\mathbb{i}}$ and ${\mathbb{V}}_{\mathbb{j}}$ are coalition partners in $\pi$; that is, neither ${\mathbb{V}}_{\mathbb{i}}$ nor ${\mathbb{V}}_{\mathbb{j}}$ is a dominating set of $\mathbb{G}$, but ${\mathbb{V}}_{\mathbb{i}}\cup {\mathbb{V}}_{\mathbb{j}}$ is a dominating set of $\mathbb{G}$.

1.1. Literature Review

This paper distinguishes itself by extending coalition theory to fuzzy graph structures, providing a unique framework for modeling cooperative behaviors in uncertain environments such as smart cities, which are not addressed in traditional coalition graph models.

Table 1. Comparison of key studies in graph theory and their relevance to fuzzy coalition graphs.

Reference	Key Results	Relevance to Current Paper
T.W. Haynes et al. (2020) [1]	Introduced fundamental definitions and concepts in coalition theory with standard graphs.	Provided a foundational understanding of coalition formation and domination principles, adapted for fuzzy graphs in our study.
B. Ganesan et al. (2022) [2]	Examined integrity under strong domination in both crisp and fuzzy graphs.	Contributed insights on structural robustness, relevant for coalition stability in fuzzy graphs.
O.T. Manjusha, M.S. Sunitha (2015) [3]	Defined strong domination metrics within fuzzy graphs.	Formed the basis for developing fuzzy coalition domination, enhancing the understanding of dominance in fuzzy structures.
T.W. Haynes et al. (2021) [5]	Established bounds on coalition numbers for different graph classes.	Informed approach to identifying coalition numbers in fuzzy graphs, especially relevant in estimating coalition sets.
T.W. Haynes et al. (2023) [6]	Explored structural properties and concepts of self-coalition in graphs.	Guided the development of coalition structures in fuzzy graphs, focusing on self-coalition applicability in uncertain systems.
T.W. Haynes et al. (2023) [7]	Analyzed coalition properties in specific graph structures like paths and cycles.	Aided in adapting coalition theory to different fuzzy graph structures, allowing broader applications with various graph types.
T.W. Haynes et al. (2023) [4]	Established the general properties of coalition graphs.	Provided the basis for defining and extending coalition properties specifically to fuzzy graphs.
O. Ore (1962) [8]	Offered key concepts foundational to all graph theory research.	Supported the theoretical grounding for adapting classic graph concepts to fuzzy graph settings.
X. Shi et al. (2022) [9]	Developed energy metrics for picture fuzzy graphs.	Helped the exploration of energy properties within fuzzy coalition graphs, useful for assessing collaboration strength.
Z. Shao et al. (2020) [10]	Applied vague graphs to real-world decision-making in medical contexts.	Inspired practical applications of fuzzy coalition graphs for decision-making and resource-sharing, especially for smart city cybersecurity.

highlights the novel focus on fuzzy coalition graphs and their practical applications in security and resource sharing across interconnected infrastructures.

1.2. Motivation

Fuzzy coalition graphs provide a valuable approach to understanding complex networks where the connections between components vary in strength or certainty. Traditional graph theory often falls short in capturing real-world networks where the relationships are not uniform but instead change in intensity or reliability. Fuzzy coalition graphs address this issue by enabling groups of vertices to collaborate in controlling the network, rather than relying on a single set. This is especially useful in systems like social networks, communication grids, and biological networks, where individual elements cannot fully dominate the network alone, but groups can work together to achieve control. This method supports the study of group dynamics, decision-making, and resilience in networks, particularly when information is incomplete or the network is evolving.

The primary hypothesis is that fuzzy coalition graphs can more effectively model real-world networks by capturing the teamwork and interdependence among elements. Additionally, we hypothesize that these graphs can improve the distribution of resources and decision-making in complex networks, such as smart cities or essential infrastructure systems. Testing these ideas aims to show that fuzzy coalition graphs can enhance our understanding and management of systems that depend on cooperation and flexibility in changing conditions.

1.3. Methodology

In this research, we focus on the concept of fuzzy coalition graphs $f{u}_{cg}$, which are designed to represent coalitions in settings where individual entities cannot fully dominate or control the system alone. The paper targets a class of problems in which interconnected agents (such as companies, infrastructure systems, or nations) need to form coalitions for a common objective, with an emphasis on minimizing individual risks through cooperation. This is particularly relevant for smart city infrastructures, where entities like transportation, energy, and healthcare systems are interconnected but vulnerable to cyber-attacks. By using $f{u}_{cg}$, this paper demonstrates how collaboration can lead to resilient, comprehensive cybersecurity strategies. This research addresses the challenge of forming coalitions that minimize risk in interconnected systems, such as cybersecurity infrastructures in smart cities. The class of solutions explored here involves coalition structures that enhance collective security by ensuring that no single entity has to defend itself alone. Our approach uses fuzzy coalition graphs ($f{u}_{cg}$) to represent these coalitions, with each vertex representing a critical infrastructure and each strong arc representing a secure connection or cybersecurity partnership. The methodology establishes key boundary conditions essential for coalition formation. First, we ensure mutual dependency for risk reduction, where systems form coalitions only when they cannot independently secure themselves. This dependency defines the necessity for each partnership, ensuring that coalition members are actively supporting each other’s cybersecurity needs. Additionally, connectivity is a priority, each coalition connects directly with at least one other critical system, minimizing isolated vulnerabilities and enabling a robust, interlinked defense network. Lastly, the risk distribution is balanced, with each coalition designed to share cybersecurity resources equitably, ensuring that no single system is overburdened by risk. This methodological structure provides a systematic approach to forming coalitions, ensuring that each coalition meets our defined boundary conditions for risk minimization. The result is an efficient coalition network that promotes optimal cybersecurity collaboration and effective risk-sharing across interconnected systems, exemplified in the smart city scenario used in this research.

In this study, we investigate fuzzy coalitions that comprise $st{d}_{sets}$ within $f{u}_{graphs}.$ Our method can be used for sets with different attributes, allowing for easier modeling of complex systems. Section 1.1 provides a concise review of the existing literature on fuzzy graph theory, focusing on concepts such as strong domination, coalition formations, and their applications in interconnected systems. In Section 1.2, we discuss our motivation for studying fuzzy coalition graphs and their relevance in modeling real-world systems characterized by complexity and interdependency. In Section 1.3, the methodology involves constructing fuzzy coalition partitions in fuzzy graphs by identifying strong arcs, evaluating strong domination, and determining coalition partners and $\mathbb{F}\left(\mathbb{G}\right)$-partitions. In Section 2, we introduce fuzzy coalitions and fuzzy coalition partitions. A fuzzy coalition is when different sets of vertices join together to form a strongly dominating set, even if they cannot do so alone. We also explain how to group the vertices of a fuzzy graph into subsets that can work together. In Section 3, we define the fuzzy coalition graph ($f{u}_{cg}$) and show that any fuzzy graph $\mathbb{G}$ can be represented as the $f{u}_{cg}$ of another fuzzy graph $\mathbb{H}$. This shows how useful the $f{u}_{cg}$ structure is. We also provide an algorithm for finding fuzzy coalition graphs, which helps identify partners and build the $f{u}_{cg}$ more easily. In Section 4, we discuss how fuzzy coalition graphs can be used in real life, such as for social networks, communication networks, and traffic management. In social networks, these graphs can represent influential groups. In communication networks, they can help improve connections between nodes. In traffic management, they can help manage congestion by showing how different routes work together. Overall, fuzzy coalition graphs make it easier to solve problems with complex relationships.

2. Bounds for Fuzzy Coalition and Fuzzy Coalition Partition in Fuzzy Graphs

In this section, we derive bounds for $\mathbb{F}\left(\mathbb{G}\right)$ and identify $f{u}_{graphs}$ $\mathbb{G}$ having relatively small values for $\mathbb{F}\left(\mathbb{G}\right).$

Definition 1.

In a $f{u}_{graph}$ $\mathbb{G},$ a fuzzy coalition is formed by two disjoint vertex sets, ${\mathbb{V}}_{\mathtt{1}}$ and ${\mathbb{V}}_{\mathtt{2}},$ neither of which is a $st{d}_{set}$ but whose union ${\mathbb{V}}_{\mathtt{1}}\cup {\mathbb{V}}_{\mathtt{2}}$ is a $st{d}_{set}$. ${\mathbb{V}}_{\mathtt{1}}$ and ${\mathbb{V}}_{\mathtt{2}}$ are referred to as coalition partners.

Definition 2.

A fuzzy coalition partition, also known as an $\mathbb{f}-$ partition, in a $f{u}_{graph}$ $\mathbb{G}$ has a partition $\Pi =\{{\mathbb{V}}_{\mathtt{1}},{\mathbb{V}}_{\mathtt{2}},\cdots ,{\mathbb{V}}_{\mathbb{r}}\}$, where each subset in $\Pi$ is either a singleton $st{d}_{set}$ or does not act as a $st{d}_{set}$ by itself but makes a fuzzy coalition with another subset ${\mathbb{V}}_{\mathbb{j}}$ in $\Pi .$ The $f{u}_{cn}$ $\mathbb{F}\left(\mathbb{G}\right)$ is defined as the maximum order $\mathbb{r}$ of subsets in an $\mathbb{f}-$ partition of $\mathbb{G},$ and a $\mathbb{f}-$ partition of $\mathbb{G}$ having order $\mathbb{F}\left(\mathbb{G}\right)$ is called a $\mathbb{F}\left(\mathbb{G}\right)-$ partition.

Example 1.

Consider a $f{u}_{graph}$ $\mathbb{G}$ with vertices $\mathbb{V}=({\mathbb{v}}_{\mathtt{1}},{\mathbb{v}}_{\mathtt{2}},{\mathbb{v}}_{\mathtt{3}},{\mathbb{v}}_{\mathtt{4}},{\mathbb{v}}_{\mathtt{5}})$, where the membership values of the vertices are $\mu \left({\mathbb{v}}_{\mathtt{1}}\right)=\mathtt{0}.\mathtt{7}$, $\mu \left({\mathbb{v}}_{\mathtt{2}}\right)=\mathtt{0}.\mathtt{8}$, $\mu \left({\mathbb{v}}_{\mathtt{3}}\right)=\mathtt{0}.\mathtt{9}$, $\mu \left({\mathbb{v}}_{\mathtt{4}}\right)=\mathtt{0}.\mathtt{9}$, $\mu \left({\mathbb{v}}_{\mathtt{5}}\right)=\mathtt{0}.\mathtt{5}$. The edges have the following membership values $\sigma ({\mathbb{v}}_{\mathtt{1}},{\mathbb{v}}_{\mathtt{2}})=\mathtt{0}.\mathtt{7}$, $\sigma ({\mathbb{v}}_{\mathtt{1}},{\mathbb{v}}_{\mathtt{3}})=\mathtt{0}.\mathtt{6}$, $\sigma ({\mathbb{v}}_{\mathtt{2}},{\mathbb{v}}_{\mathtt{4}})=\mathtt{0}.\mathtt{7}$, $\sigma ({\mathbb{v}}_{\mathtt{2}},{\mathbb{v}}_{\mathtt{5}})=\mathtt{0}.\mathtt{5}$, $\sigma ({\mathbb{v}}_{\mathtt{3}},{\mathbb{v}}_{\mathtt{4}})=\mathtt{0}.\mathtt{9}$, $\sigma ({\mathbb{v}}_{\mathtt{4}},{\mathbb{v}}_{\mathtt{5}})=\mathtt{0}.\mathtt{8}$, which are given below in Figure 1.

