# Novel Concepts of Domination in Vague Graphs with Application in Medicine

Authors: Xiaoli Qiang; Maryam Akhoundi; Zheng Kou; Xinyue Liu; Saeed Kosari
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

VG can manage the uncertainty relevant to the inconsistent and indeterminate information of all real-world problems, in which FGs possibly will not succeed in bringing about satisfactory results. The previous definitions’ restrictions in FGs have made us present new definitions in VGs. A wide range of applications have been attributed to the domination in graph theory for several fields such as facility location problem, school bus routing, modeling biological networks, and coding theory. Therefore, in this research, we study several concepts of domination, such as restrained dominating set (RDS), perfect dominating set (PDS), global restrained dominating set (GRDS), totalk-dominating set, and equitable dominating set (EDS) in VGs and also introduce their properties by some examples. Finally, we try to represent the application and importance of domination in the field of medical science and discuss the topic in today’s world, namely, the corona vaccine.


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## Body

## 1. Introduction

Graph theory began its adventure from the well-known “Konigsberg bridge problem.” This problem is frequently believed to have been the beginning of graph theory. In 1739, Euler finally elucidated this problem using graphs. Even though graph theory is an extraordinarily old concept, its growing utilization in operations research, chemistry, genetics, electrical engineering, geography, sociology, and so forth has reserved it fresh. In recent times, graph principle has been utilized in communication system (mobile, internet, etc.), computer layout, and so forth. In graph theory, it is far considered that the nodes, edges, weights, and so on are definite. To be exact, there may be no question concerning the existence of these objects. However, the real world sits on a plethora of uncertainties, indicating that, in some conditions, it is believed that the nodes, edges, and weights may additionally be or may not be certain. For instance, the vehicle travel time or vehicle capacity on a road network may not be identified or known exactly. To embody such graphs, Rosenfeld [1] brought up the idea of the “fuzzy graph” in 1975. Similar to set theory, the historical past of FG theory is the fuzzy set theory advanced by Zadeh [2] in 1965. Roy and Biswas investigated the importance of interval-valued fuzzy sets on medical diagnosis [3].The notion of vague set theory, generalization of Zadeh’s fuzzy set theory, was introduced by Gau and Buehrer [4] in 1993. The concepts of rough set, soft set, bipolar soft set, and neutrosophic set were introduced in [5–9]. Kauffman [10] represented FGs based on Zadeh’s fuzzy relation [11]. Mordeson et al. [12–14] described some results in FGs. Akram et al. [15–17] developed several concepts and results on FGs. Samanta et al. [18–21] represented FCGs and some remarks on BFGs. Shao et al. [22–28] investigated new concepts in VGs and fuzzy graphs. VG notion was defined by Ramakrishna in [29]. Borzooei and Rashmanlou [30–34] analyzed new concepts of VGs. Rashmanlou et al. [35–40] investigated new results in VGs. Ghorai and Pal [41] studied regular product vague graphs and product vague line graphs. A VG is referred to as a generalized structure of an FG that delivers more exactness, adaptability, and compatibility to a system when matched with systems running on FGs. Also, a PVG is able to concentrate on determining the uncertainty coupled with the inconsistent and indeterminate information of any real-world problem, where FGs may not lead to adequate results.Domination in VGs theory is one of the most widely used topics in other sciences, including psychology, computer science, nervous systems, artificial intelligence, decision-making theory, and combinations. Although the dominance of FGs has been stated by some researchers, due to the fact that VGs are wider and are more widely used than FGs, it is observed today that they are used in many branches of engineering and medical sciences. Likewise, they have been used in many applications for the formulation and solution of many problems in various areas of science and technology exemplified by computer networks, combinatorial analyses, physics, and so forth. In 1962, Ore [42] represented “domination” for undirected graphs, and he described the definition of minimum-DSs of nodes in a graph. A. Somasundaram and S. Somasundaram [43] introduced the DS and IDS in FGs. Gani et al. [44, 45] represented the fuzzy-DS and independent-DS notion utilizing strong arcs. The IDN and IR-DN in graphs are defined by Cockayne et al. [46] and Haynes et al. [47]. Parvathi and Thamizhendhi [48] described domination in intuitionistic fuzzy graphs. Jan et al. [49–51] investigated new concepts in interval-valued fuzzy graphs and cubic bipolar fuzzy graphs. Talebi et al. [52–54] introduced some results of domination in VGs, as well as new concepts in interval-valued intuitionistic fuzzy competition graph. So, in this research, we introduce different concepts of domination, such as RDS, PDS, GRDS, EDS, and total k-dominating set in VGS. In the end, an application of domination in medical immunization is introduced.


