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

Authors: Irfan Nazeer; Tabasam Rashid; Juan Luis Garcia Guirao
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

Fuzzy graphs (FGs), broadly known as fuzzy incidence graphs (FIGs), have been recognized as being an effective tool to tackle real-world 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.


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## Body

## 1. Introduction

The 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. 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 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 [5–10].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 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 [28–34].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 edgexy 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 vertexu 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 C1,C2,C3,C4, and C5 have friendship with each other. If C1 dominates C5 or some other countries concerning the education system, then it is not necessary that C5 or other countries also dominate C1. This encourages us to introduce the concept of domination for FIGs.The remainder of this article is formulated as follows. Section2 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. These definitions are taken from [11, 17].


## 2. Preliminaries

A mappingμ: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, where φ:M⟶0,1 is FS, χ:M×M⟶0,1 is a fuzzy relation on the FSφ such that χu,v≤φu∧φv for all u,v∈M, and M is a finite set. OG=∑u∈Mφu is called the order of a graph, and SG=∑u,v∈Mχu,v is called the size of G. An FG is complete if χu,v=φu∧φv for all u,v∈V. A complete FG is represented by Kφ. In an FG, if χuv=φu∧φv, then u dominates v and v dominates u. A subset M of V is named as DS in G if for each v which does not belong to M, ∃u∈M such that u dominates v. The domination number (DN) of G is the lowest cardinality of a DS among all DSs in G. The DN of G is expressed by γG or γ. A DS M of an FG is minimal DS if no proper subset of M is a DS of an FG. Nu=v∈V|χuv=φu∧φv is said to be the neighborhood of u, and Nu∪u is called the close neighborhood of u. For an FG, we can generalize a degree of a node in distinct methods. The 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 dEn. δEG=∧dEn|n∈V shows the lowest ED, and ΔEG=∨dEn|n∈V represents the highest ED. The neighborhood degree of n is defined by ∑m∈Nnφm, and it is represented by dNn. δNG and ΔNG express the lowest and highest neighborhood degree, respectively. In an FG, a node m is called an isolated node if χmn<φm∧φn for all mn∈χ.Definition 1.
(see [36]). Let G be a simple graph having node set V and edge set E. Then, 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.Figure 1
A FIG.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 ψg,h≤φg∧χh 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.Remark 1.
(see [36]). If φu>0, then u is in the support of φ where u∈V. If χuv>0, then uv is in the support of χ where uv∈V×V, and if ψu,uv>0, then u,uv is in the support of ψ where u,uv∈V×E. φ∗, χ∗, and ψ∗ are representing supports of φ, χ, and ψ, respectively.
If the value of an IPψu,uv or ψv,vu is not given in the FIG, then its value will be equal to zero. Also, two vertices u and v are connected in the FIG if there exists a path such that u,u,uv,uv,v,uv,v between u and v.Definition 3.
(see [36]). A FIG is said to be CFIG if ψi,ij=φi∧χij for each ψi,ij∈ψ∗. Also, ψi,ij=ψj,ji for each i,j∈φ∗. It is denoted by K∗.Definition 4.
(see [39]). Let G be a FIG; the incidence degree di of a node u∈φ∗ is defined as diu=∑u≠vψu,uv.
The lowestdi of G is defined by ΩG=mindiv|v∈V.
The highestdi of G is defined by ΔG=maxdi|v∈V.


