A Study on Fuzzy Resolving Domination Sets and Their Application in Network Theory

Abstract

Consider a simple connected fuzzy graph (FG) G and consider an ordered fuzzy subset H = {(u₁, σ(u₁)), (u₂, σ(u₂)), …(u_k, σ(u_k))}, |H| ≥ 2 of a fuzzy graph; then, the representation of σ − H is an ordered k-tuple with regard to H of G. If any two elements of σ − H do not have any distinct representation with regard to H, then this subset is called a fuzzy resolving set (FRS) and the smallest cardinality of this set is known as a fuzzy resolving number (FRN) and it is denoted by Fr(G). Similarly, consider a subset S such that for any u∈S, ∃v∈V − S, then S is called a fuzzy dominating set only if u is a strong arc. Now, again consider a subset F which is both a resolving and dominating set, then it is called a fuzzy resolving domination set (FRDS) and the smallest cardinality of this set is known as the fuzzy resolving domination number (FRDN) and it is denoted by F_γr(G). We have defined a few basic properties and theorems based on this FRDN and also developed an application for social network connection. Moreover, a few related statements and illustrations are discussed in order to strengthen the concept.

1. Introduction

A fuzzy graph (FG) was lengthened to an intuitionistic fuzzy graph (IFG) by Atanassov [1]. In 2000, Chartrand introduced an upper dimension of graphs [2] and resolvability in graphs [3], which was later extended to resolvability in fuzzy graphs by Shanmugapriya and Jiny. Gani and Chandrasekaran developed domination in FG [4,5] and they also combined with many others and carried out extensive research in the field of fuzzy graphs, such as investigations of the order and size of a fuzzy graph and their strong and weak domination [6]. In recent years, a resolving restrained domination set of graphs [7] was developed by Gerald and Monsanto Helen in 2021. Moderson et al. [8,9] developed a fuzzy graph with a wide range of applications. The fuzzy graph theory was derived from the fuzzy set by Kauffman in 1973, and Rosenfled [10] extended the fuzzy graph theory in 1975. Shanmugapriya and Jiny have also introduced a modified FRN [11] and the properties of FRSs [12]. In this article, we will find an FRDS in order to evaluate the properties of FRSs. A fuzzy set is more prone to uncertainty and vagueness in the set. Later, many real-life applications are developed using this concept. Graph theory preceded fuzzy sets, which were largely developed in the 21st century. Fuzzy sets were evaluated by Zadeh [13] in 1965, whereas graph theory was proclaimed by Euler in 1936. A fuzzy system was developed to be used in the case of uncertainty and vagueness in many real-life situations. Later, it was developed by many researchers. A fuzzy graph is represented by giving a membership value to all the vertices and edges of a graph [14]. An application of fuzzy sets includes traffic light problems, network theory, mathematical biology, identification of drugs, cancer detection, examination schedules, and image capturing. Fuzzy concepts find wide-ranging applications in multi-criteria decision making [15,16,17]. Many applications are developed using the set clustering parameter. There exists many different pieces of information in real-world problems that can be interpreted using different types of graphs such as a fuzzy graph, an IFG, and an NFG. The membership value is a specific single value that lies between zero and one. Representing a problem as a graph gives more convenient results and is very efficient in solving the problem quickly and also at a perfect time. Graph theory is very useful for representing a complicated problem in a simple graph using nodes and arcs. We have tried applying an FRS and FDS in combination and evaluated an FRDS in this paper.

Let there be a finite, simple, undirected, and connected FG. Let us consider the vertices $v_1,v_2,\dots v_n$; the diameter is denoted as $diam\left(v_1,v_n\right)$. The significance of the relationship between them and the greatest weight of all connections between them is denoted by what we express in this work. Slater, Harry et al. introduce the notion of resolving numbers in graph theory. The applications of resolving a set of graphs are drug discovery, robot navigation, coin-weighing solutions, network discovery, and mastermind games.

The FRDS has been newly developed using FRSs and FDSs and we are hopeful that this area of research will be fruitful and and that this concept will be helpful in solving many problems. We have solved a problem based on social network theory to find an effective set.

In this article, we define a new fuzzy resolving parameter, such as the FRDS and its number, the fuzzy resolving restrained domination set (FRRDS). We have also defined the properties of FRDSs and an application in network theory using FRDSs.

2. Preliminaries

In this section, we have provided a basic definition to deal with the fuzzy resolving domination set.

Definition 1.