The edges $({\mathbb{v}}_{\mathtt{1}},{\mathbb{v}}_{\mathtt{2}}),({\mathbb{v}}_{\mathtt{2}},{\mathbb{v}}_{\mathtt{4}}),({\mathbb{v}}_{\mathtt{3}},{\mathbb{v}}_{\mathtt{4}}),({\mathbb{v}}_{\mathtt{4}},{\mathbb{v}}_{\mathtt{5}})$ are strong arcs, since their weights are at least as great as the strength of connectedness between the respective vertices.

The Partition $\Pi =\{\{{\mathbb{v}}_{\mathtt{1}},{\mathbb{v}}_{\mathtt{2}}\},\{{\mathbb{v}}_{\mathtt{3}},{\mathbb{v}}_{\mathtt{5}}\},\left\{{\mathbb{v}}_{\mathtt{4}}\right\}\}$ is a $\mathbb{f}-ptn$ of $\mathbb{G}$. No set of $\Pi$ is a $st{d}_{sets}$. But $\{{\mathbb{v}}_{\mathtt{1}},{\mathbb{v}}_{\mathtt{2}}\}$ and $\left\{{\mathbb{v}}_{\mathtt{4}}\right\}$ form $f{u}_c$; $\{{\mathbb{v}}_{\mathtt{1}},{\mathbb{v}}_{\mathtt{2}}\}$ and $\{{\mathbb{v}}_{\mathtt{3}},{\mathbb{v}}_{\mathtt{5}}\}$ form $f{u}_c$. Thus every set forms a $f{u}_c$ with at least one other set. Hence, $\mathbb{F}\left(\mathbb{G}\right)=\mathtt{3}$, and $\Pi$ is an $\mathbb{F}\left(\mathbb{G}\right)-ptn$.

Theorem 1.

Every $f{u}_{graph}$ $\mathbb{G}$ can be partitioned into an $\mathbb{f}-ptn.$

Proof. 

Consider $\Pi =\{{\mathbb{V}}_{\mathtt{1}},{\mathbb{V}}_{\mathtt{2}},\cdots ,{\mathbb{V}}_{\mathbb{r}}\}$ as a $s{t}_{dom}$ partition of $\mathbb{V}$ of a $f{u}_{graph}$ $\mathbb{G}$ and that $\mathbb{r}={\gamma }_{\mathbb{s}}\left(\mathbb{G}\right)$ is the $st{d}_{number}$ of $\mathtt{𝔾}.$ We assume that each ${\mathbb{V}}_{\mathbb{i}}$, where $1\le \mathbb{i}\le \mathbb{r}-\mathtt{1}$ is a minimal $st{d}_{set}$ in $\mathbb{G}.$ If any ${\mathbb{V}}_{\mathbb{i}}$ is not a minimal, we can enhance it by taking a minimal $st{d}_{subset}$ ${\mathbb{V}}_{\mathbb{i}}^{\prime }\subseteq {\mathbb{V}}_{\mathbb{i}}$ and adding the difference ${\mathbb{V}}_{\mathbb{i}}-{\mathbb{V}}_{\mathbb{i}}^{\prime }$ in the set ${\mathbb{V}}_{\mathbb{r}}.$ In a $f{u}_{graph}$ $\mathbb{G},$ when a minimal $st{d}_{set}$ ${\mathbb{V}}_{\mathbb{i}}$ where $\mathtt{1}\le \mathbb{i}\le \mathbb{r}-\mathtt{1}$ is not a singleton, it is split into two non-empty subsets, ${\mathbb{V}}_{\mathbb{i},\mathtt{1}}$ and ${\mathbb{V}}_{\mathbb{i},\mathtt{2}}.$ These subsets, while not strongly dominating on their own, will together form a $f{u}_c$ that strongly dominates the $f{u}_{graph}$. By following this procedure for each ${\mathbb{V}}_{\mathbb{i}}$, where $\mathtt{1}\le \mathbb{i}\le \mathbb{r}-\mathtt{1},$, a new $f{u}_c$ ${\Pi }^{\prime }$ of sets is generated. In this collection, every set is either a singleton $st{d}_{set}$ or not a $st{d}_{set}$ that pairs with another set in ${\Pi }^{\prime }$ to form a $f{u}_c.$

If ${\mathbb{V}}_{\mathbb{r}}$ is a singleton that strongly dominates, combining it into ${\Pi }^{\prime }$ instantly produces an $\mathbb{f}-ptn$ of $\mathbb{V}$ with an of at least $\mathbb{r}={\gamma }_{\mathbb{s}}\left(\mathbb{G}\right).$ On the other hand, if ${\mathbb{V}}_{\mathbb{r}}$ is a minimal $st{d}_{set}$ that contains more than one vertex, it can be divided into two non-$st{d}_{set}$ and included in ${\Pi }^{\prime },$, thus creating an $\mathbb{f}-ptn$ of $\mathbb{G}$ with an of at least $\mathbb{r}+\mathtt{1},$, which exceeds ${\gamma }_{\mathbb{s}}\left(\mathbb{G}\right).$ If ${\mathbb{V}}_{\mathbb{r}}$ is not a minimal $st{d}_{set},$, then let ${\mathbb{V}}_{\mathbb{r}}^{\prime }\subseteq {\mathbb{V}}_{\mathbb{r}}$ be a minimal $st{d}_{subset}.$ Partition ${\mathbb{V}}_{\mathbb{r}}^{\prime }$ into two non-empty, non-$st{d}_{sets}$ ${\mathbb{V}}_{\mathbb{r},\mathtt{1}}$ and ${\mathbb{V}}_{\mathbb{r},\mathtt{2}},$ which form a $f{u}_c.$ Define ${\mathbb{W}}_{\mathbb{r}}={\mathbb{V}}_{\mathbb{r}}-{\mathbb{V}}_{\mathbb{r}}^{\prime }$ and add ${\mathbb{V}}_{\mathbb{r},\mathtt{1}}$ and ${\mathbb{V}}_{\mathbb{r},\mathtt{2}}$ to ${\Pi }^{\prime }.$

Since ${\mathbb{W}}_{\mathbb{r}}$ cannot be a $st{d}_{set}$ (otherwise, $\mathbb{G}$ would have more than ${\gamma }_{\mathbb{s}}\left(\mathbb{G}\right)$ disjoint $st{d}_{set},$, which contradicts the definition of ${\gamma }_{\mathbb{s}}\left(\mathbb{G}\right).$), there are two cases to consider:

Case 1.

If ${\mathbb{W}}_{\mathbb{r}}$ forms a $f{u}_c$ with any other non-$st{d}_{set}$ in ${\Pi }^{\prime },$ then including ${\mathbb{W}}_{\mathbb{r}}$ to ${\Pi }^{\prime }$ produces an $\mathbb{f}-ptn$ of $\mathbb{G}$ with an of at least $\mathbb{r}+\mathtt{2},$ which is greater than${\gamma }_{\mathbb{s}}\left(\mathbb{G}\right).$

Case 2.

${\mathbb{W}}_{\mathbb{r}}$ does not form a $f{u}_c$ with any set in ${\Pi }^{\prime }$ and replace it with the union ${\mathbb{V}}_{\mathbb{r},\mathtt{2}}\cup {\mathbb{W}}_{\mathbb{r}}.$ This modification ensures that ${\Pi }^{\prime }$ still represents a $\mathbb{f}-ptn$ of $\mathbb{G},$ with at least $\mathbb{r}+\mathtt{1}>{\gamma }_{\mathbb{s}}\left(\mathbb{G}\right).$

In all cases, we have constructed a $\mathbb{f}-ptn$ for $\mathbb{G},$ thereby proving that every $f{u}_{graph}$ $\mathbb{G}$ indeed has a $\mathbb{f}-ptn.$    □

Proposition 1.

For a $f{u}_{graph}$ $\mathbb{G}$ of $\mathbb{n},$ then $\mathtt{1}\le \mathbb{F}\left(\mathbb{G}\right)\le \mathbb{n}.$

Proof. 

Let $\mathbb{G}=(\mathbb{V},\sigma ,\mu )$ be a $f{u}_{graph}$ with $\mathbb{V}$ such that $\left|\mathbb{V}\right|=\mathbb{n},$ where $\mathbb{n}$ represents the no. of vertices in $\mathbb{G}.$ The $f{u}_{cn}$ $\mathbb{F}\left(\mathbb{G}\right)$ is defined as the max - number $\mathbb{r}$ for which there exists a $\mathbb{f}-ptn$, $\Pi =\{{\mathbb{V}}_{\mathtt{1}},{\mathbb{V}}_{\mathtt{2}},\cdots ,{\mathbb{V}}_{\mathbb{r}}\}$ of $\mathbb{V}$. Each ${\mathbb{V}}_{\mathbb{i}}$ is either a singleton $st{d}_{set}$ or a non-$st{d}_{set}$ that pairs with another set in the partition to form a $f{u}_c.$

• Step 1: $\mathbb{F}\left(\mathbb{G}\right)\ge \mathtt{1}$

To establish the $l.b$, consider the trivial partition $\Pi =\left\{\mathbb{V}\right\},$ where the whole $\mathbb{V}$ is treated as a single set. Since $\mathbb{V}$ itself is trivially a $st{d}_{set}$ in the $f{u}_{graph}$ $\mathbb{G},$ this partition is a valid $\mathbb{f}-ptn.$ Consequently, the $f{u}_{cn}$ $\mathbb{F}\left(\mathbb{G}\right)$ must satisfy:

$$\mathbb{F}\left(\mathbb{G}\right)\ge \mathtt{1}.$$

• Step 2: $\mathbb{F}\left(\mathbb{G}\right)\le \mathbb{n}$

For the $u.b$, consider the partition $\Pi =\{\left\{{\mathbb{v}}_{\mathtt{1}}\right\},\left\{{\mathbb{v}}_{\mathtt{2}}\right\},\cdots ,\left\{{\mathbb{v}}_{\mathbb{n}}\right\}\},$ where each vertex ${\mathbb{v}}_{\mathbb{i}}\in \mathbb{V}$ forms its own singleton set $\left\{{\mathbb{v}}_{\mathbb{i}}\right\}.$ This partition consists of $\mathbb{n}$ disjoint subsets of $\mathbb{V},$ and since $\left|\mathbb{V}\right|=\mathbb{n},$ this represents the maximum possible number of disjoint subsets that can be formed from $\mathbb{V}.$ Since each vertex in a $f{u}_{graph}$ can form a singleton $st{d}_{set}$ on its own, this partition is a valid $\mathbb{f}-ptn.$ Therefore, the $f{u}_{cn}$ $\mathbb{F}\left(\mathbb{G}\right)$ cannot exceed $\mathbb{n}.$ Thus, $\mathbb{F}\left(\mathbb{G}\right)\le \mathbb{n}.$

Hence, $\mathtt{1}\le \mathbb{F}\left(\mathbb{G}\right)\le \mathbb{n}.$    □

Proposition 2.

