Abstract
Fuzzy graphs (FGs), broadly known as fuzzy incidence graphs (FIGs), have been recognized as being an effective tool to tackle realworld problems in which vague data and information are essential. Dominating sets (DSs) have multiple applications in diverse areas of life. In wireless networking, DSs are being used to find efficient routes with ad hoc mobile networks. In this paper, we extend the concept of domination of FGs to the FIGs and show some of their important properties. We propose the idea of order, size, and domination in FIGs. Two types of domination, namely, strong fuzzy incidence domination and weak fuzzy incidence domination, for FIGs are discussed. A relationship between strong fuzzy incidence domination and weak fuzzy incidence domination for complete fuzzy incidence graphs (CFIGs) is also introduced. An algorithm to find a fuzzy incidence dominating set (FIDS) and a fuzzy incidence domination number (FIDN) is discussed. Finally, an application of fuzzy incidence domination (FID) is provided to choose the best medical lab among different labs for the conduction of tests for the coronavirus.
1. Introduction
The graph concept stands as one of the most dominant and widely employed tools for multiple realworld problem representation, modeling, and analysis. To indicate the objects and the relations between them, the graph vertices or nodes and edges or arcs are applied, respectively. In graphs, the notion of domination first took place in the game of chess during the 1850s. In Europe, lovers of chess thought carefully about the complication of fixing the fewer numbers of queens that can be laid down on a chessboard so that all the squares are engaged by a queen. The idea of domination began with Claude Berge and Ore [1]. Domination in graphs has plenty of applications in a variety of areas. Concepts from DSs appear in problems involving finding sets of representatives in monitoring communication networks. Hosamani et al. [2] proposed an idea of cdominating energy in graphs and obtained various properties of cdominating energy. Harisha et al. [3] discussed degree sequence and join operations in graphs. Mathew et al. [4] introduced vertex rough graphs and the membership function of vertex rough graphs. For more detailed study on graphs, we may suggest to the reader [5–10].
When there are uncertainty and ambiguity among relationships of different objects, the graphs are not enough. This lack in graphs led Zadeh [11] to propose an idea of fuzzy sets (FSs). FSs are beneficial when there is inconclusiveness among different objects. After the landmark work of Zadeh on FSs, Rosenfeld [12] gave the notion of FGs. The study of FGs opens a new door for many researchers to participate in this field such as Gani and Ahamed [13] examined the properties of different kinds of degree, order, and size of FGs and compared the relationship between degree, order, and size of FGs. Akram [14] presented an idea of bipolar FGs and investigated their various properties. Bhattacharya [15] gave the notion of eccentricity and center of FGs. Rashmanlou and Jun [16] introduced complete intervalvalued FGs. Somasundaram and Somasundaram [17] presented an outstanding idea of domination in FGs. Somasundaram [18] discussed various types of products such as union, join, composition, and Cartesian product and their domination parameters for FGs. Gani and Ahamed [19] introduced strong and weak domination in FGs. Manjusha and Sunitha [20, 21] suggested the idea of total domination using strong arcs and strong domination in FGs, respectively. Ponnappan [22] et al. proposed the novel idea of edge domination, total edge domination, the edge domination number, and the total edge domination number for certain classes of FGs. Dharmalingam and Rani [23] proposed an idea of equitable domination in FGs. Dharmalingam and Nithya [24, 25] discussed the concept of excellent domination and very excellent domination in FGs. Wang et al. [26, 27] addressed the control and filtering problems for Markov jump singularity perturbed systems approximated by Takagi–Sugeno fuzzy models and stability analysis on singularity perturbed descriptor systems. For some other significant and constructive works on FGs, we may refer to [28–34].