## 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.
AssumeG=φ,χ,ψ is a FIG. Then, OG=∑u≠v,u,v∈Vψu,uv is called the order of G, and SG=∑e∈χ∗χe is called the size of G.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 ofO(G) of FIGs.
AssumeG=φ,χ,ψ is a FIG having φ=x,y,z; φx=0.5,φy=0.6,φz=0.9; χxy=0.5,χxz=0.4,χyz=0.5;ψx,xy=0.4,ψy,yx=0.3,ψx,xz=0.3,ψz,zx=0.4, ψy,yz=0.5,andψz,zy=0.4. Then, OG=2.3 is the total annual profit of these factories with SG=1.4.Proposition 1.
In a FIGG=φ,χ,ψ, SG≤OG.Proof.
LetG=φ,χ,ψ be a FIG with one vertex. Then, OG=SG=0, i.e.,(1)OG=SG.
It is a trivial case. AssumeG with more than one vertex. OG 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.(2)SG<OG.
From equations (1) and (2), we get(3)SG≤OG.Proposition 2.
For anyFIG, the following inequality holds: OG≥SG≥ΔG≥ΩG.Proof.
AssumeG=φ,χ,ψ is a FIG with a nonempty vertex set. Since ΩG represents lowest di and ΔG denotes highest di of G,(4)ΔG≥ΩG.
We knowOG=∑u≠v,u,v∈Vψu,uv and SG=∑e∈χ∗χe.
By the definition of the size ofG, SG=∑e∈χ∗χe≥∨div|v∈V, i.e.,(5)ΔG≤SG.
By Proposition1,(6)SG≤OG.
From inequalities (4)–(6), we obtained OG≥SG≥ΔG≥ΩG.
Mordeson has shown∑u∈σ∗diu≤2∑e∈μ∗μe [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.
Thedi sum of all vertices in a FIG is equal to twice the average sum of all the IPs, i.e.,(7)∑v∈φ∗div=2∑u,v∈Vψu,uv+ψv,vu2.Proof.
LetG=φ,χ,ψ be a FIG, where V=v1,v2,v3,…,vn, φ⊆V, χ⊆E, and ψ⊆V×E.
Sincediv=∑u≠vψu,uv,(8)div1=ψv1,v1v2+ψv1,v1v3+⋯+ψv1,v1vndiv2=ψv2,v2v1+ψv2,v2v3+⋯+ψv2,v2vn⋮divn=ψvn,vnv1+ψvn,vnv2+⋯+ψvn,vnvn−1.
This implies∑v∈Vdiv=div1+div2+⋯+divn.(9)∑v∈Vdiv=ψv1,v1v2+ψv1,v1v3+⋯+ψv1,v1vn+ψv2,v2v1+ψv2,v2v3+⋯+ψv2,v2vn+⋯+ψvn,vnv1+ψvn,vnv2+⋯+ψvn,vnvn−1,∑v∈Vdiv=22ψv1,v1v2+ψv1,v1v3+⋯+ψv1,v1vn+ψv2,v2v1+ψv2,v2v3+⋯+ψv2,v2vn+⋯+ψvn,vnv1+ψvn,vnv2+⋯+ψvn,vnvn−1.
By rearranging the terms,(10)∑v∈Vdiv=2ψv1,v1v2+ψv2,v2v12+ψv1,v1v3+ψv3,v3v12+⋯+ψv1,v1vn+ψvn,vnv12+ψv2,v2v3+ψv3,v3v22+ψv2,v2v4+ψv4,v4v22+⋯+ψv2,v2vn+ψvn,vnv22+⋯+ψvn−1,vn−1vn+ψvn,vnvn−12,∑v∈Vdiv=2∑u,v∈Vψu,uv+ψv,vu2.Example 2.
A FIGG=φ,χ,ψ is provided in Figure 2 with φ=x,y,z. We have ∑divi=2.1 and ∑u,v∈Vψu,uv+ψv,vu/2=1.05. This implies ∑divi=2∑u,v∈Vψu,uv+ψv,vu/2.Figure 2
A FIG with∑divi=2.1=2∑u,v∈Vψu,uv/2=21.05.