## 2. Preliminaries

In this section, some basic concepts of VGs are reviewed to facilitate the next sections.A graph denotes a pairG∗=V,E satisfying E⊆V×V. The elements of V and E are the nodes and edges of the graph G∗, correspondingly.An FG has the form ofξ=γ,ν, where γ:V⟶0,1 and ν:V×V⟶0,1 as is defined by νab≤γa∧γb, ∀a,b∈V, and ν is a symmetric fuzzy relation on γ and ∧ denotes the minimum.Definition 1.
(see [4]). A VSA is a pair tA,fA on set V where tA and fA are used as real valued functions which can be defined on V⟶0,1, so that tAa+fAa≤1,∀a∈V. The interval tAa,1−fAa is considered as the vague value of a in A.Definition 2.
(see [29]). A pair ξ=A,B is said to be a VG on a crisp graph G∗, where A=tA,fA is a VS on V and B=tB,fB is a VS on E⊆V×V such that tBab≤mintAa,tAb and fBab≥maxfAa,fAb, for each edge ab∈E.Definition 3.
(see [35]). A VG ξ is called complete VG if tBab=mintAa,tAb and fBab=maxfAa,fAb,∀a,b∈V.Definition 4.
(see [35]). The complement of a VG ξ=A,B is a VG ξ¯=A¯,B¯, where A¯=A=tA¯,fA¯ and B¯=tB¯,fB¯ are defined by the following:V¯=VtA¯a=tAa,fA¯a=fAa,∀a∈VtB¯ab=0,iftBab>0,mintAa,tAb,iftBab=0.fB¯ab=0,iffBab>0,maxfAa,fAb,iffBab=0.Definition 5.
(see [31]). A vague path in a VG ξ=A,B is a sequence of distinct nodes b1,b2,…,bnso that eithertBbkbk+1>0 or fBbkbk+1>0,∀1≤k≤n−1. It was shown by Pn.Definition 6.
An edgeab of a VG ξ is called an effective edge if tBab=tAa∧tAb and fBab=fAa∨fAb. Otherwise, it is called a noneffective edge.Definition 7.
(see [31]). An edge ab in a VG ξ is called a strong edge if tBab≥tB∞ab and fBab≤fB∞ab.Definition 8.
(see [31]). Let ξ=A,B be a VG. Let a,b∈V, then a dominates b in ξ, if there exists a strong edge between a and b.Definition 9.
(see [32]). The cartesian product of two VGs ξ1=A1,B1 and ξ2=A2,B2 of the graphs G1∗=V1,E1 and G2∗=V2,E2, denoted by ξ1×ξ2=A1×A2,B1×B2, is defined as follows:(1)itA1×tA2a,b=mintA1a,tA2b,fA1×fA2a,b=maxfA1a,fA2b,for alla,b∈V,iitB1×tB2a,a1a,b1=mintA1a,tB2a1b1,fB1×fB2a,a1a,b1=maxfA1a,fB2a1b1,for alla∈V1,a1b1∈E2,iiitB1×tB2a,a1b,a1=mintB1ab,tA2a1,fB1×fB2a,a1b,a1=maxfB1ab,fA2a1,for alla1∈V2,ab∈E1.Definition 10.
(see [35]). A node a in a VG ξ is said to be an isolated node if tBab=0 and fBab=0, for all b∈V and a≠b. That is, Na=∅.Definition 11.
(see [31]). S⊆V is called a DS in ξ if, ∀a∈V−S, ∃b∈S, so that a dominates b.Definition 12.
(see [31]). Let ξ=A,B be a VG. The vertex cardinality of S⊆V is defined as(2)S=∑a∈S1+tAa−fAa2.
Notations are shown in Table1.Table 1
Some basic notations.
NotationMeaningFGFuzzy graphVSVague setξVague graphDSDominating setRDSRestrained dominating setIDSIndependent dominating setIDNIndependent dominating numberIR-DNIrredundance dominating numberK-DSK-dominating setPDSPerfect dominating setPDNPerfect dominating numberK-DNK-dominating numberT-KDNTotal K-dominating numberMI-PDSMinimal perfect dominating setT-KDSTotal K-dominating setRDNRestrained dominating numberGRDNGlobal restrained dominating numberCVGComplete vague graphEDSEquitable dominating setGRDSGlobal restrained dominating set