$G\left(V,\sigma ,\mu \right)$is an FG where µ is a symmetric fuzzy relation on  $\sigma : V\to \left[0, 1\right]$ and  $\mu :V\times V\to \left[0, 1\right]$ such that

$$\mu \left(u,v\right)\le \sigma \left(u\right)\Lambda \sigma \left(v\right), \forall u,v\in V$$

Definition 2.

An FG  $G$ is regarded as a complete fuzzy graph (CFG) if  $\mu \left(u,v\right)=\sigma \left(u\right)\Lambda \sigma \left(v\right)\forall u,v\in V$ and  ${\mu }^{\infty }\left(u,v\right)=\mu \left(u,v\right), \forall u,v\in V.$

Definition 3.

The strength or membership value of the path’s weakest edge is referred to as the path’s weight. The greatest weight among all possible pathways between  $v_1$ and  $v_n$ is given by  ${\mu }^{\infty }\left(v_1,v_n\right)$ and it represents the weight of the connectedness between  $v_1$ and  $v_n$.

Definition 4.

An  $n\times n$ matrix, which is regarded as  $X_{ij}=\mu \left(v_i,v_j\right)$ for  $i\ne j$ and  $X_{ij}=\sigma \left(v_i\right)$ when  $i=j$, is called an adjacency matrix  $A$ of  $G$.

Definition 5.

In a fuzzy graph, an arc  $\left(u,y\right)$ is regarded as a strong arc if  ${\mu }^{\infty }\left(u,y\right)=\mu \left(u,y\right)$.

Definition 6.

The representation of  $\left(z,\sigma \left(z\right)\right)ϵ\sigma -H=\left\{\left(u_{k+1},\sigma \left(u_{k+1}\right)\right),\left(u_{k+2},\sigma \left(u_{k+2}\right)\right),\dots \left(u_n,\sigma \left(u_n\right)\right)\right\}$ with regards to  $H$ is  $\left\{w\left(z_j,u_1\right),w\left(z_j,u_2\right),\dots w\left(z_j,u_k\right)\right\}$, where  $j=k+1, k+2,\dots n$ are written in row form. This matrix is called the fuzzy resolving matrix of order  $n-k\times k$ and it is represented as  $R_{n-k\times k}$.

Definition 7.

Consider an ordered fuzzy subset  $H=\left\{\left(u_1,\sigma \left(u_1\right)\right), \left(u_2, \sigma \left(u_2\right)\right),\dots \left(u_k,\sigma \left(u_k\right)\right)\right\}$, $\left|H\right|\ge 2$, and the representation of  $\left(z,\sigma \left(z\right)\right)ϵ\sigma -H=\left\{\left(u_{k+1},\sigma \left(u_{k+1}\right)\right),\left(u_{k+2},\sigma \left(u_{k+2}\right)\right),\dots \left(u_n,\sigma \left(u_n\right)\right)\right\}$ and  $\left\{w\left(z,u_1\right),w\left(z,u_2\right),\dots w\left(z,u_k\right)\right\}$, where  $w\left(z,y\right)$ is the importance of the relationship between  $z$ and  $y$. If each pair of elements in the fuzzy subset  $H$ has a different representation with regard to  $H$ of  $G$, the set is referred to as an FRS. The FRN is represented as  $Fr\left(G\right)$, the minimal cardinality of the FRS.

Definition 8.

A set  $K$ of vertices  $V$ is a dominating set if every vertex of  $V\backslash K$ is adjacent to any vertex of  $K$. The smallest cardinality of this set is called the fuzzy domination number (FDN) and it is denoted by  $F_{\gamma }\left(G\right).$

Definition 9.

Consider  $G$ a cycle and if a cycle has more than one weakest arc, it is described as having a fuzzy cycle.

Definition 10.

If any two elements of a fuzzy graph  $G$ have unique representations with regard to  $H$, the FRS of  $G$ is treated as a fuzzy super resolving set (FSRS), and we also write the FSRN, which is indicated by  $Sr\left(G\right)$, and the super resolving matrix, which is indicated by  $S_{n\times k}$, is the lowest cardinality of an FSRS of an FG  $G$.

3. Fuzzy Resolving Domination

Let $G$ be a simple connected FG and let $S$ be a subset of vertices from $G$ which is both fuzzy resolving and fuzzy dominating; this is known as a fuzzy resolving domination set (FRDS). The smallest cardinality of this set is defined as the fuzzy resolving domination number which is denoted by $F_{\gamma r}\left(G\right)$.