## 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 effectivedi, minimum effective di, maximum effective di, neighborhood incidence degree Ndi, minimum Ndi, maximum Ndi, FIDS, and FIDN for FIGs. The main benefit of our work is that if we know effective di and Ndi 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 di, minimum effective di, maximum effective di, Ndi, minimum Ndi, maximum Ndi, 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) ifψi,ij=φi∧χij for all i∈V, ij∈E.Definition 7.
Open incidence neighborhood (IN) is defined asINi=j∈V|ψi,ij=φi∧χij. Closed incidence neighborhood of i is FINi=FINi∪i.
For a FIG,di of a node can be generalized in distinct ways.Definition 8.
The effectivedi of a node m is described as dEIPm=∑ψm,mn. The minimum effective di is denoted by δdEIPG=mindEIPm|m∈V. The maximum effective di is denoted by ΔdEIPG=maxdEIPm|m∈V.Definition 9.
Ndi of a node m is expressed as Ndim=∑n∈INmφn. The minimum Ndi is defined by δdING=mindINm|m∈V. The maximum Ndi is defined by ΔdING=maxdINm|m∈V.Definition 10.
A vertexi in a FIG G=φ,χ,ψ dominates vertex j if ψi,ij=φi∧χij, and a vertex j dominates i if ψj,ij=φj∧χij. The set of these types of vertices is called a FIDS of the FIG.Definition 11.
A FIDSD is called a minimal FIDS of G if no proper subset of D is a FIDS of G.Definition 12.
The FIDN is the minimum fuzzy cardinality (FC) of the FIDS among all FIDSs inG. It is represented by γFI.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 Figure3 with(11)φ=w,x,y,z,φw=0.5,φx=0.3,φy=0.5,φz=1,χwx=0.3,χwy=0.5,χxy=0.2,χyz=0.5,ψw,wx=0.3,ψw,wy=0.5,ψx,xw=0.2,ψx,xy=0.2,ψy,yw=0.5,ψy,yx=0.2,ψy,yz=0.5,ψz,zy=0.4.
The minimal FIDS isQ=y, and FIDN=γFI=φy=0.5. This shows y is the city through which the flow of traffic to all other cities is maximum.Figure 3
FIG withγFI = 0.5.Remark 2.
(1)
For anyu,v∈V, if u dominates v, then it is not necessary that v dominates u.(2)
Ifψu,uv<φu∧χuv,∀u∈V, uv∈E. This implies V is the unique FIDS of G. Conversely, if V is the only FIDS of G, then ψu,uv<φu∧χuv,∀u∈V, uv∈E.(3)
For the CFIG,i is a FIDS for every i belonging to V, and we have γFIK∗=minx∈Vφx.Definition 13.
A nodem of FIG G=φ,χ,ψ is named as an isolated node if ψm,mn<φm∧χmn,∀n∈V−m, i.e., FINm=∅. Therefore, in the FIG, no node is dominated by an isolated node, but an isolated node dominates itself.Definition 14.
AssumeG=φ,χ,ψ is a FIG; the complement of G=φ,χ,ψ is a FIG which is represented by G¯=φ¯,χ¯,ψ¯ and is defined as(i)
φ¯a=φa(ii)
χ¯ab=φa∧φb−χab(iii)
ψ¯a,ab=minmaxφa,χab−ψa,ab,φa∧φb−χabExample 4.
AssumeG=φ,χ,ψ is a FIG having φ=w,x,y,z; φw=0.3,φx=0.5,φy=0.9,φz=0.7; χwx=0.3,χwy=0.2,χxy=0.4,χyz=0.6; and ψw,wx=0.3,ψx,xw=0.3,ψx,xy=0.3,ψy,yx=0.2,ψw,wy=0.1,ψy,yw=0.1, ψy,yz=0.5,ψz,zy=0.5.
Its complementG¯ is shown in Figure 4.Figure 4
GraphG is complement to G¯.Theorem 1.
For any FIG,G=φ,χ,ψ2p>γFI+γ¯FI, where γFI and γ¯FI are the FIDN of G and G¯, 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.
AssumeG=φ,χ,ψ is a FIG, and let i and j be the nodes of G. Then, i strongly dominates j or j weakly dominates i if the following two conditions are satisfied:(i)
dii≥dij(ii)
ψi,ij=φi∧χij
We callj strongly dominates i or i weakly dominates j if(i)
dij≥dii(ii)
ψj,ji=φj∧χjiDefinition 16.
A setR⊆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.
The lowest FC of a strong FIDS is uttered as the strong FIDN, and it is represented byγSFIG or γSFI, and the lowest FC of a weak FIDS is named as the weak FIDN, and it is represented by γWFIG or γWFI.Example 5.
AssumeG=φ,χ,ψ is a FIG as shown in Figure 5 having φ=w,x,y,z; φw=1,φx=0.7,φy=0.5,φz=1, χwx=0.6,χwy=0.4,χxy=0.5,χyz=0.4,ψw,wx=0.5,ψx,xw=0.6,ψx,xy=0.4, ψy,yx=0.5,ψw,wy=0.4,ψy,yw=0.4,ψy,yz=0.4,andψz,zy=0.4. Assume R=y. We have V−R=w,x,z. Here, y strongly fuzzy incidence dominates w,x, and z because diy=1.3 is greater than di of all the remaining vertices, i.e., diw=1.0, dix=1.0, and diz=0.4. There are no other strong FIDSs. Thus, the only strong FIDS is R=y. Therefore, γSFI=0.5. We have a weak FIDS which is R1=w,x,z with γWFI=φw+φx+φz=1+0.7+1=2.7.Figure 5
G having γSFI<γWFI.Remark 3.
IfG is not a CFIG, then γSFI<γWFI.Theorem 2.
For any CFIG withψi,ij=φi∧χij for all i∈V and ij∈E, the inequality given in the following always holds:(12)γWFI≤γSFI.Proof.
LetG=φ,χ,ψ be a CFIG with ψi,ij=φi∧χij. Assume for every wi∈V, φwi are the same. Since G is a CFIG with χwiwj=φwi∧φwj for all wi,wj∈V and ψwi,wiwj=φwi∧χwiwj for all wi∈V, wiwj∈E.
Thus, everywi∈V is a strong as well as weak FIDS; therefore,(13)γSFI=γWFI.
Assume for allwi∈V, φwi are not the same. In a CFIG with diwi≥diwj, 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 γWFI=φwi with diwi≤diwj for all wi, wj∈V and ψwi,wij=φwi∧χwiwj for all wi∈V, wiwj∈E.
Certainly, the strong FIDS has a node set other than that node set. This implies(14)γWFI<γSFI,and from equations (13) and (14), we get γWFI≤γSFI.Example 6.
AssumeG=φ,χ,ψ is a CFIG having φ=x,y,z; φx=0.5,φy=0.3,φz=0.8; χxy=0.3,χxz=0.5,χyz=0.3; and ψx,xy=0.3,ψy,yx=0.3,ψy,yz=0.3,ψz,zy=0.3,ψx,xz=0.5,ψz,zx=0.5. Here, D1=x is a strong FIDS which strongly dominates y,z and D2=z is another strong FIDS because it also strongly dominates x,y. Therefore, γSFI=0.5 and γWFI=0.3.Theorem 3.
For a CFIG, the inequalities given in the following are true:(i)
γFI≤γSFI≤OG—highest di of G(ii)
γFI≤γWFI≤OG—lowest di of GProof.
(i)
From Definitions15–17, we have(15)γFI≤γSFI.We knowOG=p, the sum of di of the FIG. Also,(16)OG−not including the highestdiof the FIG=OG−ΔG.From equations (15) and (16), γFI≤γSFI≤OG—highest di of G.(ii)
From Definitions15–17, the weight of γFI of the FIG is less than or equal to γWFI of the FIG as elements of F weakly dominate any one of elements of V−F. Therefore, the weak FIDN will be greater than or equal to γFI.(17)γWFIG≥γFIG.
Also,(18)OG−δG=p−δG.
From equations (17) and (18), we get γFI≤γWFI≤OG—lowest di of G.Example 7.
AssumeG=φ,χ,ψ is a CFIG having φ=x,y,z; φx=0.8,φy=0.3,φz=0.9; χxy=0.3,χxz=0.8,χyz=0.3; and ψx,xy=0.3,ψy,yx=0.3,ψy,yz=0.3,ψz,zy=0.3,ψx,xz=0.8,ψz,zx=0.8. dix=1.1,diy=0.6,diz=1.1,γFI=0.3,γSFI=0.8,γWFI=0.3, order of G=2.8, highest di of G=1.1, and lowest di of G=0.6. 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 real-life example to select the best medical lab among different labs (Algorithm1).Algorithm 1: Steps to find the FIDS and FIDN.
Step 1: input the vertex setV, edge set E⊆V×V, and IP set I⊆V×E.Step 2: sortV such that V=φli≤φli+1≤⋯≤ln.Step 3: LOOP forj=1 to the number of verticesLetFIDSj=∅, NDS=∅, W=∅, and Sum = ∅;Puti=jStep 4: LOOP Forvi∈VLOOP Foru∈V−vi∪WIfφvi∧χviu=ψvi,viuThenFIDSj=vi∪FIDSj,W=u∪W,V−FIDSj∪W.END LOOP.Ifvi∈FIDSjThen go to Step 4.ElseNDS=vi∪NDS.END LOOP IfV=∅.Step 5: LOOP Forvi∈WLOOP Foru∈NDSIfφvi∧χviu=ψvi,viuThenFIDSj=FIDSj∪vi and NDS=NDS−uEND LOOPEND LOOPStep 6: IfNDS≠∅ Then FIDSj=FIDSj∪NDS.Step 7: Sum = [Sum SumFIDSj];j=j+1Ifj≤ number of verticesThen Go To Step 3ElseEND LOOP (Step 3)Step 8: FIDN is the minimum value of array Sum.


