Domination of Fuzzy Incidence Graphs with the Algorithm and Application for the Selection of a Medical Lab

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.


Introduction
e graph concept stands as one of the most dominant and widely employed tools for multiple real-world 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. e 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 c-dominating energy in graphs and obtained various properties of c-dominating 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][6][7][8][9][10].
When there are uncertainty and ambiguity among relationships of different objects, the graphs are not enough.
is 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. e 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 interval-valued 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 H ∞ 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][29][30][31][32][33][34].
ere is a drawback of FGs because they are unspeakable to give any clue of the effect of a vertex on an edge. is 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. e 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. is 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. ey 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]. e 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. at is why we propose these ideas for FIGs. rough these ideas, we can study different characteristics of FIGs. In graphs and FGs, if vertex u dominates vertex v, then v also dominates u, but in FIGs, if vertex u dominates v, then it is not necessary that v also dominates u. For example, five different countries C 1 , C 2 , C 3 , C 4 , and C 5 have friendship with each other. If C 1 dominates C 5 or some other countries concerning the education system, then it is not necessary that C 5 or other countries also dominate C 1 . is encourages us to introduce the concept of domination for FIGs. e 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 min and ∨ or max, respectively. Some of the basic definitions and results are given in the following to comprehend the remaining contents of the article. ese definitions are taken from [11,17].

Preliminaries
A mapping μ: Z ⟶ [0, 1] is called a fuzzy subset (FS) of Z. An FG with M as the underlying set is a pair G � (φ, χ), For an FG, we can generalize a degree of a node in distinct methods. e sum of the weights of the EEs incident at node n is said to be the effective degree (ED) of node n. It is shown by dE(n). δ E (G) � ∧ dE(n)|n ∈ V { } shows the lowest ED, and Δ E (G) � ∨ dE(n)|n ∈ V { } represents the highest ED. e neighborhood degree of n is defined by m∈N(n) φ(m), and it is represented by dN(n). δ N (G) and Δ N (G) express the lowest and highest neighborhood degree, respectively. In an FG, a node m is called an isolated node if χ(mn) < φ(m)∧φ(n) for all mn ∈ χ.
Definition 1 (see [36]). Let G be a simple graph having node set V and edge set E. en, G � (V, E, I) is named as an incidence graph (IG), where I⊆V × E. Figure 1 represents an IG. (u, uv) is said to be an incidence pair (IP) if (u, uv) ∈ IG.
Definition 2 (see [36]). Let G be a graph with a vertex set V and an edge set E and φ and χ be FSs of a vertex set and an edge set, respectively. Consider ψ to be a FS of V × E. If ψ(g, h) ≤ φ(g)∧χ(h) for every g ∈ V and h ∈ E, then ψ is named as fuzzy incidence (FI) of G, and (φ, χ) is known as the fuzzy subgraph of G; if ψ is FI of G, then G � (φ, χ, ψ) is known as a FIG of G.

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.
Example 1. Suppose three garment factories are working with each other on the basis of some pros and cons. e 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. is can be done by using the definition of O(G) of FIGs.

Proposition 1. In a FIG
It is a trivial case. Assume G with more than one vertex. O(G) is the sum of all IPs of G. 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.

□ Proposition 2. For any FIG, the following inequality holds:
We By the definition of the size of G, By Proposition 1, From inequalities (4) Mordeson has shown u∈σ * (d i (u)) ≤ 2 e∈μ * μ(e) [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. Proof.
Mathematical Problems in Engineering By rearranging the terms, □ Example 2. A FIG G � (φ, χ, ψ) is provided in Figure 2 with

Domination in Fuzzy Incidence Graphs
Definition 9. Nd i of a node m is expressed as   Figure 3 with e minimal FIDS is Q � y , and FIDN � c FI � φ(y) � 0.5. is shows y is the city through which the flow of traffic to all other cities is maximum.  FIG which is represented by G �  (φ, χ, ψ) and is defined as

Mathematical Problems in Engineering
Its complement G is shown in Figure 4. For any FIG, G � (φ, χ, ψ)

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. e 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. a FIG, and let i and j be the nodes of G. en, i strongly dominates j or j weakly dominates i if the following two conditions are satisfied: We call j strongly dominates i or i weakly dominates j if Definition 16. A set R⊆V is a strong FIDS if each node in V − R is strongly fuzzy incidence dominated by at least one node in R. In a similar way, R is called a weak FIDS if each node in V − R is weakly fuzzy incidence dominated by at least one node in R.

Definition 17.
e lowest FC of a strong FIDS is uttered as the strong FIDN, and it is represented by c SFI (G) or c SFI , and the lowest FC of a weak FIDS is named as the weak FIDN, and it is represented by c WFI (G) or c WFI . Figure 5 having

Example 5. Assume G � (φ, χ, ψ) is a FIG as shown in
Here, y strongly fuzzy incidence dominates w, x, and z because d i (y) � 1.3 is greater than d i of all the remaining vertices, i.e., d i (w) � 1.0, d i (x) � 1.0, and d i (z) � 0.4. ere are no other strong FIDSs. us, the only strong FIDS is R � y . erefore, c SFI � 0.5. We have a weak FIDS which is

Remark 3.
If G is not a CFIG, then c SFI < c WFI .