Let $G$ be a simple connected FG and any set $S\subseteq V\left(G\right)$ is strictly an FRDS of $G$ if it is an FRDS and $N_G\left(u\right){\displaystyle \cap }S\ne S$ for all $u\in V\left(G\right)\backslash S$. The smallest cardinality of this set is called a strictly fuzzy resolving domination number (SFRDN) and it is denoted by $F{S}_{\gamma r}\left(G\right)$.

Let $G$ be a connected FG. Let $F\subseteq V\left(G\right)$ be a fuzzy resolving restrained dominating set (FRRDS) of $G$ if $F$ is an FRDS and $F=V\left(G\right)$ or $\langle V\left(G\right)\backslash F\rangle$ has no single vertex and the smallest cardinality of this set is a fuzzy resolving restrained domination number which is denoted by $F{R}_{\gamma r}\left(G\right)$.

Let $F\subseteq V\left(G\right)$ be a fuzzy super resolving dominating set (FSRDS) of $G$ if $F$ is an FSRS and dominating set. The smallest cardinality of this set is called the fuzzy super resolving domination number (FSRDN) which is denoted by $F{S}_{\gamma r}\left(G\right)$.

3.1. Illustration

In the graph below (Figure 1), the connectedness matrix is given by

$$\left|\begin{array}{ccccc}0.7& 0.7& 0.6& 0.6& 0.6\\ 0.7& 0.8& 0.6& 0.8& 0.7\\ 0.6& 0.6& 0.6& 0.6& 0.6\\ 0.6& 0.8& 0.6& 1& 0.7\\ 0.6& 0.7& 0.6& 0.7& 0.9\end{array}\right|$$

The fuzzy resolving sets are $\left\{R_1,R_2\right\}, \left\{R_1,R_4\right\},\left\{R_1,R_5\right\} and \left\{R_2,R_4\right\}$, as they have no similar values.

Hence, $Fr\left(G\right)=2$.

In these sets, the fuzzy dominating sets are $\left\{R_1,R_2\right\} and \left\{R_2,R_4\right\}.$

Hence, $F_{\gamma }\left(G\right)=2$.

Therefore, $F_{\gamma r}\left(G\right)=2$.

For this graph, an FRRDS does not exist since $N_G\left(R_1\right){\displaystyle \cap }S=S$, corresponding to the set $\left\{R_2,R_4\right\}.$ In this graph, an FRRDN exists since $V-\left\{R_2,R_4\right\}$ has no isolated vertex. Also, the above two FRDSs are FSRDSs.

3.2. Remarks

Remark 1.

If G is a connected FG, then  $2\le F_{\gamma r}\le n-1$.

Proof. 

An FRS cannot be a single vertex or n-vertex as there will be no other vertices to check for a different representation with respect to the set.   □

Remark 2.

Let   $\left\{x,v\right\}\in F_{\gamma }$ and$\left\{x,v\right\}\in Fr$ if and only if   $\left\{x,v\right\}\in F_{\gamma r}$.

Proof. 

Let $F_{\gamma }, Fr$ and $F_{\gamma r}$ be the fuzzy domination number, FRN and FRDN respectively. By the definition of an FRDS, it is obvious that both vertices will also be present in $F_{\gamma r}$.   □

3.3. Theorem 1

If   $F_{\gamma }=m, Fr=n$, then

If   $m>n$, then   $F_{\gamma r}=m$

If   $m=n$, then   $F_{\gamma r}=m=n$

If   $m<n$, then   $F_{\gamma r}=n$

Proof. 

Let $F_{\gamma }, Fr$ and $F_{\gamma r}$ be the fuzzy domination number, FRN and FRDN respectively.

(i)

If $m>n$, then we can always find a resolving set with $m$ vertices but we cannot find a domination set with $n$ vertices. Therefore, $F_{\gamma r}=m$.

(ii)

If $m=n$ then by the definition of fuzzy resolving domination, it is both a resolving and dominating set. Therefore, $F_{\gamma r}=m=n$.

(iii)

If $m<n$, then it is easy to find the dominating set with $n$ vertices but we could not find a resolving set with $m$ vertices. Therefore, $F_{\gamma r}=n$, which proves the theorem.

□

3.4. Theorem 2

If   $G$   is a CFG, then   $F_{\gamma r}\left(G\right)$   is at least   $2$   for all   $n \ge 3$   and   $F{R}_{\gamma r}\left(G\right)$   exists for   $n \ge 4$.