There is a drawback of FGs because they are unspeakable to give any clue of the effect of a vertex on an edge. This deficiency in FGs opens a new door to introduce the idea of FIGs. FIGs are more impressive than FGs because in FGs, the membership value (MSV) of an edge xy is always equal to the MSV of an edge yx, but in FIGs, the MSV of (x, xy) may or may not be equal to (y, xy). For example, if nodes represent various residence colonies and arcs express roads connecting these colonies, we can have an FG expressing the extent of traffic from one colony to another. The colony with a large number of colonists will have a large number of ramps in the colony. So, if x and y are two colonies and xy is a road connecting them, then (x, xy) could indicate the ramp system from the road xy to the colony x. For the unweighted graph, both x and y will have an impact of 1 on xy. For the directed graph, the effect of x on xy represented by (x, xy) is 1, whereas (y, xy) is 0. This concept is generalized through FIGs. Dinesh [35] gave the notion of FIGs. Mathew and Mordeson [36] discussed connectivity concepts and other structural properties of FIGs. Mordeson and Mathew [37] developed the idea of fuzzy end nodes and fuzzy incidence cut vertices in FIGs. Mordeson et al. [38] proposed an innovative idea of vague FIGs and their eccentricity. They applied the vague FIGs to problems involving human trafficking and illegal immigration. For a more detailed and comprehensive study on FIGs, we may suggest to the reader [39, 40].
The main motivation of our work is that the degree, order, size, and domination in graphs and FGs are discussed by many researchers, but no one has explored these ideas for FIGs. That is why we propose these ideas for FIGs. Through these ideas, we can study different characteristics of FIGs. In graphs and FGs, if vertex u dominates vertex , then also dominates u, but in FIGs, if vertex u dominates , then it is not necessary that also dominates u. For example, five different countries , and have friendship with each other. If dominates or some other countries concerning the education system, then it is not necessary that or other countries also dominate . This encourages us to introduce the concept of domination for FIGs.
The remainder of this article is formulated as follows. Section 2 provides some preliminary results. Order, size, and their relationship in FIGs are discussed in Section 3. FID and complement of FIGs are given in Section 4. Section 5 demonstrates the types of domination such as strong FID and weak FID. A relationship between strong and weak FID for CFIGs is also discussed in the same section. Section 6 provides an application of FID in the selection of the best medical lab among different medical labs. A comparative analysis of our study with a previous study is discussed in Section 7. Conclusions and prospects are elaborated in Section 8.
In this paper, minimum and maximum operators are represented by or and or , respectively. Some of the basic definitions and results are given in the following to comprehend the remaining contents of the article. These definitions are taken from [11, 17].
2. Preliminaries
A mapping is called a fuzzy subset (FS) of Z. An FG with as the underlying set is a pair , where is , is a fuzzy relation on the such that for all , and is a finite set. is called the order of a graph, and is called the size of . An FG is complete if for all . A complete FG is represented by . In an FG, if , then dominates and dominates . A subset of is named as DS in if for each which does not belong to , such that dominates . The domination number (DN) of is the lowest cardinality of a DS among all DSs in . The DN of is expressed by or . A DS of an FG is minimal DS if no proper subset of is a DS of an FG. is said to be the neighborhood of , and is called the close neighborhood of . For an FG, we can generalize a degree of a node in distinct methods. The sum of the weights of the incident at node is said to be the effective degree (ED) of node . It is shown by . shows the lowest ED, and represents the highest ED. The neighborhood degree of is defined by , and it is represented by . and express the lowest and highest neighborhood degree, respectively. In an FG, a node is called an isolated node if for all .
Definition 1. (see [36]). Let be a simple graph having node set and edge set . Then, is named as an incidence graph (IG), where . Figure 1 represents an IG. is said to be an incidence pair (IP) if .
Definition 2. (see [36]). Let be a graph with a vertex set and an edge set and and be FSs of a vertex set and an edge set, respectively. Consider to be a FS of . If for every and , then is named as fuzzy incidence (FI) of , and is known as the fuzzy subgraph of ; if is FI of , then is known as a FIG of .
Remark 1. (see [36]). If , then is in the support of where . If , then is in the support of where , and if , then is in the support of where . , , and are representing supports of , , and , respectively.
If the value of an IP or is not given in the FIG, then its value will be equal to zero. Also, two vertices and are connected in the FIG if there exists a path such that between and .
Definition 3. (see [36]). A FIG is said to be CFIG if for each . Also, for each . It is denoted by .
Definition 4. (see [39]). Let be a FIG; the incidence degree of a node is defined as .