Theorem 2. For any CFIG with ψ(i, ij) � φ(i)∧χ(ij) for all
i ∈ V and ij ∈ E, the inequality given in the following always holds: x (0.5) y (0.9) Proof. Let G � (φ, χ, ψ) be a CFIG with ψ(i, ij) � φ(i)∧χ(ij). Assume for every w i ∈ V, φ(w i ) are the same. Since G is a CFIG with χ(w i w j ) � φ(w i )∧φ(w j ) for all w i , w j ∈ V and ψ(w i , w i w j ) � φ(w i )∧χ(w i w j ) for all w i ∈ V, w i w j ∈ E. us, every w i ∈ V is a strong as well as weak FIDS; therefore, Assume for all w i ∈ V, φ(w i ) are not the same. In a CFIG with d i (w i ) ≥ d i (w j ), 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 Certainly, the strong FIDS has a node set other than that node set. is implies and from equations (13) and (14), we get c WFI ≤ c SFI .
is a strong FIDS which strongly dominates y, z and D 2 � z { } is another strong FIDS because it also strongly dominates x, y . erefore, c SFI � 0.5 and c WFI � 0.3.

Theorem 3.
For a CFIG, the inequalities given in the following are true: We From equations (15) and (16) erefore, the weak FIDN will be greater than or equal to c FI . Also, From equations (17) and (18), , and lowest d i of G � 0.6. Hence, eorem 3 can be verified.

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 real-life example to select the best medical lab among different labs (Algorithm 1). Here, in our study, we are not mentioning the original names of these labs; therefore, consider the labs l 1 , l 2 , l 3 , l 4 , l 5 , and l 6 . In FIGs, the vertices show the labs, and edges show the contract conditions among the labs to share the facilities or test kits. e IPs show the transferring of patients from one lab to another lab due to the lack of resources (machinery, equipment, kits, and doctors). e 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 G � (φ, χ, ψ) is a FIG shown in Figure 6 having Step 1: input the vertex set V, edge set E⊆V × V, and IP set I⊆V × E.
Step 3: LOOP for j � 1 to the number of vertices Let FIDS j � ∅, NDS � ∅, W � ∅, and Sum � ∅; If v i ∈ FIDS j en go to Step 4. Step Step   Algorithm 2 is applied.
is shows that patients can visit any one of the labs from this set. e government should provide the resources to labs l 3 , l 4 , and l 6 for the proper and easy conduction of tests for coronavirus.

Comparative Analysis
In an FG, if vertex u dominates vertex v, then vertex u also dominates vertex v, but in FIGs, it is not necessary that if vertex u dominates vertex v, then vertex v also dominates vertex u. erefore, FIGs are more interesting than FGs. Here, we are comparing our study with the previous study. In Figure 6, a FIG represents six different medical labs working with each other for conducting the coronavirus tests. We get FIDS � l 3 , l 4 , l 6 comprising three different medical labs which are best for the tests of coronavirus with c FI � 1. In Figure 6, if we remove all the IPs, we get an FG. In the case of an FG, l 1 is dominating l 2 and l 3 because χ(l 1 l 2 ) � 0.8 � φ(l 1 )∧φ(l 2 ) � 0.8∧0.9 and e rest of the labs are not dominating each other. erefore, DS D � l 1 , l 4 , l 5 , l 6 with DN 2. In the case of an FG, the government needs to work on four labs, but in the case of the FIG, the government needs to work on three labs. So, the government should work on three labs instead of four labs. In this way, the government can save its money and time.
is shows that our study is better than the previous studies.
Step 7: Sum � [φ(l 3 ) + φ(l 4 ) + φ(l 6 ) � 0.3 + 0.2 + 0.5 � 1, 1, 1, 1, 1, 1] For j � 2, 3, 4, 5, 6 FIDS � l 3 , l 4 , l 6 For j � 7 As j > number of vertices END LOOP (Step 3) Step 8: FIDN � 1. Mathematical Problems in Engineering of the FIG is explained. Our study will help us to find the order, size, FIDS, FIDN, d i , effective d i , and Nd i for the union, join, composition, tensor product, normal product, and the Cartesian product of FIGs if we know the order, size, FIDS, FIDN, d i , effective d i , and Nd i of two FIGs. e results explained in this paper may be used to study different FIG invariants. An algorithm to find the FIDS and FIDN is provided. An application of the FIDS for FIGs is provided to choose the best medical lab for conducting tests of coronavirus among different medical labs. Our objective is to extend our research work to Hamiltonian FIGs, bipolar FIGs, the coloring of FIGs, and the chromaticity of FIGs. Further work on these ideas will be reported in upcoming papers.

Data Availability
All the data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.