Proof. 

Consider a CFG $G$ which is $\mu \left(u, v\right)= \min \left\{\mathsf{\sigma }\left(u\right), \mathsf{\sigma }\left(v\right)\right\}$ and all the edges are strong. Therefore, any two elements in the sets are going to be a fuzzy domination set. By the definition of resolving sets, the representations of $\sigma -H$ are all different. Therefore, it is always easy to find a resolving set of cardinalities of more than 2. Hence, $F_{\gamma r}$ is at least 2 for all $n \ge 3$. Similarly, by the definition of a complete fuzzy graph, if we remove any set of vertices from the original graph, the remaining graph will be connected. Hence, $V\left(G\right)\backslash H$ contains no isolated vertex which proves that FRRDSs exist for all complete fuzzy graphs with $n \ge 4$. □

3.5. Theorem 3

Consider a fuzzy star graph; if the function μ is not a constant for all the edges, then $F_{\gamma r}\left(G\right)=2$.

Proof. 

Consider a fuzzy star graph $G$ in which all the edges are strong. Additionally, since $\mu$ is not a constant function, then the membership value of all the edges is not the same. It is easy to find a fuzzy domination set with a cardinality of 2. Similarly, we can also find a fuzzy resolving set since the representation of the edge set values must be distinct. Hence, the fuzzy resolving domination number $F_{\gamma r}\left(G\right)=2$.   □

3.6. Theorem 4

The union of two FRDSs is also an FRDS.

Proof. 

Consider two fuzzy resolving domination sets $R_1=\left\{u,v\right\}$ and $R_2=\left\{w,x\right\}$ with respect to the fuzzy graph $G$, which means that they are fuzzy resolving and fuzzy domination sets, respectively. The union of two FRSs is also an FRS. If the vertices in the sets contain strong arcs, then $R_1{\displaystyle \cup }{R}_2$ is also an FRDS.   □

3.7. Theorem 5

The FRDN of a fuzzy graph G does not need to be an FSRDN of G. However, the converse is true.

Proof. 

Let $H=\{{\mathsf{\sigma }}_1,{\mathsf{\sigma }}_2,\dots {\mathsf{\sigma }}_m\}$ be a fuzzy resolving domination set and here $m$ is the FRDN of an FG $G$ since m is the smallest cardinality of the given set H. Similarly, there will be many resolving subsets and domination sets containing all the vertices of $G$, and they will have different resolving numbers and domination sets according to $G$. We also need to consider the vertices of $H$ for an FSRDS, which may be different from or the same as the sets of the FRDS. Hence, an FRDN need not be an FSRDN of $G$.   □

Now, let $H$ be an SRDS of $G$. It is easy to see that the same resolving domination subsets will be true for the resolving subset of a fuzzy graph. Hence, an FRDN and FSRDN of a fuzzy graph are the same. Therefore, the converse of this statement is also true.

3.8. Corollary 1

The intersection of two FRDSs need not be an FRDS.

Proof. 

Consider two FRDS with cardinality mn H₁ and H₂. The intersection of both of these sets will be less than or equal to $\mathrm{mn}$. It is evident that the intersection of two sets need not be an FRDS, which proves the statement.   □

3.9. Theorem 6

Let G be an FG with   $\left|V\right|=4$   and let   $G^{*}$   be a four fuzzy cycle. If the function μ is not a constant, then the FRDN and FRRDN are either 2 or do not exist.

Proof. 

We already have a theorem which states that “Let $G\left(V, \mathsf{\sigma }, \mathsf{\mu }\right)$ be an FG with $\left|V\right|=4$ and let $G^{*}$ be a cycle. If the function $\mu$ is not a constant, then $Fr\left(G\right)=2$.” We have three different cases to prove this theorem.   □

Case 1.

If $G$ has the two weakest arcs, then the remaining two arcs are strong. These strong arcs will cover three or four vertices in the fuzzy graph; it is then obvious to take 2 vertices in the pair to obtain a dominating set. Hence, $F_{\gamma }\left(G\right)=2$, but these two vertices have an equal chance of being a fuzzy resolving set. In that case, $F_{\gamma r}\left(G\right)=2$; otherwise, the vertices do not match with the other, implying that an FRDS does not exist in this case. If $V\backslash H$ is connected and has no isolated vertex, then the cardinality of FRRDS is 2; otherwise, an FRRDS does not exist.

Case 2.