If $\mathbb{G}$ is a $f{u}_{graph}$ without $ful{l}_{ver}$ and $is{o}_{ver}$, then ${\gamma }_{\mathbb{s}}\left(\mathbb{G}\right)\le \mathbb{F}\left(\mathbb{G}\right).$

Proof. 

Consider $\mathbb{G}$ as a $f{u}_{graph}$ without $is{o}_{ver}.$ We start with a minimal $st{d}_{set}$ $\mathbb{D}$ of $\mathbb{G},$ where $\left|\mathbb{D}\right|={\gamma }_{\mathbb{s}}\left(\mathbb{G}\right).$ We know that $\mathbb{D}$ ensures that every vertex in $\mathbb{G}$ is either in $\mathbb{D}$ or strongly dominated by at least one vertex in $\mathbb{D}.$ Since $\mathbb{G}$ has no $is{o}_{ver}$, every vertex is neighbor to at least one vertex in $\mathbb{D},$ and no single vertex can strongly dominate the entire $f{u}_{graph}$, which implies that ${\gamma }_{\mathbb{s}}\left(\mathbb{G}\right)$ is at least $\mathtt{1}$ and at most $\left|\mathbb{V}\right|-\mathtt{1}.$ To form a $f{u}_{cp}$, we consider the set $\mathbb{D}$ and its complement in $\mathbb{V}.$ Each vertex in $\mathbb{D}$ can be a singleton $st{d}_{set}$. For the remaining vertices, each can be paired with vertices in $\mathbb{D}$ to create coalitions. This process generates at least ${\gamma }_{\mathbb{s}}\left(\mathbb{G}\right)$ subsets, as each vertex in $\mathbb{D}$ forms a separate subset, and the remaining vertices are arranged in $f{u}_c$ s with these subsets. Since the $f{u}_{cn}$ $\mathbb{F}\left(\mathbb{G}\right)$ is the max number of sets in a valid $f{u}_{cp},$ and our partition with ${\gamma }_{\mathbb{s}}\left(\mathbb{G}\right)$ subsets is valid, it follows that ${\gamma }_{\mathbb{s}}\left(\mathbb{G}\right)\le \mathbb{F}\left(\mathbb{G}\right).$ Thus, ${\gamma }_{\mathbb{s}}\left(\mathbb{G}\right)$ is bounded above by $\mathbb{F}\left(\mathbb{G}\right),$ completing this proof.    □

Proposition 3.

If $\mathbb{G}$ is a $f{u}_{graph}$ with $\mathbb{n}$ vertices and $\mathbb{G}\ne {\mathbb{K}}_{\mathbb{n}}$, and if there are $\mathbb{k}$ vertices in $\mathbb{G}$ with degree $\mathbb{n}-\mathtt{1},$ then the inequality $\mathtt{3}\le \mathbb{k}+\mathtt{2}\le \mathbb{F}\left(\mathbb{G}\right).$

Proof. 

Since $\mathbb{G}\ne {\mathbb{K}}_{\mathbb{n}},$ at least three sets are required in any $f{u}_{cp}$, because two sets alone cannot strongly dominate the entire $f{u}_{graph}$. This gives the $l.b$ $\mathbb{F}\left(\mathbb{G}\right)\ge \mathtt{3}.$ For the $\mathbb{k}$ vertices of degree $\mathbb{n}-\mathtt{1},$ each forms a singleton $st{d}_{set},$ contributing $\mathbb{k}$ sets. The remaining $\mathbb{n}-\mathbb{k}$ vertices, having degrees less than $\mathbb{n}-\mathtt{1},$ must form at least two $f{u}_c$ to strongly dominate the $f{u}_{graph}.$ Thus, the $f{u}_{cn}$ is at least $\mathbb{k}+\mathtt{2},$ giving $\mathbb{F}\left(\mathbb{G}\right)\ge \mathbb{k}+\mathtt{2}.$ Combining these, we conclude $\mathtt{3}\le \mathbb{k}+\mathtt{2}\le \mathbb{F}\left(\mathbb{G}\right).$    □

The limits provided in Proposition 3 are precise for fuzzy star graphs ${\mathbb{K}}_{\mathtt{1},\mathbb{n}-\mathtt{1}}$ with $\mathbb{n}\ge \mathtt{3},$ as $\mathbb{F}\left({\mathbb{K}}_{\mathtt{1},\mathbb{n}-\mathtt{1}}\right)=\mathtt{3}$.

Now, let us analyze $f{u}_{graphs}$ that contain $is{o}_{v}er$. Let $\mathbb{G}=(\mathbb{V},\sigma ,\mu )$ be a $f{u}_{graph}$ of $\mathbb{n}\ge \mathtt{2}$ with a non-empty subset $\mathbb{I}\subseteq \mathbb{V}$ consisting of $is{o}_{ver}.$ Let $\Pi =\{{\mathbb{V}}_{\mathtt{1}},{\mathbb{V}}_{\mathtt{2}},\cdots ,{\mathbb{V}}_{\mathbb{r}}\}$ represent a $f{u}_{cp}$ of $\mathbb{G}.$ Since $\mathbb{G}$ has $is{o}_{ver},$ none of the sets in $\Pi$ can be singleton $st{d}_{sets}.$ Therefore, each set ${\mathbb{V}}_{\mathbb{i}}$ in $\Pi$ cannot strongly dominate the $f{u}_{graph}$ individually and must form a $f{u}_c$ with another set ${\mathbb{V}}_{\mathbb{j}}.$ If more than two sets in $\Pi$ include vertices, this implies they cannot form $f{u}_c$. As a result, the $is{o}_{ver}$ from $\mathbb{I}$ can only be included in at most two sets in the partition. Assume ${\mathbb{V}}_{\mathtt{1}}\cap \mathbb{I}\ne \varnothing$ and ${\mathbb{V}}_{\mathtt{2}}\cap \mathbb{I}\ne \varnothing .$ Then, for all $\mathbb{j}\ge \mathtt{3},$ ${\mathbb{V}}_{\mathbb{j}}\cap \mathbb{I}=\varnothing .$ Consequently, ${\mathbb{V}}_{\mathbb{j}}$ for $\mathbb{j}\ge \mathtt{3}$ cannot participate in any $f{u}_c$ involving $is{o}_{ver}$ in a $f{u}_{graph}$ and can only be part of a maximum of two sets in a $\mathbb{f}-ptn.$

Observation 1.

The following is true if $\mathbb{G}$ is a $f{u}_{graph}$ of $\mathbb{n}\ge \mathtt{2}$ with a non-empty set I of $is{o}_{ver}.$

1. 

$\mathbb{F}\left(\mathbb{G}\right)=\mathtt{2}.$

2. 

$\mathbb{F}\left(\mathbb{G}\right)\ge \mathtt{3}$ and $\mathbb{I}\subseteq {\mathbb{V}}_{\mathbb{j}}$ are satisfied for some $\mathbb{j}.$ by the partition $\Pi =\{{\mathbb{V}}_{\mathtt{1}},{\mathbb{V}}_{\mathtt{2}},\cdots ,{\mathbb{V}}_{\mathbb{r}}\}$ of $\mathbb{G}.$

Theorem 2.

If $\mathbb{G}$ is a $f{u}_{graph}$ without $ful{l}_{ver}$ and min - deg $\delta \left(\mathbb{G}\right)\le \mathtt{1},$ then $\mathtt{2}+\delta \left(\mathbb{G}\right)\le F\left(\mathbb{G}\right).$

Proof. 

Let $\mathbb{G}$ be a $f{u}_{graph}$ of $\mathbb{n}$ with $\mathtt{1}\le \delta \left(\mathbb{G}\right)\le \Delta \left(\mathbb{G}\right)\le \mathbb{n}-\mathtt{2},$ and let $\mathbb{v}$ be a vertex with degrees $\delta \left(\mathbb{G}\right)=\mathbb{r}\ge \mathtt{1}.$ Let $\mathbb{N}\left(\mathbb{v}\right)=\{{\mathbb{v}}_{\mathtt{1}},{\mathbb{v}}_{\mathtt{2}},\cdots ,{\mathbb{v}}_{\mathbb{r}}\}$ be the neighbors of $\mathbb{v}.$ Since $\mathbb{G}$ has no $ful{l}_{ver},$ $\mathbb{V}\left(\mathbb{G}\right)-\mathbb{N}\left[\mathbb{v}\right]\ne \varnothing .$ Define a partition $\Pi$ of $\mathbb{G}$ where ${\mathbb{V}}_{\mathbb{i}}=\left\{{\mathbb{v}}_{\mathbb{i}}\right\}$ for $\mathtt{1}\le \mathbb{i}\le \mathbb{r},$ ${\mathbb{V}}_{\mathbb{r}+\mathtt{1}}=\left\{\mathbb{v}\right\},$ and ${\mathbb{V}}_{\mathbb{r}+\mathtt{2}}=\mathbb{V}\left(\mathbb{G}\right)-\mathbb{N}\left[\mathbb{v}\right].$ To show that $\Pi$ is a $\mathbb{f}-ptn$ of $\mathbb{G},$, observe that none of the sets in $\Pi$ are a $st{d}_{set},$, as $\mathbb{G}$ has no $ful{l}_{ver}.$ The neighbors of $\mathbb{v}$ are divided into two groups: those without neighbors outside $\mathbb{N}\left[\mathbb{v}\right]$ and those with at least one neighbor outside $\mathbb{N}\left[\mathbb{v}\right].$ The first group strongly dominates $\mathbb{N}\left(\mathbb{v}\right),$ and the second group strongly dominates the first group. The set $\mathbb{V}\left(\mathbb{G}\right)-\mathbb{N}\left[\mathbb{v}\right]$ strongly dominates the second group. Thus, every set ${\mathbb{V}}_{\mathbb{i}}$ for $\mathtt{1}\le \mathbb{i}\le \mathbb{r}$ is a $fu{C}_{partner}$ of ${\mathbb{V}}_{\mathbb{r}+\mathtt{2}},$ confirming that $\Pi$ is a $\mathbb{f}-ptn$ of $\mathbb{G}$ with $\mathbb{r}+\mathtt{2}=\delta \left(\mathbb{G}\right)+\mathtt{2}.$ Hence, $\mathtt{2}+\delta \left(\mathbb{G}\right)\le \mathbb{F}\left(\mathbb{G}\right).$    □

Proposition 4.

The graph $\mathbb{G}\ne \overline{{\mathbb{K}}_{\mathbb{n}}}$ has a non-empty set $\mathbb{I}$ of $is{o}_{ver}.$ Let ${\mathbb{G}}^{\prime }=\mathbb{G}[\mathbb{V}-\mathbb{I}]A.$ It follows that $\mathbb{F}\left(\mathbb{G}\right)\ge \mathtt{2}+\delta \left({\mathbb{G}}^{\prime }\right).$

Proof. 