## 3. Certain Notions of Domination in Vague Graphs

Definition 13.
Letξ=A,B be a VG and let k≥1 be an integer. A subset S⊆V is called a K-DS of ξ if, for each node a∈V−S, there exists an a−b vague path which includes at least k effective edges for b∈S. The K-DN of ξ, denoted by γkξ, is described as the minimum cardinality among all K-DS in ξ.Definition 14.
Letξ=A,B be a VG and let k≥1 be an integer. A subset S⊆V is called a T-KDS of ξ if, for each node a∈V, ∃ an a−b vague path that includes at least k effective edges for b∈S. The T-KDN of ξ, demonstrated by γtkξ, is described as the minimum cardinality between all T-KDS in ξ.Example 1.
Consider an example of a T-2DSof VG ξ shown in Figure 1.
It is clear from Figure1 that S=a,f is a minimal T-2DS of VG ξ. The T-2DN of ξ is γkξ=0.8.Figure 1
Total 2-dominating set ofξ.Definition 15.
Letξ be a VG. A set S⊆V is called an RDS of ξ if each node in V−S dominates a node in S and also a node in V−S. The RDN of ξ, demonstrated by γrξ, is described as the minimum cardinality of an RDS in ξ.Example 2.
Consider a VGξ=A,B as shown in Figure 2. It is obvious that S=c,d is an RDS of ξ. The RDN of ξ is γrξ=0.75.Figure 2
RDS ofξ.Definition 16.
Letξ be a VG. A set S⊆V is called GRDS of ξ if it is an RDS of both ξ and ξ¯. The GRDN of ξ, denoted by γgrξ, is described as the minimum cardinality of a GRDS in ξ.Example 3.
Considerξ and ξ¯ as shown in Figures 3 and 4. It is easy to see that S1=a,b and S2=c,d are GRDSs of ξ. The GRDN of ξ is γgrξ=0.9.Figure 3
Global RD set ofξ.Figure 4
ξ¯.Theorem 1.
Suppose thatξ=A,B is a CVG; then γrξ=V.Proof.
Assume thatξ=A,B is a CVG. Then, tBab=mintAa,tAb and fBab=maxfAa,fAb. Let S be the MI-RDS of ξ. Then, each node in V−S dominates a node in S and also a node in V−S. Hence, each node dominates all other nodes. So, γrξ=V.Definition 17.
Letξ=A,B be a VG. A subset S⊆V is called a PDS of ξ if, for each node b∈V−S, there exists exactly one node a∈S so that a dominates b.Definition 18.
We say that a PDSS is an MI-PDS if, for every a∈S, the set S−a is not a PDS in ξ. The minimum cardinality among all MI-PDSs is called the PDN of ξ and it was shown by γpifξ or simply γpif.Example 4.
Consider a VGξ=A,B as shown in Figure 5. It is obvious that S=a,d is an MI-PDS. The PDN of ξ is γpif=0.9.Figure 5
VGξ.Theorem 2.
Every DS in CVGξ=A,B is a PDS.Proof.
LetS be an MI-DS of a VG ξ. Since ξ is complete, each edge in ξ is one effective edge and each node b∈V−S is neighbor to exactly one node a∈S. Hence, each DS in ξ is a PDS.Definition 19.
The strong product of two VGsξ1=A1,B1 and ξ2=A2,B2 of the graphs G1∗=V1,E1 and G2∗=V2,E2, where V1∩V2=∅, denoted by G1×G2=A1×A2,B1×B2, is defined as follows:(3)itA1⊠tA2a,b=mintA1a,tA2b,fA1⊠fA2a,b=maxfA1a,fA2b,∀a,b∈V,iitB1⊠tB2a,a1a,b1=mintA1a,tB2a1b1,fB1⊠fB2a,a1a,b1=maxfA1a,fB2a1b1,for alla∈V1anda1b1∈E2,iiitB1⊠tB2a,a1b,a1=mintB1ab,tA2a1,fB1⊠fB2a,a1b,a1=maxfB1ab,fA2a1,∀a1∈V2andab∈E1,ivtB1⊠tB2a,a1b,b1=mintB1ab,tB2a1b1,fB1⊠fB2a,a1b,b1=maxfB1ab,fB2a1b1,∀ab∈E1anda1b1∈E2.Theorem 3.
Letξ1=A1,B1 and ξ2=A2,B2 be two VGs with V1∩V2=∅. The strong product ξ=ξ1×ξ2 remains connected even after removal of all noneffective edges in it.Proof.
Assume thatξ=ξ1×ξ2 is a strong product of two VGs ξ1 and ξ2. Let e=a,a1b,b1 be a noneffective edge in ξ; that is, tBa,a1b,b1<mintAa,a1,tAb,b1, and fBa,a1b,b1>maxfAa,a1,fAb,b1. Let ξ′=ξ−e, and suppose that ξ′ is disconnected. The edge e disconnects the graph into more than one component. Hence, there is no path among a,a1 and b,b1 except the edge e=a,a1,b,b1 in ξ′. This implies that tBa,a1b,b1=tAa,a1,tAb,b1 and fBa,a1b,b1=fAa,a1,fAb,b1, which is a contradiction. So, ξ′ is connected.Remark 1.