If $G$ has the three weakest arcs, then the remaining arc is strong. This strong arc will cover only two vertices in the FG. Since $G^{*}$ is an f-cycle, it will cover the remaining two vertices, which comprise a fuzzy domination set. Hence, $F_{\gamma }\left(G\right)=2$, but these two vertices have an equal chance of being a fuzzy resolving set. In that case, $F_{\gamma r}\left(G\right)=2$; otherwise, the vertices do not match with the other, implying that an FRDS does not exist in this case. If $V\backslash H$ is connected and has no isolated vertex, then the cardinality of FRRDS is 2; otherwise, FRRDS does not exist.

Case 3.

If $G$ has the four weakest arcs, then there is no strong arc in the graph which implies that the fuzzy domination set does not exist. Hence, an FRDS and FRRDS do not exist.

By combining all the cases, the FRDN and FRRDN are either 2 or do not exist.

3.10. Theorem 7

Let G be an FG with   $\left|V\right|=5$   and let   $G^{*}$   be a cycle. If the function   $\mu$   is not a constant, then the FDN is either 2 or 3 or does not exist.

Proof. 

We have four different cases to prove this theorem.   □

Case 4.

If $G$ has the two weakest arcs, then the remaining three arcs are strong. If the two weakest arcs are adjacent to each other, then we have four vertices which will be strong. We can easily find any two vertices that will dominate all the other vertices. Hence, the FDN is 2 in this case. If the two arcs are not adjacent to each other, then this leaves all the vertices as strong. So, it is easy to find any two vertices that will dominate the others. Hence, the FDN is 2.

Case 5.

If $G$ has the three weakest arcs, then the remaining two arcs are strong. If the three weakest arcs are adjacent to each other, then we have three vertices which will be strong. We can easily find any pair of two vertices that will dominate all the other vertices. Hence, the FDN is 2 in this case. If the two arcs are adjacent to each other and the remaining one is not adjacent, then this leaves the four vertices as strong. So, it is easy to find any two vertices that will dominate others. Hence, the FDN is 2.

Case 6.

If $G$ has the four weakest arcs, then the remaining arc in the graph is strong. This leaves only two vertices as strong and will not cover all the other vertices. Hence, the FDS does not exist in this case.

Case 7.

If $G$ has the five weakest arcs, then there is no strong arc in the graph which implies that the fuzzy domination set does not exist. Hence, FDN does not exist.

By combining all the cases, the FDN is either 2 or 3 or does not exist.

In the general case, if $G$ is an FG and $G^{*}$ is a five cycle, then the FDN varies from 2 to $n-1$. In some cases, it does not exist.

3.11. Theorem 8

If   $H$   is an FRDS of an FG, then   $G-H$   is not necessarily an FRDS.

Proof. 

Consider $H=\left\{g_1,g_2,\dots g_n\right\}$ as an FRDS with $n\ge 3$. Then, $\mathsf{\sigma }-H$ are the remaining vertices from the graph $G$. The representation of these vertices depends on the edge membership value of the subset $H$. If the representations are unique, then the set may satisfy the FRDS criteria. Hence, the representation values may or may not be distinct with respect to the vertices of $G$. Therefore, $G-H$ does not need to be an FRDS, which concludes the proof.   □

3.12. Theorem 9

If$G\cong G^{\prime }$, then$F_{\gamma r}\left(G\right)=F_{\gamma r}\left(G^{\prime }\right)$.

Proof. 

If $G\cong G \prime$, then $\exists j:V\to V^{\prime }$, which is a bijective map that satisfies (i) $\mathsf{\sigma }\left(v\right)= {\mathsf{\sigma }}^{\prime }\left(j\left(v\right)\right) \forall v\in V$ and (ii) $\mathsf{\mu }\left(u,v\right)= {\mathsf{\mu }}^{\prime }\left(j\left(u\right), j\left(v\right)\right) \forall u,v\in V$. Consider $G$ and assume that $F_{\gamma r}\left(G\right)=k$ and $H=\left\{g_1,g_2,\dots g_n\right\}$ are the corresponding FRDS, then the vertices contain a strong arc and all the representations are distinct. Since $G\cong G^{\prime }$, $G$ and $G^{\prime }$ satisfy the same condition as strong arcs. Hence, $G$ and $G^{\prime }$ have the same FRDN.   □

4. Application in Real Life Scenario

Fuzzy graphs have a major role in real-life application-based problems and recent research is only based on application development. In social network problems, it is mostly domination that plays a vital role in identifying the specific sets which will be useful for the problem. Moreover, resolving sets help identify the unique properties of the sets compared with all the other vertices. Combining these sets will enable us to find a unique and distinct set, which will be a more efficient method to solve the problem. Finding the best route based on the distinct properties of all the cities is now achieved using FRDSs. In addition, we can apply this method to many different problems to obtain an accurate solution. A social network approach helps to transform big problems into simpler ones.