Consider a $f{u}_{graph}$ $\mathbb{G}$ with $is{o}_{ver}$ $\mathbb{I}.$ The induced $f{u}_{sg}$ ${\mathbb{G}}^{\prime },$ which is $\mathbb{G}$ restricted to $\mathbb{V}-\mathbb{I},$ has a min - deg $\delta \left({\mathbb{G}}^{\prime }\right).$ According to Theorem 2 for $f{u}_{graphs}$ with no $ful{l}_{ver},$ if the min - deg of ${\mathbb{G}}^{\prime }$ is $\delta \left({\mathbb{G}}^{\prime }\right),$ then the $f{u}_{cn}$ of ${\mathbb{G}}^{\prime }$ satisfies $\mathtt{2}+\delta \left({\mathbb{G}}^{\prime }\right)\le F\left({\mathbb{G}}^{\prime }\right).$ Since the $is{o}_{ver}$ in $\mathbb{G}$ does not affect the strongly dominating properties of ${\mathbb{G}}^{\prime },$ the $f{u}_{cn}$ $\mathbb{F}\left(\mathbb{G}\right)$ of the full graph must accommodate at least the $f{u}_{cn}$ required by ${\mathbb{G}}^{\prime }.$ Thus, the $\mathbb{F}\left(\mathbb{G}\right)$ of the original $f{u}_{graph}$ $\mathbb{G}$ must be at least $\mathtt{2}+\delta \left({\mathbb{G}}^{\prime }\right).$ Therefore, we have $\mathtt{2}+\delta \left({\mathbb{G}}^{\prime }\right)\le \mathbb{F}\left(\mathbb{G}\right).$    □

Proposition 5.

For a $f{u}_{graph}$ $\mathbb{G}$ with no $ful{l}_{ver},$ if $\mathbb{v}$ is a simplicial vertex, then $\mathbb{deg}\left(\mathbb{V}\right)+\mathtt{2}\le \mathbb{F}\left(\mathbb{G}\right).$

Proof. 

Let $\mathbb{G}$ be a $f{u}_{graph}$, where $\mathbb{v}$ is a simplicial vertex. Since $\mathbb{G}$ has no $ful{l}_{ver},$ $\mathbb{v}$ cannot strongly dominate the entire $f{u}_{graph}$ on its own. The $\mathbb{deg}\left(\mathbb{V}\right)$ represents its number of neighbors. To fully strongly dominate $\mathbb{G},$ at least $\mathtt{2}+\mathbb{deg}\left(\mathbb{V}\right)$ sets are required, as $\mathbb{G}$ and its neighbors cover part of the $f{u}_{graph},$ but additional sets are necessary to strongly dominate the remaining vertices. Therefore, the $f{u}_{cn}$ satisfies $\mathtt{2}+\mathbb{deg}\left(\mathbb{V}\right)\le \mathbb{F}\left(\mathbb{G}\right).$    □

Theorem 3.

Let $\mathbb{G}$ be a $f{u}_{graph}$ with max - deg $\Delta \left(\mathbb{G}\right),$ and let $\Pi$ be an $\mathbb{F}\left(\mathbb{G}\right)-ptn.$ If $\mathbb{X}\in \Pi ,$ then $\mathbb{X}$ is part of at most $\Delta \left(\mathbb{G}\right)+\mathtt{1}$ $f{u}_c.$

Proof. 

Consider $\Pi$ as an $\mathbb{F}\left(\mathbb{G}\right)-ptn$ of the $f{u}_{graph}$ $\mathbb{G},$ and let $\mathbb{X}\in \Pi$ be a subset within the partition. If $\mathbb{X}$ is a $st{d}_{set},$ it must consist of a single vertex and cannot participate in any $f{u}_c.$ Now, assuming that $\mathbb{X}$ is not a $st{d}_{set}$, let $\mathbb{x}\in \mathbb{V}$ be not strongly dominated by $\mathbb{X}.$ Any set in $\Pi$ that forms a $f{u}_c$ with $\mathbb{X}$ is restricted by the size of $\mathbb{N}\left[\mathbb{x}\right],$ which is $\mathbb{N}\left[\mathbb{x}\right]=\mathbb{deg}\left(\mathbb{x}\right)+\mathtt{1}.$ Given that the max - deg is $\mathbb{G}$ in $\Delta \left(\mathbb{G}\right),$ it follows that $\mathbb{X}$ can be part of at most $\Delta \left(\mathbb{G}\right)+\mathtt{1}$ $f{u}_c.$    □

Proposition 6.

If $\mathbb{G}$ is a $f{u}_{graph}$ of $\mathbb{n}\ge \mathtt{2}$ with max - deg $\Delta \left(\mathbb{G}\right)$ and a non-empty set of $is{o}_{ver}$ $\mathbb{I},$ then $\mathbb{F}\left(\mathbb{G}\right)=\mathtt{2}$ or $\mathbb{F}\left(\mathbb{G}\right)\le \Delta \left(\mathbb{G}\right)+\mathtt{2}.$

Proof. 

Let $\mathbb{G}$ be a $f{u}_{graph}$ of $\mathbb{n}\ge 2$ with max - deg $\Delta \left(\mathbb{G}\right)$ and a non-empty set $\mathbb{I}$ of $is{o}_{ver}.$ No set containing vertices from $\mathbb{I}$ can strongly dominate the $f{u}_{graph}.$ Therefore, the $f{u}_{cn}$ $\mathbb{F}\left(\mathbb{G}\right)$ must include at least two sets that strongly dominate the non-$is{o}_{ver}$, while also accounting for $\mathbb{I}.$ If the graph has only $is{o}_{ver}$ and one other set, the $f{u}_{cn}$ is exactly $\mathbb{F}\left(\mathbb{G}\right)=\mathtt{2}.$ However, if there are additional vertices with degrees up to $\Delta \left(\mathbb{G}\right),$ the $f{u}_{cn}$ is constrained by the maximum degree, resulting in $\mathbb{F}\left(\mathbb{G}\right)\le \Delta \left(\mathbb{G}\right)+\mathtt{2}.$ Thus, $\mathbb{F}\left(\mathbb{G}\right)$ is either $\mathtt{2}$ or $\mathbb{F}\left(\mathbb{G}\right)\le \Delta \left(\mathbb{G}\right)+\mathtt{2},$ depending on the presence of non-$is{o}_{ver}.$    □

Next, we conclude by identifying the $f{u}_{graphs}$ with small $f{u}_{cn}$. Specifically, we describe the structure of $f{u}_{graphs}$ $\mathbb{G}$ for which $\mathbb{F}\left(\mathbb{G}\right)\in \{\mathtt{1},\mathtt{2}\},$ as well as those isolate-free $f{u}_{graphs}$, where $\mathbb{F}\left(\mathbb{G}\right)=\mathtt{3}.$ These findings offer a clear characterization of such $f{u}_{graphs},$ highlighting their $f{u}_{cn}$ properties in terms of minimal values.

Theorem 4.

Let $\mathbb{G}$ be an $\mathbb{n}$ $f{u}_{graph}.$ Then,

1. 

$\mathbb{F}\left(\mathbb{G}\right)=\mathtt{1}$ if and only if $\mathbb{G}={\mathbb{K}}_{\mathtt{1}}$

2. 

$\mathbb{F}\left(\mathbb{G}\right)=\mathtt{2}$ if and only if $\mathbb{G}={\overline{\mathbb{K}}}_n$ or $\mathbb{G}={\mathbb{K}}_{\mathtt{2}}$ for $\mathbb{n}\ge \mathtt{2}.$

Proof. 

(1) It is an easy process to prove that the only $f{u}_{graph}$ with $f{u}_{cn}$, $\mathbb{F}\left(\mathbb{G}\right)=\mathtt{1}$ is trivial $f{u}_{graph}$, ${\mathbb{K}}_{\mathtt{1}}$.

(2)

Based on the previous discussions $\mathbb{F}\left(\mathbb{G}\right)=\mathtt{2}$, if for $\mathbb{n}\ge \mathtt{2}$, $\mathbb{G}={\mathbb{K}}_{\mathtt{2}}$ or $\mathbb{G}=\overline{{\mathbb{K}}_{\mathbb{n}}}$. On the other hand, suppose that $\mathbb{F}\left(\mathbb{G}\right)=\mathtt{2}.$ $\mathbb{G}$ must have $\mathbb{n}\ge \mathtt{2}$ according to $\left(1\right).$ For any $\mathbb{n}\ge \mathtt{2},$ the results hold if $\mathbb{G}={\mathbb{K}}_{\mathtt{2}}$ or $\mathbb{G}=\overline{{\mathbb{K}}_{\mathbb{n}}}.$ Therefore, we can conclude that $\mathbb{G}$ has at least one edge and $\mathbb{n}\ge \mathtt{3}.$

□

We can assume $\mathbb{G}$ is incomplete, since $\mathbb{F}\left({\mathbb{K}}_{\mathbb{n}}\right)=\mathbb{n}.$ Additionally, according to Proposition 3, $\mathbb{G}$ has no vertices of $\mathbb{n}-\mathtt{1}$ degree. There is a contradiction if $\mathbb{n}=\mathtt{3}$: ${\mathbb{P}}_{\mathtt{2}}\cup {\mathbb{K}}_{\mathtt{1}}$ and $\mathbb{F}\left(\mathbb{G}\right)=\mathtt{3}.$ Therefore, $\mathbb{n}\ge \mathtt{4}$ can be assumed. At least one $is{o}_{ver}$ exists in $\mathbb{G}$ according to Proposition 2. Let $\mathbb{I}$ represent the collection of $is{o}_{ver}$ of $\mathbb{G}$. With a min - deg of at least $\mathtt{1},$ ${\mathbb{G}}^{\prime }=\mathbb{G}[\mathbb{V}-\mathbb{I}]$ is a $f{u}_{sg}$ of $\mathbb{G}.$

Take note that there is not a minimal $st{d}_{set}$ of ${\mathbb{G}}^{\prime }$ for any of $\mathbb{I},$ ${\mathbb{V}}_{\mathtt{1}}.$ The complement ${\mathbb{V}}_{\mathtt{2}}=\mathbb{V}\left({\mathbb{G}}^{\prime }\right)-{\mathbb{V}}_{\mathtt{1}}$ is a substantially $st{d}_{set}$ of ${\mathbb{G}}^{\prime }$ according to Ore’s Theorem. While ${\mathbb{V}}_{\mathtt{1}}\cup \mathbb{I}$ and ${\mathbb{V}}_{\mathtt{2}}\cup \mathbb{I}$ are individually $st{d}_{set}$ of $\mathbb{G},$ note that none of $\mathbb{I},$ ${\mathbb{V}}_{\mathtt{1}},$ or ${\mathbb{V}}_{\mathtt{2}}$ strongly dominate $\mathbb{G}.$ Consequently, there is a contradiction: the partition $\{\mathbb{I},{\mathbb{V}}_{\mathtt{1}},{\mathbb{V}}_{\mathtt{2}}\}$ of $\mathbb{G}$ is a $\mathbb{f}-ptn$ of $\mathbb{G}$ with $\mathtt{3}.$

Theorem 5.