The lowest of is defined by .
The highest of is defined by .
3. Relationship between the Order and Size of Fuzzy Incidence Graphs
In this section, we define the order and size of the FIG. We also discuss a relationship between the order and the size of the FIG.
Definition 5. Assume is a FIG. Then, is called the order of , and is called the size of .
Example 1. Suppose three garment factories are working with each other on the basis of some pros and cons. The MSV of the vertices represents the number of workers working in a factory, the MSV of an edge indicates the contract policies of these factories with each other, and the MSV of an IP shows the annual profit of a factory. We want to calculate the total annual profit of these factories. This can be done by using the definition of O(G) of FIGs.
Assume is a FIG having ; ; , . Then, is the total annual profit of these factories with .
Proposition 1. In a FIG , .
Proof. Let be a with one vertex. Then, , i.e.,It is a trivial case. Assume with more than one vertex. is the sum of all IPs of . Since IPs are 2 times of edges, the total sum of all the MSVs of the IPs will always be greater than the total sum of all MSVs of the edges.From equations (1) and (2), we get
Proposition 2. For any , the following inequality holds: .
Proof. Assume is a FIG with a nonempty vertex set. Since represents lowest and denotes highest of ,We know and .
By the definition of the size of , , i.e.,By Proposition 1,From inequalities (4)–(6), we obtained .
Mordeson has shown [39]. In his result, there is an inequality. We are going to propose this type of result with equality but in the form of IPs.
Proposition 3. The sum of all vertices in a FIG is equal to twice the average sum of all the IPs, i.e.,
Proof. Let be a FIG, where , , , and .
Since ,This implies .By rearranging the terms,
Example 2. A FIG is provided in Figure 2 with . We have and . This implies .
4. Domination in Fuzzy Incidence Graphs
Domination in graphs has plenty of uses in various fields. In this section, we have introduced the concepts of effective , minimum effective , maximum effective , neighborhood incidence degree , minimum , maximum , FIDS, and FIDN for FIGs. The main benefit of our work is that if we know effective and of the FIG, then we can study the further properties of the FIG. Also, in future, with the help of these ideas, we will be able to find effective , minimum effective , maximum effective , , minimum , maximum , FIDS, and FIDN in union, join, composition, normal product, tensor product, and Cartesian product of FIGs.
Definition 6. An IP of a FIG is named as an effective incidence pair (EIP) if for all , .
Definition 7. Open incidence neighborhood (IN) is defined as . Closed incidence neighborhood of is .
For a FIG, of a node can be generalized in distinct ways.
Definition 8. The effective of a node is described as . The minimum effective is denoted by . The maximum effective is denoted by .
Definition 9. of a node is expressed as . The minimum is defined by . The maximum is defined by .
Definition 10. A vertex in a FIG dominates vertex if , and a vertex dominates if . The set of these types of vertices is called a FIDS of the FIG.
Definition 11. A FIDS D is called a minimal FIDS of if no proper subset of D is a FIDS of .
Definition 12. The FIDN is the minimum fuzzy cardinality (FC) of the FIDS among all FIDSs in . It is represented by .
Example 3. Let us look at a practical example involving a FIDS and FIDN. As an illustrative case, consider a FIG in which vertices show different cities and the MSV of the vertices indicates the population of the cities. The MSV of the edges shows the mutual understanding of all these cities with each other, and the MSV of the IPs represents the traffic flow from one city to another city. FIDS will help us to find which city/cities has the maximum traffic flow to all other cities.
A FIG is shown in Figure 3 withThe minimal FIDS is , and . This shows is the city through which the flow of traffic to all other cities is maximum.
Remark 2. (1)For any , if dominates , then it is not necessary that dominates .(2)If , . This implies is the unique FIDS of . Conversely, if is the only FIDS of , then , .(3)For the CFIG, is a FIDS for every belonging to , and we have .
Definition 13. A node of FIG is named as an isolated node if , i.e., . Therefore, in the FIG, no node is dominated by an isolated node, but an isolated node dominates itself.