Consider a social networking service such as Facebook, Instagram or Twitter. Let us assume that there is a group of people following a few useful pages, such as pages dealing with social awareness, health tips and so on. Let us presume that the group of eight people are mutually or independently connected and follow these four useful pages. To depict the people and the ties that bind them, we use vertices and edges. We have also discussed the strong value based on their connection to one another. A few people share good information with their friends to keep in touch with them. The posts people share will either be helpful or useless, but we can identify those who offer the most knowledgeable content that will be helpful to others using the FRDS. The vertex membership value represents the strength of the amount of knowledge they already have and the edge represents the strength of the knowledge they share with other people and to other pages. The process of finding the resolving sets and dominating sets enables us to find the strength of two vertices as well as the distinct properties of all the vertices. This will help us identify the major dominating and resolving set, which will be useful for all the other vertices, and they do not have to follow all the other pages. In the diagram shown in Figure 2, the triangle-marked vertices are the fuzzy resolving dominating sets and the vertices in these sets have much more knowledge than the other vertices. The persons or pages we obtain as a fuzzy resolving domination set are enough to provide all the information that will be useful to others. Other people do not have to follow all the pages unless they have followed the remaining useful pages.

A pervasive question is “What is the use of applying Resolving sets rather than Dominating sets?” The dominating sets involve only strong arcs but resolving sets help find the unique helpful sets using strong arcs. Combining these sets will enable us to find more accurate sets for many different problems. Similarly, we can apply this concept to many real-life problems. If we consider a group of schools in rural areas, we can find the best and most convenient school for the children using FRDSs so that they can find which school is good and also the shortest distance for them to reach the school.

Consider

$$\begin{gathered} \sigma \left(A\right)=0.9,\sigma \left(B\right)=0.8,\sigma \left(C\right)=0.7,\sigma \left(D\right)=0.6,\sigma \left(E\right)=0.6,\sigma \left(F\right)=0.5,\sigma \left(G\right)=0.6,\sigma \left(H\right)=0.7, \\ \sigma \left(I\right)=0.9,\sigma \left(J\right)=0.8,\sigma \left(K\right)=0.5,\sigma \left(L\right)=0.4. \end{gathered}$$

and

$$\begin{gathered} \mu \left(AB\right)=0.7,\mu \left(AD\right)=0.6,\mu \left(AK\right)=0.4,\mu \left(AL\right)=0.2,\mu \left(BC\right)=0.6,\mu \left(BE\right)=0.6,\mu \left(BF\right)=0.1, \\ \mu \left(CD\right)=0.6,\mu \left(CG\right)=0.5,\mu \left(CH\right)=0.7,\mu \left(DJ\right)=0.3,\mu \left(DI\right)=0.4,\mu \left(EF\right)=0.3,\mu \left(EK\right)=0.3, \\ \mu \left(FG\right)=0.4,\mu \left(GH\right)=0.5,\mu \left(HI\right)=0.5,\mu \left(IJ\right)=0.7,\mu \left(JL\right)=0.3,\mu \left(KL\right)=0.2. \end{gathered}$$

We need to find a representation of all the vertices with respect to the FRS, which indicates that only a few vertices are enough to obtain an FRDS and these vertices help find more accurate sets.

The FRDS is $\left\{A, C,E,I\right\}$ since the values of the representations are all different.

5. Conclusions

This research study evaluated some new notions, such as the FRDN and FRRDN of a fuzzy graph, whose properties are discussed. Additionally, we have also developed an application based on the network theory concept using the FRDS. A few limitations should be considered while interpreting the results given above. This application has a significant impact on a variety of real-world problems and aids in the identification of an accurate set that will be beneficial for the issue and serve as the best solution. This idea can be used to determine the best treatment, identify the best particular things, and more. In the future, we would like to investigate other aspects of resolving fuzzy sets. In addition, resolving fuzzy sets act as the best tool to find an accurate solution and many research papers in this field are being written. We would also like to elaborate on the study of FRDSs in the future.