Let $\mathbb{G}$ be a $f{u}_{graph}$ with no $is{o}_{ver}$ having $\mathbb{n}$ vertices. If $\mathbb{G}={\mathbb{K}}_{\mathtt{3}}$ or $\mathbb{G}\cong {\mathbb{K}}_{\mathtt{1},\mathbb{n}-\mathtt{1}}$ for $\mathbb{n}\ge \mathtt{3},$ then $\mathbb{F}\left(\mathbb{G}\right)=\mathtt{3}.$

Proof. 

Using the previous context, it is straightforward to conclude that, for $\mathbb{n}\ge \mathtt{3}$, $\mathbb{F}\left({\mathbb{K}}_{\mathtt{3}}\right)=\mathbb{F}\left({\mathbb{K}}_{\mathtt{1},\mathbb{n}-\mathtt{1}}\right)=\mathtt{3}$. Now, suppose that $\mathbb{F}\left(\mathbb{G}\right)=\mathtt{3}$. From Theorem 4, we can obtain that $\mathbb{n}\ge \mathtt{3}$ and $\mathbb{G}\ne \overline{{\mathbb{K}}_{\mathbb{n}}}$. The graph $\mathbb{G}$ must belong to the set $\{{\mathbb{P}}_{\mathtt{3}}\cong {\mathbb{K}}_{\mathtt{1},\mathtt{2}},{\mathbb{K}}_{\mathtt{3}}\}$, since for these graphs $\mathbb{F}\left(\mathbb{G}\right)=\mathtt{3}$, as expected when $\mathbb{n}=\mathtt{3}$. Now, assume $\mathbb{n}\ge \mathtt{4}$. Proposition 2 implies that $\mathbb{G}$ must contain at least one $ful{l}_{ver}$, since it does not have any $is{o}_{ver}$. Let $\mathbb{a}$ be a $ful{l}_{ver}$ in $\mathbb{G}$. Proposition 3 states that if $\mathbb{a}$ is the only $ful{l}_{ver}$ in $\mathbb{G}$, then $\mathbb{F}\left(\mathbb{G}\right)\ge \mathtt{4}$. Now, examine the $f{u}_{sg}$ induced by $\mathbb{V}-\left\{\mathbb{a}\right\}$. Notice that $\mathbb{G}[\mathbb{V}-\{\mathbb{a}\left\}\right]$ is not strongly dominated by any vertex in $\mathbb{V}-\left\{\mathbb{a}\right\}$. Therefore, a possible $\mathbb{F}\left(\mathbb{G}\right)$-partition of $\mathbb{G}$ is ${\Pi }^{\prime }\cup \left\{\mathbb{a}\right\}$. It follows that $\mathbb{F}\left(\mathbb{G}\right[\mathbb{V}-\left\{\mathbb{a}\right\}\left]\right)=\mathtt{2}$, since $\mathbb{F}\left(\mathbb{G}\right)=\mathtt{3}$. Theorem 4 suggests that $\mathbb{G}[\mathbb{V}-\{\mathbb{a}\left\}\right]$ must be the complement of a complete $f{u}_{graph}$, that is ${\overline{\mathbb{K}}}_{\mathbb{n}-\mathtt{1}}$, for $\mathbb{n}\ge \mathtt{4}$. Thus, for $\mathbb{n}\ge \mathtt{4}$, $\mathbb{G}\cong {\mathbb{K}}_{\mathtt{1},\mathbb{n}-\mathtt{1}}$.    □

3. Fuzzy Coalition Graphs and Its Algorithm

In Section 3, we define $f{u}_{cg}$ and show that every fuzzy graph $\mathbb{G}$ corresponds to a $f{u}_{cg}$ $\mathbb{H}$. The $f{u}_{cg}$ is constructed by identifying subsets of vertices in G that form $f{u}_c$, where these subsets together act as $st{d}_{sets}$. We demonstrate that each vertex in $\mathbb{H}$ represents a $f{u}_c$ in $\mathbb{G}$, with edges connecting $fu{C}_{partners}$. This correspondence ensures that the key properties of strong domination are preserved in the $f{u}_{cg}$. Thus, the $f{u}_{cg}$ simplifies the analysis of $f{u}_c$ dynamics in $f{u}_{cg}$. We introduce an algorithm that facilitates this transformation, which identifies subsets of vertices within $\mathbb{G}$ that act as coalitions and ensures that the corresponding graph $\mathbb{H}$ preserves the properties necessary for coalition analysis.

Definition 3.

Let $\mathbb{G}:(\Pi ,\sigma ,\mu )$ be a $f{u}_{graph}$ with $\Pi =\{{\mathbb{V}}_{\mathtt{1}},{\mathbb{V}}_{\mathtt{2}},\cdots ,{\mathbb{V}}_{\mathbb{k}}\}$. The fuzzy coalition graph $\mathbb{FG}(\mathbb{G},\Pi )$ of $\mathbb{G}$ is the $f{u}_{graph}$ with $\{{\mathbb{V}}_{\mathtt{1}},{\mathbb{V}}_{\mathtt{2}},\cdots ,{\mathbb{V}}_{\mathbb{k}}\}$, where $\sigma \left({\mathbb{V}}_{\mathbb{k}}\right)=\sigma \left({\mathbb{v}}_{\mathbb{i}}\right)\wedge \sigma \left({\mathbb{v}}_{\mathbb{j}}\right)$, corresponding one-to-one with the sets of $\Pi ,$ and the two vertices ${\mathbb{V}}_{\mathbb{i}}$ and ${\mathbb{V}}_{\mathbb{j}}$ are adjacent in $\mathbb{FG}(\mathbb{G},\Pi )$ if and only if the sets ${\mathbb{V}}_{\mathbb{i}}$ and ${\mathbb{V}}_{\mathbb{j}}$ are $fu{C}_{partners}$ in $\Pi$; that is, neither ${\mathbb{V}}_{\mathbb{i}}$ nor ${\mathbb{V}}_{\mathbb{j}}$ are $st{d}_{sets}$ of $\mathbb{G},$ but ${\mathbb{V}}_{\mathbb{i}}\cup {\mathbb{V}}_{\mathbb{j}}$ is a $st{d}_{set}$ of $\mathbb{G}.$

Example 2.

The complete $f{u}_{graph}$ ${\mathbb{K}}_{\mathbb{n}}$ has exactly one $\mathbb{f}-ptn$, namely its singleton partition ${\Pi }_{\mathtt{1}}$, for which $\mathbb{FG}({\mathbb{K}}_{\mathbb{n}};{\Pi }_{\mathtt{1}})\cong \overline{{\mathbb{K}}_{\mathbb{n}}}$. That is, the $f{u}_c$ of ${\mathbb{K}}_{\mathbb{n}}$ with its singleton partition is isomorphic to the complement $\overline{{\mathbb{K}}_{\mathbb{n}}}$ of ${\mathbb{K}}_{\mathbb{n}}$. The singleton $\mathbb{f}-ptn$ ${\Pi }_{\mathtt{1}}$ of the cycle ${\mathbb{C}}_{\mathtt{5}}$ gives $\mathbb{FG}({\mathbb{C}}_{\mathtt{5}};{\Pi }_{\mathtt{1}})\cong {\mathbb{C}}_{\mathtt{5}}$ refer in Figure 2, while the $\mathbb{f}-ptn$ ${\Pi }_{\mathtt{2}}=\{\{{\mathbb{v}}_{\mathtt{1}},{\mathbb{v}}_{\mathtt{2}}\},\{{\mathbb{v}}_{\mathtt{3}},{\mathbb{v}}_{\mathtt{4}}\},\left\{{\mathbb{v}}_{\mathtt{5}}\right\}\}$ of ${\mathbb{C}}_{\mathtt{5}}$ results in $\mathbb{FG}({\mathbb{C}}_{\mathtt{5}};{\Pi }_{\mathtt{2}})\cong {\mathbb{K}}_{\mathtt{3}}$ refer in Figure 3.

Theorem 6.

For any $f{u}_{graph}$ $\mathbb{G},$ there is a $f{u}_{graph}$ $\mathbb{H}$ and some $\mathbb{f}-ptn$ $\Pi$ of $\mathbb{H},$ such that $\mathbb{FG}(\mathbb{H},\Pi )\simeq \mathbb{G}.$

Proof. 

Let $\mathbb{G}$ be a $f{u}_{graph}$ with the $\mathbb{V}\left(\mathbb{G}\right)=\{{\mathbb{v}}_{\mathtt{1}},{\mathbb{v}}_{\mathtt{2}},\cdots ,{\mathbb{v}}_{\mathbb{n}}\}\cup \{{\mathbb{u}}_{\mathtt{1}},{\mathbb{u}}_{\mathtt{2}},\cdots ,{\mathbb{u}}_{\mathbb{t}}\},$ where each vertex ${\mathbb{v}}_{\mathbb{i}}$ has a degree at least $\mathtt{1},$ and each vertex ${\mathbb{u}}_{\mathbb{i}}$ is an isolate. Let ${\mathbb{G}}^{\prime }=\mathbb{G}-\{{\mathbb{u}}_{\mathtt{1}},{\mathbb{u}}_{\mathtt{2}},\cdots ,{\mathbb{u}}_{\mathbb{t}}\},$ $\mathbb{E}\left({\mathbb{G}}^{\prime }\right)=\mathbb{E}\left(\mathbb{G}\right)=\{{\mathbb{e}}_{\mathtt{1}},{\mathbb{e}}_{\mathtt{2}},\cdots ,{\mathbb{e}}_{\mathbb{m}}\},$ and $\mathbb{E}\left(\overline{\mathbb{G}}\right)=\{{\mathbb{c}}_{\mathtt{1}},{\mathbb{c}}_{\mathtt{2}},\cdots ,{\mathbb{c}}_{\mathbb{p}}\},$ where $\mathbb{m}+\mathbb{p}=\left({\displaystyle \genfrac{}{}{0pt}{}{\mathbb{n}+\mathbb{t}}{\mathtt{2}}}\right).$ We will construct a $f{u}_{graph}$ $\mathbb{H}$ and a $\mathbb{f}-ptn$ $\Pi$ of $\mathbb{H},$ such that the $f{u}_c$ graph of $\mathbb{H}$ and $\Pi$ is a isomorphic to $\mathbb{G},$ $\mathbb{FG}(\mathbb{H},\Pi )\simeq \mathbb{G}.$

We begin the construction of $\mathbb{H}$ with a complete $f{u}_{graph}$ ${\mathbb{K}}_{\mathbb{n}},$ whose vertices are labeled $\{{\mathbb{v}}_{\mathtt{1}},{\mathbb{v}}_{\mathtt{2}},\cdots ,{\mathbb{v}}_{\mathbb{n}}\}$ corresponding to the $\mathbb{n}$ non-$is{o}_{ver}$ of $\mathbb{G}.$ We will refer to these as the $\mathbb{n}$ base vertices of $\mathbb{H},$ since we will add additional vertices to $\mathbb{H}.$ Our partition $\Pi$ of $\mathbb{H}$ begins as a singleton partition, and we denote the sets of $\Pi$ as ${\mathbb{V}}_{\mathbb{i}}=\left\{{\mathbb{v}}_{\mathbb{i}}\right\},$ for each ${\mathbb{v}}_{\mathbb{i}}\in \mathbb{V}\left({\mathbb{G}}^{\prime }\right).$ We will add to the sets in the partition as we build $\mathbb{H}.$

For each edge ${\mathbb{e}}_{\mathbb{i}}\in \mathbb{E}\left(\mathbb{G}\right),$ we proceed as follows. Let ${\mathbb{e}}_{\mathbb{i}}={\mathbb{v}}_{\mathbb{j}}{\mathbb{v}}_{\mathbb{k}}.$

1.