### 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 labsl1, l2, l3, l4, l5, and l6. 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.AssumeG=φ,χ,ψ is a FIG shown in Figure 6 having(19)φ=l1,l2,l3,l4,l5,l6;φl1=0.8,φl2=0.9,φl3=0.3,φl4=0.2,φl5=0.5,φl6=0.5,χl1l2=0.8,χl1l3=0.3,χl2l3=0.2,χl3l4=0.1,χl2l6=0.4,χl4l5=0.1,ψl1,l1l2=0.6,ψl2,l2l1=0.5,ψl1,l1l3=0.2,ψl3,l3l1=0.3,ψl2,l2l3=0.1,ψl3,l3l2=0.2,ψl3,l3l4=0.1,ψl4,l4l3=0.05,ψl2,l2l6=0.2,ψl6,l6l2=0.3,ψl4,l4l5=0.1,ψl5,l5l4=0.03.Figure 6
A FIG with γFI=1.Algorithm2 is applied.Algorithm 2:Application of Algorithm 1.
Step 1:V=φl1=0.8,φl2=0.9,φl3=0.3,φl4=0.2,φl5=0.5,φl6=0.5, E=χl1l2=0.8,χl1l3=0.3,χl2l3=0.2,χl2l6=0.4,χl3l4=0.1,χl4l5=0.1, and I=ψl1,l1l2=0.6,ψl2,l1l2=0.5,ψl1,l1l3=0.2,ψl3,l1l3=0.3,ψl2,l2l3=0.1,ψl3,l2l3=0.2,ψl2,l2l6=0.2,ψl6,l2l6=0.3,ψl3,l3l4=0.1,ψl4,l3l4=0.05,ψl4,l4l5=0.1,ψl5,l4l5=0.6.Step 2:V=φl4=0.2,φl3=0.3,φl5=0.5,φl6=0.5,φl1=0.8,φl2=0.9.Step 3: forj=1FIDS1=∅, NDS=∅, W=∅, and Sum=∅.puti=1Step 4:v1=φl4, l4 dominates l5, and l3 dominates l1,l2. Therefore, FIDS1=l3,l4, W=l5,l1,l2, and NDS=l6.Step 5: forl5,l1,l2∈W dominates l6∈NDS. This implies FIDS1=l3,l4,.Step 6: sinceNDS≠∅, therefore, FIDS1=l3,l4,l6.Step 7:Sum=φl3+φl4+φl6=0.3+0.2+0.5=1,1,1,1,1,1Forj=2,3,4,5,6FIDS=l3,l4,l6Forj=7Asj> number of verticesEND LOOP (Step 3)Step 8:FIDN=1.This shows that patients can visit any one of the labs from this set. The government should provide the resources to labsl3,l4, and l6 for the proper and easy conduction of tests for coronavirus.