The strong productξ1⊠ξ2 of two connected vague graphs is a connected vague graph.Theorem 4.
If a nodea dominates a node b in ξ1 and a node a1 dominates a node b1 in ξ2, then the node ab does not dominate the node a1b1 in ξ1×ξ2.Proof.
Suppose thata dominates b in ξ1. Then ∃ an effective edge ab in ξ1, i.e., tBab=mintAa,tAb and fBab=maxfAa,fAb. Similarly, assume that a node a1 dominates a node b1 in ξ2, so that tBa1b1=mintAa1,tAb1 and fBa1b1=maxfAa1,fAb1. Now, by definition of Cartesian product, there does not exist any edge among the nodes a,a1 and b,b1 in ξ1×ξ2; i.e., tBa,a1b,b1=0 and fBa,a1b,b1=0. Therefore, a,a1 does not dominate b,b1 in ξ1×ξ2.Definition 20.
The direct product of two VGsξ1=A1,B1 and ξ2=A2,B2 of the graphs G1∗=V1,E1 and G2∗=V2,E2, where V1∩V2=∅, denoted by ξ1⊗ξ2=A1⊗A2,B1⊗B2, is defined as follows:(4)itA1⊗tA2a,b=mintA1a,tA2b,fA1⊗fA2a,b=maxfA1a,fA2b,∀a,b∈V,iitB1⊗tB2a,a1b,b1=mintB1ab,tB2a1b1,fB1⊗fB2a,a1b,b1=maxfB1ab,fB2a1b1,∀ab∈E1anda1b1∈E2.Note 1.
If a nodea dominates a node b in ξ1 and a node a1 dominates a node b in ξ2, then the node a,a1 dominates the node b,b1 in ξ1⊗ξ2. It is shown in Figure 6.Figure 6
Direct product ofξ1 and ξ2.Example 5.
Consider a VG as in Figure6. It is obvious that the node a dominates b in ξ1 and e dominates f in ξ2. Likewise, the node a,e dominates the node b,f in ξ1×ξ2.Definition 21.
Letξ=A,B be a VG. A subset S⊆V is called an EDS of ξ if, for each node b∈V−S, ∃ a node a∈S so that ab∈E, degta−degtb≤1, tBab=mintAa,tAb, degfa−degfb≥1, and fBab=maxfAa,fAb. The EDN of ξ, denoted by γeξ, is defined as the minimum cardinality of an EDS S.Example 6.
Consider a VGξ=A,B, as shown in Figure 7.
It is obvious that the MI-EDS of a VGξ is S=a. The EDN is γeξ=0.45. Note that an EDS S is called an MI-EDS of ξ, if, for each node b∈S, the set S−b is not an EDS.Figure 7
Equitable dominating set ofξ.Theorem 5.
Letξ1 and ξ2 be VGs on nonempty sets V1 and V2, respectively. Then,(5)γeξ1⊗ξ2≤minS1×V1,S2×V2.Proof.
Assume thatS1 and S2 are EDSs of minimum cardinality of ξ1 and ξ2, respectively. Then, for each node b∈V−S1, ∃ a node a∈S1 so that degtb−degta≤1, tBab=mintAa,tAb, degfb−degfa≥1, and fBab=maxfAa,fAb. Similarly, for each node c∈V−S2, ∃ a node d∈S2 so that degtd−degtc≤1, tBcd=mintAc,tAd, degfd−degfc≥1, and fBcd=maxfAc,fAd. That is, a dominates b in ξ1 and d dominates c in ξ2. Therefore, by Note 1, the node a,d dominates the node b,c in ξ1⊗ξ2. Thus, degta,d−degtb,c≤1, tBa,db,c=mintAa,d,tAb,c, degfa,d−degfb,c≥1, and fBa,db,c=maxfAa,d,fAb,c. So, γeξ1⊗ξ2≤minS1×V1,S2×V2.Theorem 6.
LetS1 and S2 be K-DSs of connected VGs ξ1=A1,B1 and ξ2=A2,B2, respectively; then 1ξ1×ξ2 is connected. (2) If S1 is a connected K-DS of ξ1, then S1×V2 is a connected K-DS of ξ1×ξ2. (3) If S2 is a connected K-DS of ξ2, then V1×S2 is a connected K-DS of ξ1×ξ2.Proof.
To prove thatξ1×ξ2 is connected, consider any two arbitrary distinct nodes a,d and b,c of V1×V2. Then, by definition of Cartesian product, ∃ a path between these two nodes in the following cases:(1)
Ifa=b, then, since ξ2 is a connected VG, ∃ a path P:d,d1,d2,…,c so that tBef>0 and fBef>0 for any two nodes e,f of vague path P. Hence, tBa,da,c=mintA1a,tB2cd and fBa,da,c=maxfA1a,fB2cd. So, P′:a,da,d1,a,d2,…\\,a,c is the vague path between a,d,b,c in ξ1×ξ2.(2)