Add two new vertices ${\mathbb{x}}_{\mathbb{i},\mathbb{j}}$ and ${\mathbb{x}}_{\mathbb{i},\mathbb{k}}$ to $\mathbb{H}.$

2.

Add edges between ${\mathbb{x}}_{\mathbb{i},\mathbb{j}}$ and all of the base vertices except for ${\mathbb{v}}_{\mathbb{j}}.$ Similarly, add edges between ${\mathbb{x}}_{\mathbb{i},\mathbb{k}}$ and all of the base vertices except for ${\mathbb{v}}_{\mathbb{k}}.$ Thus, each of the vertices ${\mathbb{x}}_{\mathbb{i},\mathbb{j}}$ and ${\mathbb{x}}_{\mathbb{i},\mathbb{k}}$ has degree $\mathbb{n}-\mathtt{1}$ in $\mathbb{H}.$

3.

Add vertex ${\mathbb{x}}_{\mathbb{i},\mathbb{j}}$ to the set ${\mathbb{V}}_{\mathbb{k}}$ and add vertex ${\mathbb{x}}_{\mathbb{i},\mathbb{k}}$ to the set ${\mathbb{V}}_{\mathbb{j}}.$

For each edge ${\mathbb{c}}_{\mathbb{i}}={\mathbb{v}}_{\mathbb{j}}{\mathbb{v}}_{\mathbb{k}}\in \mathbb{E}\left(\overline{\mathbb{G}}\right),$ we proceed as follows.

4.

Add one new vertex ${\mathbb{a}}_{\mathbb{j},\mathbb{k}}$ to $\mathbb{H}.$

5.

Add edges between ${\mathbb{a}}_{\mathbb{j},\mathbb{k}}$ and all of the base vertices except for ${\mathbb{v}}_{\mathbb{j}}$ and ${\mathbb{v}}_{\mathbb{k}}.$ Thus, vertex ${\mathbb{a}}_{\mathbb{j},\mathbb{k}}$ has degree $\mathbb{n}-\mathtt{2}$ in $\mathbb{H}.$

6.

Add the vertex ${\mathbb{a}}_{\mathbb{j},\mathbb{k}}$ to any set of the partition $\Pi$ except for ${\mathbb{V}}_{\mathbb{j}}$ and ${\mathbb{V}}_{\mathbb{k}}.$

Finally, if ${\mathbb{u}}_{\mathbb{i}}$ is isolated in $\mathbb{G},$ then it necessarily corresponds to a strongly dominating vertex in $\mathbb{H}.$ Thus, we conclude our construction of $\mathbb{H}$ by adding $\mathbb{t}$ vertices to $\mathbb{H}$ labeled ${\mathbb{u}}_{\mathtt{1}},{\mathbb{u}}_{\mathtt{2}},\cdots ,{\mathbb{u}}_{\mathbb{t}}$ corresponding to the isolates in $\mathbb{G}.$ Then, add edges such that each of these vertices is a strongly dominating vertex in $\mathbb{H}.$ We extend $\Pi$ so that each ${\mathbb{u}}_{\mathbb{i}}$ is in a singleton set ${\mathbb{U}}_{\mathbb{i}}=\left\{{\mathbb{u}}_{\mathbb{i}}\right\}$ of $\Pi$ for $\mathbb{i}\in \left[\mathbb{t}\right].$

7.

Each edge in $\mathbb{H}$ is connected to either at least one base vertex or a strongly dominating vertex.

8.

No set ${\mathbb{V}}_{\mathbb{j}}$ in the partition is a $st{d}_{set}$ of $\mathbb{H}.$ This follows since the degree of ${\mathbb{v}}_{\mathbb{i}}$ in $\mathbb{G}$ is at least one, in which case there exists at least one edge ${\mathbb{e}}_{\mathbb{i}}={\mathbb{v}}_{\mathbb{j}}{\mathbb{v}}_{\mathbb{k}}$ incident to ${\mathbb{v}}_{\mathbb{j}},$ and therefore no vertex in ${\mathbb{V}}_{\mathbb{j}}$ can strongly dominate ${\mathbb{u}}_{\mathbb{i},\mathbb{j}}$ in $\mathbb{H}.$

9.

If ${\mathbb{v}}_{\mathbb{j}}{\mathbb{v}}_{\mathbb{k}}$ is an edge in $\mathbb{G},$ then ${\mathbb{V}}_{\mathbb{j}}$ and ${\mathbb{V}}_{\mathbb{k}}$ form a coalition in $\mathbb{H}$ because the base vertices ${\mathbb{v}}_{\mathbb{j}}$ and ${\mathbb{v}}_{\mathbb{k}}$ collectively strongly dominate every vertex in $\mathbb{H}.$

10.

If ${\mathbb{v}}_{\mathbb{j}}{\mathbb{v}}_{\mathbb{k}}$ is not an edge in $\mathbb{G},$ then ${\mathbb{V}}_{\mathbb{j}}$ and ${\mathbb{V}}_{\mathbb{k}}$ do not form a $f{u}_c$ in $\mathbb{H}$ because of the vertex ${\mathbb{a}}_{\mathbb{j},\mathbb{k}}$ which is not adjacent to either ${\mathbb{v}}_{\mathbb{j}}$ or ${\mathbb{v}}_{\mathbb{k}},$ and is thus not strongly dominated by any vertex in ${\mathbb{V}}_{\mathbb{j}}\cup {\mathbb{V}}_{\mathbb{k}}.$

11.

Each ${\mathbb{u}}_{\mathbb{i}}$ is a strongly dominating vertex in $\mathbb{H},$ so ${\mathbb{u}}_{\mathbb{i}}$ is an isolated in $\mathbb{G}.$

It follows that $\Pi$ is a $\mathbb{f}-ptn$ of $\mathbb{H}$ and that $\mathbb{FG}(\mathbb{H},\Pi )\simeq \mathbb{G}$    □

Proposition 7.

For every $f{u}_{graph}$ $\mathbb{G},$ there is a $f{u}_{graph}$ $\mathbb{H}$ and a $\mathbb{f}-ptn$ $\Pi =\{{\mathbb{V}}_{\mathtt{1}},{\mathbb{V}}_{\mathtt{2}},\cdots ,{\mathbb{V}}_{\mathbb{k}}\}$ of $\mathbb{H},$ such that $\mathbb{FG}(\mathbb{G},\Pi )\simeq \mathbb{G}$ and every set ${\mathbb{V}}_{\mathbb{i}}\in \Pi$ is an independent set.

Proof. 

Using Theorem 6, consider $\Pi =\{{\mathbb{V}}_{\mathtt{1}},{\mathbb{V}}_{\mathtt{2}},\cdots ,{\mathbb{V}}_{\mathbb{n}},{\mathbb{U}}_{\mathtt{1}},{\mathbb{U}}_{\mathtt{2}},\cdots ,{\mathbb{U}}_{\mathbb{t}}\}$ as the $\mathbb{f}-ptn$ of the $f{u}_{graph}$ $\mathbb{H}$ such that $\mathbb{FG}(\mathbb{H},\Pi )\simeq \mathbb{G}.$ Let ${\mathbb{H}}^{\prime }$ be the $f{u}_{graph}$ constructed from the $f{u}_{graph}$ $\mathbb{H}$ by deleting all edges between two vertices in the same set ${\mathbb{V}}_{\mathbb{i}}$ in $\Pi ,$ for all $\mathtt{1}\le \mathbb{i}\le \mathbb{k}.$ Let ${\Pi }^{\prime }=\{{\mathbb{V}}_{\mathtt{1}}^{\prime },{\mathbb{V}}_{\mathtt{2}}^{\prime },\cdots ,{\mathbb{V}}_{\mathbb{k}}^{\prime }\}$ be the vertex partition of ${\mathbb{H}}^{\prime },$ where ∀$\mathtt{1}\le \mathbb{i}\le \mathbb{k},$ ${\mathbb{V}}_{\mathbb{i}}^{\prime }={\mathbb{V}}_{\mathbb{i}}.$ It is easy to see that ${\mathbb{V}}_{\mathbb{i}}\cup {\mathbb{V}}_{\mathbb{j}}$ forms a $f{u}_c$ in ${\mathbb{H}}^{\prime }.$ Therefore, $\mathbb{FG}({\mathbb{H}}^{\prime },\Pi )\simeq \mathbb{G}.$    □

Algorithm

The Algorithm 1 for constructing the $f{u}_{cg}$ is outlined as follows:

(1)

We initialize an empty partition, $\Pi \ne \varnothing$ to group the coalition partners based on strong arcs.

(2)

Coalition partners are identified by evaluating pairs of vertex sets; if neither set forms a strongly dominating set alone but their union does, they are marked as coalition partners.

(3)

The $f{u}_{cp}$ $\Pi$ is then built by evaluating each vertex to determine if it can strongly dominate individually or if it requires a coalition.

(4)

The $f{u}_{cg},\mathbb{FG}(\mathbb{G},\Pi )$ is constructed by adding edges between coalition partners who together form a strongly dominating set.