## 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 labsl1, l2, l3, l4, l5, and l6. 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.AssumeG=φ,χ,ψ is a FIG shown in Figure 6 having(19)φ=l1,l2,l3,l4,l5,l6;φl1=0.8,φl2=0.9,φl3=0.3,φl4=0.2,φl5=0.5,φl6=0.5,χl1l2=0.8,χl1l3=0.3,χl2l3=0.2,χl3l4=0.1,χl2l6=0.4,χl4l5=0.1,ψl1,l1l2=0.6,ψl2,l2l1=0.5,ψl1,l1l3=0.2,ψl3,l3l1=0.3,ψl2,l2l3=0.1,ψl3,l3l2=0.2,ψl3,l3l4=0.1,ψl4,l4l3=0.05,ψl2,l2l6=0.2,ψl6,l6l2=0.3,ψl4,l4l5=0.1,ψl5,l5l4=0.03.Figure 6
A FIG with γFI=1.Algorithm2 is applied.Algorithm 2:Application of Algorithm 1.
Step 1:V=φl1=0.8,φl2=0.9,φl3=0.3,φl4=0.2,φl5=0.5,φl6=0.5, E=χl1l2=0.8,χl1l3=0.3,χl2l3=0.2,χl2l6=0.4,χl3l4=0.1,χl4l5=0.1, and I=ψl1,l1l2=0.6,ψl2,l1l2=0.5,ψl1,l1l3=0.2,ψl3,l1l3=0.3,ψl2,l2l3=0.1,ψl3,l2l3=0.2,ψl2,l2l6=0.2,ψl6,l2l6=0.3,ψl3,l3l4=0.1,ψl4,l3l4=0.05,ψl4,l4l5=0.1,ψl5,l4l5=0.6.Step 2:V=φl4=0.2,φl3=0.3,φl5=0.5,φl6=0.5,φl1=0.8,φl2=0.9.Step 3: forj=1FIDS1=∅, NDS=∅, W=∅, and Sum=∅.puti=1Step 4:v1=φl4, l4 dominates l5, and l3 dominates l1,l2. Therefore, FIDS1=l3,l4, W=l5,l1,l2, and NDS=l6.Step 5: forl5,l1,l2∈W dominates l6∈NDS. This implies FIDS1=l3,l4,.Step 6: sinceNDS≠∅, therefore, FIDS1=l3,l4,l6.Step 7:Sum=φl3+φl4+φl6=0.3+0.2+0.5=1,1,1,1,1,1Forj=2,3,4,5,6FIDS=l3,l4,l6Forj=7Asj> number of verticesEND LOOP (Step 3)Step 8:FIDN=1.This shows that patients can visit any one of the labs from this set. The government should provide the resources to labsl3,l4, and l6 for the proper and easy conduction of tests for coronavirus.