Definition 14. Assume is a FIG; the complement of is a FIG which is represented by and is defined as(i)(ii)(iii)
Example 4. Assume is a FIG having ; ; ; and , .
Its complement is shown in Figure 4.
Theorem 1. For any FIG, , where and are the FIDN of and , respectively.
5. Strong and Weak Domination in Fuzzy Incidence Graphs
In this section, we have discussed strong and weak FID for FIGs and give different examples to understand these concepts. The results provided in this section are based on [19, 40]. In this view, similar results related to the strong FIDN and weak FIDN in FIGs are achieved.
Definition 15. Assume is a FIG, and let and be the nodes of . Then, strongly dominates or weakly dominates if the following two conditions are satisfied:(i)(ii)We call strongly dominates or weakly dominates if(i)(ii)
Definition 16. A set is a strong FIDS if each node in is strongly fuzzy incidence dominated by at least one node in . In a similar way, is called a weak FIDS if each node in is weakly fuzzy incidence dominated by at least one node in .
Definition 17. The lowest FC of a strong FIDS is uttered as the strong FIDN, and it is represented by or , and the lowest FC of a weak FIDS is named as the weak FIDN, and it is represented by or .
Example 5. Assume is a FIG as shown in Figure 5 having ; , , . Assume . We have . Here, strongly fuzzy incidence dominates , and because is greater than of all the remaining vertices, i.e., , , and . There are no other strong FIDSs. Thus, the only strong FIDS is . Therefore, . We have a weak FIDS which is with .
Remark 3. If is not a CFIG, then .
Theorem 2. For any CFIG with for all and , the inequality given in the following always holds:
Proof. Let be a CFIG with . Assume for every , are the same. Since is a CFIG with for all and for all , .
Thus, every is a strong as well as weak FIDS; therefore,Assume for all , are not the same. In a CFIG with , from all the nodes, one of them strongly dominates all the remaining nodes; if it is smallest among all the nodes, then the FIDS with that node is called a weak FIDN that is with for all , and for all , .
Certainly, the strong FIDS has a node set other than that node set. This impliesand from equations (13) and (14), we get .
Example 6. Assume is a CFIG having ; ; ; and . Here, is a strong FIDS which strongly dominates and is another strong FIDS because it also strongly dominates . Therefore, and .
Theorem 3. For a CFIG, the inequalities given in the following are true:(i)—highest of (ii)—lowest of
Proof. (i)From Definitions 15–17, we have We know , the sum of of the . Also, From equations (15) and (16), —highest of .(ii)From Definitions 15–17, the weight of of the FIG is less than or equal to of the FIG as elements of weakly dominate any one of elements of . Therefore, the weak FIDN will be greater than or equal to .Also,From equations (17) and (18), we get —lowest of .
Example 7. Assume is a CFIG having ; ; ; and . , order of , highest of , and lowest of . Hence, Theorem 3 can be verified.
6. Algorithm and Application of FID in the Selection of the Best Medical Lab
In this section, an algorithm to find the FIDS and FIDN for the FIG is provided. We apply our algorithm in the reallife example to select the best medical lab among different labs (Algorithm 1).

6.1. Algorithm: Steps to Find the FIDS and FIDN
6.2. Application to Select the Best Medical Lab
Suppose there are six different medical labs which are working in a city for conducting tests of coronavirus. Here, in our study, we are not mentioning the original names of these labs; therefore, consider the labs , , , , , and . In FIGs, the vertices show the labs, and edges show the contract conditions among the labs to share the facilities or test kits. The IPs show the transferring of patients from one lab to another lab due to the lack of resources (machinery, equipment, kits, and doctors). The FIDS of the FIG is the set of labs which perform the tests independently. In this way, we can save the time of patients and overcome the long traveling of patients by providing the few facilities to these labs.
Assume is a FIG shown in Figure 6 having
Algorithm 2 is applied.

This shows that patients can visit any one of the labs from this set. The government should provide the resources to labs , and for the proper and easy conduction of tests for coronavirus.
7. Comparative Analysis
In an FG, if vertex dominates vertex , then vertex also dominates vertex , but in FIGs, it is not necessary that if vertex dominates vertex