Algorithm 1 Algorithm for constructing the $f{u}_{cg}$

Input:$f{u}_{graph}$$\mathbb{G}$ with vertex set $\mathbb{V}$. Output:$f{u}_{cg}$ $\mathbb{F}\mathbb{G}(\mathbb{G},\Pi )$, where $\Pi =\{{\mathbb{V}}_{\mathtt{1}},{\mathbb{V}}_{\mathtt{2}},\cdots ,{\mathbb{V}}_{\mathbb{k}}\}$ is the $f{u}_{cp}$. Initialize an empty partition $\Pi =\varnothing$. Initialize the $f{u}_{graph}$ $\mathbb{F}\mathbb{G}(\mathbb{G},\Pi )$. Step 1: Identify $fu{C}_{partners}$ (a) For each pair of vertex sets ${\mathbb{V}}_{\mathbb{I}},{\mathbb{V}}_{\mathbb{j}}\subseteq \mathbb{V}$ (where ${\mathbb{V}}_{\mathbb{I}}\ne \mathbb{V}\mathbb{j}$): i. Check if ${\mathbb{V}}_{\mathbb{I}}$ and ${\mathbb{V}}_{\mathbb{j}}$ are not $st{d}_{set}$ individually. ii. If ${\mathbb{V}}_{\mathbb{I}}\cup {\mathbb{V}}_{\mathbb{j}}$ forms a $st{d}_{set}$: Mark ${\mathbb{V}}_{\mathbb{I}}$ and ${\mathbb{V}}_{\mathbb{j}}$ as $fu{C}_{partners}$. Step 2: Construct$f{u}_{cp}$ $\Pi$ (a) For each vertex $\mathbb{v}\in \mathbb{V}$: i. If $\left\{\mathbb{v}\right\}$ is a $st{d}_{set}$, add $\left\{\mathbb{v}\right\}$ as a singleton set ${\mathbb{V}}_{\mathbb{I}}$ to $\Pi$. ii. Otherwise, find a set ${\mathbb{V}}_{\mathbb{j}}\subseteq \mathbb{V}$ such that $\left\{\mathbb{v}\right\}\cup {\mathbb{V}}_{\mathbb{j}}$ forms a ${std}_{set}$. Combine $\left\{\mathbb{v}\right\}$ with ${\mathbb{V}}_{\mathbb{j}}$ into a set, and add it to the partition $\Pi$. (b) Repeat this process to ensure $\Pi$ is the largest possible partition, where all subsets are either $st{d}_{set}$ or $f{u}_c$. Step 3: Build the$f{u}_{cg}$$\mathbb{F}\mathbb{G}(\mathbb{G},\Pi )$ (a) For each pair of sets ${\mathbb{V}}_{\mathbb{I}},{\mathbb{V}}_{\mathbb{j}} \in \Pi$: i. Check if ${\mathbb{V}}_{\mathbb{I}}$ and ${\mathbb{V}}_{\mathbb{j}}$ are $fu{C}_{partners}$ (i.e., ${\mathbb{V}}_{\mathbb{I}}\cup {\mathbb{V}}_{\mathbb{j}}$ forms a $st{d}_{set}$). ii. If true, add an edge between ${\mathbb{V}}_{\mathbb{I}}$ and ${\mathbb{V}}_{\mathbb{j}}$ in $\mathbb{F}\mathbb{G}(\mathbb{G},\Pi )$. Return the $f{u}_{cg}$ $\mathbb{F}\mathbb{G}(\mathbb{G},\Pi )$, along with the $f{u}_{cp}$ $\Pi$.

4. Application for Fuzzy Coalition Graphs

The concept of a $f{u}_{cg}$ can be applied in real-life scenarios where multiple agents (e.g., companies, individuals, or nations) need to form coalitions to achieve a common goal, particularly when individual agents cannot fully dominate or control the system alone. In a smart city, various entities (such as traffic management systems, power grids, water supply systems, hospitals, etc.) are interconnected via an IoT (Internet of Things) infrastructure. These systems are highly vulnerable to cyber-attacks, and no single entity can entirely safeguard the city’s infrastructure from these threats. Collaboration is essential.

Let us consider a $f{u}_{cg}$ for the cybersecurity collaboration of a smart city. The graph involves seven critical infrastructures that need to work together to protect the city from cyber threats. We will label these infrastructures as vertices and define the strong arcs (representing strong cybersecurity partnerships) between them.

Consider critical systems as vertices: ${\mathbb{v}}_{\mathtt{1}}$: transportation system, ${\mathbb{v}}_{\mathtt{2}}$: energy grid, ${\mathbb{v}}_{\mathtt{3}}$: water supply system, ${\mathbb{v}}_{\mathtt{4}}$: healthcare system, ${\mathbb{v}}_{\mathtt{5}}$: public safety system, ${\mathbb{v}}_{\mathtt{6}}$: waste management system, ${\mathbb{v}}_{\mathtt{7}}$:communication network.

Consider cybersecurity alliances as strong arcs:$({\mathbb{v}}_{\mathtt{1}},{\mathbb{v}}_{\mathtt{2}})$: the transportation and energy grid share cybersecurity resources, $({\mathbb{v}}_{\mathtt{2}},{\mathbb{v}}_{\mathtt{3}})$: the energy grid and water supply system have a strong cybersecurity partnership, $({\mathbb{v}}_{\mathtt{3}},{\mathbb{v}}_{\mathtt{4}})$: the water supply and healthcare systems collaborate, $({\mathbb{v}}_{\mathtt{4}},{\mathbb{v}}_{\mathtt{5}})$: the healthcare and public safety systems work together for cyber defense,$({\mathbb{v}}_{\mathtt{5}},{\mathbb{v}}_{\mathtt{6}})$: the public safety and waste management systems are strongly connected, $({\mathbb{v}}_{\mathtt{6}},{\mathbb{v}}_{\mathtt{7}})$: the waste management and communication networks are partners,$({\mathbb{v}}_{\mathtt{7}},{\mathbb{v}}_{\mathtt{1}})$: the communication network and transportation system, $({\mathbb{v}}_{\mathtt{1}},{\mathbb{v}}_{\mathtt{3}})$: the transportation and water supply system, $({\mathbb{v}}_{\mathtt{2}},{\mathbb{v}}_{\mathtt{4}})$: the energy grid and healthcare system, $({\mathbb{v}}_{\mathtt{4}},{\mathbb{v}}_{\mathtt{6}})$: the healthcare and waste management system.

The membership values for each system in the smart city’s cybersecurity coalition are assigned based on their relative importance, vulnerability to cyber-attacks, and interdependence with other systems. The transportation system $\left({\mathbb{v}}_{\mathtt{1}}\right)$ is assigned a value of $\mathtt{0}.\mathtt{75}$ due to its crucial role in the city’s functioning and its connections to other critical systems like energy and communication. The energy grid $\left({\mathbb{v}}_{\mathtt{2}}\right)$ is given the highest value of $\mathtt{0}.\mathtt{85}$, reflecting its central role in the city’s infrastructure. The water supply system $\left({\mathbb{v}}_{\mathtt{3}}\right)$, while important, has a slightly lower value of $\mathtt{0}.\mathtt{70}$, as its direct impact on the city is less severe compared to systems like energy or transportation. The healthcare system $\left({\mathbb{v}}_{\mathtt{4}}\right)$ is assigned $\mathtt{0}.\mathtt{65}$, given its importance in public health and its vulnerability to cyber threats. The public safety system $\left({\mathbb{v}}_{\mathtt{5}}\right)$ receives a value of $\mathtt{0}.\mathtt{80}$, recognizing its role in maintaining safety, but having less interconnectedness than more critical systems. The waste management system $\left({\mathbb{v}}_{\mathtt{6}}\right)$ is assigned $\mathtt{0}.\mathtt{57}$, as its failure would not immediately affect core city functions, though it still plays a vital role in maintaining public health. Finally, the communication network $\left({\mathbb{v}}_{\mathtt{7}}\right)$, crucial for coordinating all systems, is given a value of $\mathtt{0}.\mathtt{90}$, reflecting its importance in facilitating communication during crises. These values guide the formation of coalitions, ensuring the city’s infrastructure collaborates effectively to mitigate cybersecurity risks.

The membership values for the strong arcs between the systems in the smart city’s cybersecurity setup reflect how closely these systems need to work together. The arc between the transportation system $\left({\mathbb{v}}_{\mathtt{1}}\right)$ and energy grid $\left({\mathbb{v}}_{\mathtt{2}}\right)$ is given a high value of $\mathtt{0}.\mathtt{75}$, as any problem in one system can directly affect the other. The connection between the transportation system $\left({\mathbb{v}}_{\mathtt{1}}\right)$ and the communication network $\left({\mathbb{v}}_{\mathtt{7}}\right)$ has a value of $\mathtt{0}.\mathtt{70}$, showing a strong but slightly less critical collaboration. The water supply system $\left({\mathbb{v}}_{\mathtt{3}}\right)$ and healthcare system $\left({\mathbb{v}}_{\mathtt{4}}\right)$ have a value of $\mathtt{0}.\mathtt{65}$, as both are important but their cooperation is not as urgent as other systems. The healthcare system $\left({\mathbb{v}}_{\mathtt{4}}\right)$ and public safety system $\left({\mathbb{v}}_{\mathtt{5}}\right)$ have a value of $\mathtt{0}.\mathtt{60}$, indicating necessary but less intense cooperation. The arc between healthcare $\left({\mathbb{v}}_{\mathtt{4}}\right)$ and waste management $\left({\mathbb{v}}_{\mathtt{6}}\right)$ is set at $\mathtt{0}.\mathtt{57}$, showing moderate cooperation, as waste management plays a role in public health but is not as critical. Finally, the arc between waste management $\left({\mathbb{v}}_{\mathtt{6}}\right)$ and the communication network $\left({\mathbb{v}}_{\mathtt{7}}\right)$ has the lowest value of $\mathtt{0}.\mathtt{55}$, indicating a lower but still important connection. These values show the varying levels of cooperation needed for strong cybersecurity in the city’s infrastructure given in Figure 4.

The graph shows which coalitions are connected and can collaborate for overall security (

Table 2. Fuzzy coalition graph and analysis for seven critical systems.