## 7. Comparative Analysis

In an FG, if vertexu 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. Therefore, 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=l3,l4,l6 comprising three different medical labs which are best for the tests of coronavirus with γFI=1. In Figure 6, if we remove all the IPs, we get an FG. In the case of an FG, l1 is dominating l2 and l3 because χl1l2=0.8=φl1∧φl2=0.8∧0.9 and χl1l3=0.3=φl1∧φl3=0.8∧0.3. The rest of the labs are not dominating each other. Therefore, DS D=l1,l4,l5,l6 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. This shows that our study is better than the previous studies.


## 8. Conclusion

The notion of domination in graphs is vital from a theoretical as well as an application point of view. Different authors have come out with more than thirty-five domination parameters. In this paper, we initiated novel ideas of FIGs, namely, order, size, a relationship between order and size, FIDS, strong FIDS, and weak FIDS. A relationship among order, FIDS, strong FIDS, and weak FIDS is also discussed. The formula to calculatedi, effective di, and Ndi of the FIG is explained. Our study will help us to find the order, size, FIDS, FIDN, di, effective di, and Ndi for the union, join, composition, tensor product, normal product, and the Cartesian product of FIGs if we know the order, size, FIDS, FIDN, di, effective di, and Ndi of two FIGs. The 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.



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*Source: 6682502-2021-05-11.xml*