Vertices	Edges	${\mathit{fuC}}_{\mathit{partners}}$	${\mathit{fuC}}_{\mathit{set}}$	Connected Coalitions
${\mathbb{v}}_{\mathtt{1}}$	${\mathbb{v}}_{\mathtt{2}}$, ${\mathbb{v}}_{\mathtt{3}}$, ${\mathbb{v}}_{\mathtt{7}}$	$\{{\mathbb{v}}_{\mathtt{1}},{\mathbb{v}}_{\mathtt{3}}\}$	$\{{\mathbb{v}}_{\mathtt{1}},{\mathbb{v}}_{\mathtt{3}}\}$	$\{{\mathbb{v}}_{\mathtt{2}},{\mathbb{v}}_{\mathtt{4}}\}$, $\{{\mathbb{v}}_{\mathtt{5}},{\mathbb{v}}_{\mathtt{7}}\}$
${\mathbb{v}}_{\mathtt{2}}$	${\mathbb{v}}_{\mathtt{1}}$, ${\mathbb{v}}_{\mathtt{3}}$, ${\mathbb{v}}_{\mathtt{4}}$	$\{{\mathbb{v}}_{\mathtt{2}},{\mathbb{v}}_{\mathtt{4}}\}$	$\{{\mathbb{v}}_{\mathtt{2}},{\mathbb{v}}_{\mathtt{4}}\}$	$\{{\mathbb{v}}_{\mathtt{1}},{\mathbb{v}}_{\mathtt{3}}\}$, $\{{\mathbb{v}}_{\mathtt{5}},{\mathbb{v}}_{\mathtt{7}}\}$,$\left\{{\mathbb{v}}_{\mathtt{6}}\right\}$
${\mathbb{v}}_{\mathtt{3}}$	${\mathbb{v}}_{\mathtt{1}}$, ${\mathbb{v}}_{\mathtt{2}}$, ${\mathbb{v}}_{\mathtt{4}}$	$\{{\mathbb{v}}_{\mathtt{1}},{\mathbb{v}}_{\mathtt{3}}\}$	$\{{\mathbb{v}}_{\mathtt{1}},{\mathbb{v}}_{\mathtt{3}}\}$	$\{{\mathbb{v}}_{\mathtt{2}},{\mathbb{v}}_{\mathtt{4}}\}$, $\{{\mathbb{v}}_{\mathtt{5}},{\mathbb{v}}_{\mathtt{7}}\}$
${\mathbb{v}}_{\mathtt{4}}$	${\mathbb{v}}_{\mathtt{2}}$, ${\mathbb{v}}_{\mathtt{3}}$, ${\mathbb{v}}_{\mathtt{5}}$	$\{{\mathbb{v}}_{\mathtt{2}},{\mathbb{v}}_{\mathtt{4}}\}$, $\{{\mathbb{v}}_{\mathtt{4}},{\mathbb{v}}_{\mathtt{6}}\}$	$\{{\mathbb{v}}_{\mathtt{2}},{\mathbb{v}}_{\mathtt{4}}\}$	$\{{\mathbb{v}}_{\mathtt{1}},{\mathbb{v}}_{\mathtt{3}}\}$, $\{{\mathbb{v}}_{\mathtt{5}},{\mathbb{v}}_{\mathtt{7}}\}$, $\left\{{\mathbb{v}}_{\mathtt{6}}\right\}$
${\mathbb{v}}_{\mathtt{5}}$	${\mathbb{v}}_{\mathtt{4}}$, ${\mathbb{v}}_{\mathtt{6}}$, ${\mathbb{v}}_{\mathtt{7}}$	$\{{\mathbb{v}}_{\mathtt{5}},{\mathbb{v}}_{\mathtt{7}}\}$, $\{{\mathbb{v}}_{\mathtt{3}},{\mathbb{v}}_{\mathtt{5}}\}$	$\{{\mathbb{v}}_{\mathtt{5}},{\mathbb{v}}_{\mathtt{7}}\}$	$\{{\mathbb{v}}_{\mathtt{1}},{\mathbb{v}}_{\mathtt{3}}\}$, $\{{\mathbb{v}}_{\mathtt{2}},{\mathbb{v}}_{\mathtt{4}}\}$
${\mathbb{v}}_{\mathtt{6}}$	${\mathbb{v}}_{\mathtt{5}}$, ${\mathbb{v}}_{\mathtt{7}}$	$\{{\mathbb{v}}_{\mathtt{4}},{\mathbb{v}}_{\mathtt{6}}\}$	$\left\{{\mathbb{v}}_{\mathtt{6}}\right\}$	$\{{\mathbb{v}}_{\mathtt{2}},{\mathbb{v}}_{\mathtt{4}}\}$
${\mathbb{v}}_{\mathtt{7}}$	${\mathbb{v}}_{\mathtt{1}}$, ${\mathbb{v}}_{\mathtt{5}}$, ${\mathbb{v}}_{\mathtt{6}}$	$\{{\mathbb{v}}_{\mathtt{5}},{\mathbb{v}}_{\mathtt{7}}\}$	$\{{\mathbb{v}}_{\mathtt{5}},{\mathbb{v}}_{\mathtt{7}}\}$	$\{{\mathbb{v}}_{\mathtt{1}},{\mathbb{v}}_{\mathtt{3}}\}$, $\{{\mathbb{v}}_{\mathtt{2}},{\mathbb{v}}_{\mathtt{4}}\}$

Vertex: Lists the vertices representing the critical systems; Connected To: The vertices (systems) connected to this vertex; $fu{C}_{partners}$: Vertices that form a $f{u}_c$ with this vertex; $fu{C}_{set}$: The $f{u}_{cp}$ that this vertex belongs to; Connected Coalitions: Other coalitions that are connected to the coalition of this vertex.

).

$\{{\mathbb{v}}_{\mathtt{1}},{\mathbb{v}}_{\mathtt{3}}\}$ (transportation and water supply) are linked to $\{{\mathbb{v}}_{\mathtt{2}},{\mathbb{v}}_{\mathtt{4}}\}$ (energy grid and healthcare) and $\{{\mathbb{v}}_{\mathtt{5}},{\mathbb{v}}_{\mathtt{7}}\}$ (public safety and communication network). $\left\{{\mathbb{v}}_{\mathtt{6}}\right\}$ (waste management) is linked to $\{{\mathbb{v}}_{\mathtt{2}},{\mathbb{v}}_{\mathtt{4}}\}$ and $\{{\mathbb{v}}_{\mathtt{5}},{\mathbb{v}}_{\mathtt{7}}\}$, ensuring the entire network can respond to threats.

The $f{u}_{cp}$ $\Pi =\{\{{\mathbb{v}}_{\mathtt{1}},{\mathbb{v}}_{\mathtt{3}}\},\{{\mathbb{v}}_{\mathtt{2}},{\mathbb{v}}_{\mathtt{4}}\},\{{\mathbb{v}}_{\mathtt{5}},{\mathbb{v}}_{\mathtt{7}}\},\left\{{\mathbb{v}}_{\mathtt{6}}\right\}\}$.

The application of the $f{u}_c$ concept to a graph with seven vertices results in the identification of $fu{C}_{set}$s that reflect how smaller groups of systems form coalitions to dominate the entire network. However, based on the resulting $f{u}_{cg}$ and partition structure, the underlying relationships between these $fu{C}_{set}$s resemble the following graph structure. This happened because

• Four coalition sets emerged in which every set is strongly linked to the others.
• In $f{u}_{cg}$, every $fu{C}_{set}$ is adjacent to the other sets, similarly to the structure of Figure 5, which is a $f{u}_{cg}$ on $\mathtt{4}$ vertices, where every vertex is connected to any other vertex by an edge.

Thus, the coalition structure effectively forms the following graph due to the interconnected nature of the coalitions. Each system (or vertex) in the $f{u}_{cg}$ has edges connecting it to other system within the coalition.

In the fuzzy coalition partition $\{\{{\mathbb{v}}_{\mathtt{1}},{\mathbb{v}}_{\mathtt{3}}\},\{{\mathbb{v}}_{\mathtt{2}},{\mathbb{v}}_{\mathtt{4}}\},\{{\mathbb{v}}_{\mathtt{5}},{\mathbb{v}}_{\mathtt{7}}\},\left\{{\mathbb{v}}_{\mathtt{6}}\right\}\}$, each coalition is assigned a fuzzy membership value that reflects the strength of collaboration between critical systems in the smart city’s cybersecurity efforts. The coalition $\{{\mathbb{v}}_{\mathtt{1}},{\mathbb{v}}_{\mathtt{3}}\}$, representing the transportation and water supply systems, has a membership value of $\mathtt{0}.\mathtt{70}$, indicating a strong collaboration. The coalition $\{{\mathbb{v}}_{\mathtt{2}},{\mathbb{v}}_{\mathtt{4}}\}$, between the energy grid and healthcare systems, has a membership value of $\mathtt{0}.\mathtt{65}$, suggesting a moderately strong partnership. The coalition $\{{\mathbb{v}}_{\mathtt{5}},{\mathbb{v}}_{\mathtt{7}}\}$, formed by the public safety and communication network systems, has the highest membership value of $\mathtt{0}.\mathtt{80}$, reflecting a very strong relationship for cybersecurity efforts. The coalition $\left\{{\mathbb{v}}_{\mathtt{6}}\right\}$, representing the waste management system, has a membership value of $\mathtt{0}.\mathtt{57}$, indicating a weaker individual role in cybersecurity.

Additionally, the strong domination relationships between these coalitions are represented by membership values that quantify their influence on each other. For example, $\{{\mathbb{v}}_{\mathtt{1}},{\mathbb{v}}_{\mathtt{3}}\}$ strongly dominates $\{{\mathbb{v}}_{\mathtt{2}},{\mathbb{v}}_{\mathtt{4}}\}$ with a value of $\mathtt{0}.\mathtt{60}$, indicating a moderate influence of the transportation and water supply systems over the energy grid and healthcare systems. Similarly, $\{{\mathbb{v}}_{\mathtt{1}},{\mathbb{v}}_{\mathtt{3}}\}$ strongly dominates $\{{\mathbb{v}}_{\mathtt{5}},{\mathbb{v}}_{\mathtt{7}}\}$ with a value of $\mathtt{0}.\mathtt{70}$, showing a stronger impact on the public safety and communication systems. On the other hand, $\{{\mathbb{v}}_{\mathtt{2}},{\mathbb{v}}_{\mathtt{4}}\}$ strongly dominates $\{{\mathbb{v}}_{\mathtt{5}},{\mathbb{v}}_{\mathtt{7}}\}$ with a value of $\mathtt{0}.\mathtt{60}$, and $\{{\mathbb{v}}_{\mathtt{2}},{\mathbb{v}}_{\mathtt{4}}\}$ strongly dominates $\left\{{\mathbb{v}}_{\mathtt{6}}\right\}$ with a value of $\mathtt{0}.\mathtt{55}$, indicating moderate to weak dominations.

These fuzzy membership values can help model the interdependencies and vulnerabilities of the critical infrastructures, illustrating the degree of collaboration and influence between different systems, and identifying areas where cybersecurity partnerships could be optimized for better protection.

These coalitions can help the city optimally allocate cybersecurity resources. Instead of each system defending itself, these groups can share intelligence, threat detection, and countermeasures.

The Figure 5 shows the importance of collaboration across systems to achieve full coverage. No single system can protect the city on its own, but these coalitions ensure comprehensive cybersecurity.

5. Conclusions

In this research, we systematically explored the concept of $f{u}_c$ and corresponding $f{u}_c$ graphs. Beginning with the foundation of $f{u}_c$, we defined how disjoint vertex sets in a $f{u}_{graph}$ can act as coalition partners, forming $st{d}_{sets}$ through their union. This concept of $f{u}_c$ led to the introduction of $f{u}_{cp}$ ($\mathbb{f}-ptn$), where each set either strongly dominates individually or forms coalitions with others to achieve strong domination. Building on this, we introduced the $f{u}_{cg}$ $\mathbb{FG}(\mathbb{G},\Pi ),$ where each vertex represents a coalition partner from a $\mathbb{f}-ptn.$ We demonstrated that adjacency between vertices in the $f{u}_{cg}$ occurs if and only if the corresponding sets form a strongly dominating coalition in the original $f{u}_{graph}$. Moreover, we showed that for any given $f{u}_{graph}$ $\mathbb{G},$ there exists a graph $\mathbb{H}$ and a $\mathbb{f}-ptn$ such that $\mathbb{FG}(\mathbb{G},\Pi )$ is isomorphic to $\mathbb{G}.$ These results illustrate the broad applicability of $f{u}_{cg}$, showing how they provide a new perspective for understanding the structure of $f{u}_{graphs}$, particularly in settings where $s{t}_{dom}$ and $f{u}_c$ are key. The transition from the concept of $f{u}_c$ to $f{u}_{cg}$ offers a powerful tool for representing and analyzing networks, social structures, and systems characterized by uncertainty and interdependence. This research lays the groundwork for future applications in areas such as decision-making, communication networks, and fuzzy control systems.