Ifd=c, then, since ξ1 is a connected VG, ∃ a vague path Q=a,a1,a2,…,b so that tB1ab>0 and fB1ab>0 for any two nodes e,f of vague path Q. Hence, tBa,db,d=mintB1a,b,tA2d, and tBa,d,…,b,d is the vague path between a,d,b,c in ξ1×ξ2.(3)
Ifa≠b and c≠d, then, by case 1, ∃ a vague path between the nodes a,d and a,c in ξ1×ξ2. Likewise, by case 2, ∃ a vague path between the nodes a,d and b,d in ξ1×ξ2. Hence, the union of these two disjoint vague paths is a vague path between the nodes a,d and b,c in ξ1×ξ2. Now, if S1 and S2 are K-DSs of ξ1 and ξ2, respectively, then γkξ1×ξ2=minV1×S2,S1×V2. So, V1×S2 and S1×V2 are K-DSs of ξ1×ξ2 and the connectivity can be proved similarly.Theorem 7.
Letξ1 and ξ2 be VGs on nonempty sets V1 and V2, respectively. Let S1 and S2 be K-DSs of ξ1 and ξ2; then S1×V2 is an independent K-DS of ξ1×ξ2 if and only if S1 is k-independent and itB1ab<tA2d and fB1ab>fA2d, for a,b∈S1 and d∈V2; iitB2cd<tA1a and fB2cd>fA1a, for a∈S1 and c,d∈V2; and iiitB2cd<mintA2d,tA2c and fB2cd>maxfA2d,fA2c, for c,d∈V2.Proof.
To prove that each two distinct nodesa,d,b,c in S1×V1 are not neighbor, we consider three conditions. If a=b, then(6)tBa,da,c=mintA1a,tB2cd<mintA1a,mintA2c,tA2d<mintAa,d,tAa,c,fBa,da,c=maxfA1a,fB2cd>maxfA1a,maxfA2c,fA2d>maxfAa,d,fAa,c.
Ifc=d, the result is obtained by independence of a,b of S1.
Ifa≠b and c≠d, then, by definition, we have tBa,db,c=0 and fBa,db,c=0. So, a,d,b,c are not neighbors in G1×G2. Conversely, assume that iii is false. That is, ∃ nodes b,c∈V2, so that tB2bc=mintA2b,tA2c and fB2bc=maxfA2b,fA2c. Let a∈S1; then(7)tBa,ba,c=mintA1a,tB2bc=mintA1a,mintA2b,tA2c=mintAa,b,tAa,c,fBa,ba,c=maxfA1a,fB2bc=maxfA1a,maxfA2b,fA2c=maxfAa,b,fAa,c.
Hence,S1×V2 is not independent. Therefore, condition iii is true, i.e., tB2b,c<mintA2b,tA2c and fB2b,c>maxfA2b,fA2c.


## 4. Application of Domination in Medical Sciences

One year has passed since the beginning of the coronary heart disease pandemic in the world. During this year, many people have died in all countries and the lives of all people have been affected. During this period, no definitive cure for this disease has been found and many countries, in attempts to develop a corona vaccine to prevent the disease, are highly contagious. China, Russia, India, and the United States are among these countries, and of course Iran has made efforts in this regard. Most vaccines are in the final stages of production and are about to be sacrificed, and many countries have prepurchased several million doses of these vaccines at this stage. Some vaccines are artificially made from antibodies created following disease; and some other viruses have been killed or weakened. The effectiveness of the study population and less side effects are the most important issues in choosing a vaccine. Relations between countries and political issues between them are also factors affecting the type and amount of vaccines purchased. Although it has been said that the whole world should be safe and these vaccines should be given to all countries, the issues mentioned are definitely on the time required to establish comprehensive security in each country will be effective. Therefore, in this paper, we try to discuss the application and importance of domination in the field of medical sciences and discuss the topic in today’s world, namely, the corona vaccine. For this purpose, we consider five countries: Iran, China, USA, India, and Russia. In fact, we want to buy the most effective vaccine for Iran, given the effectiveness of the vaccine and the political relations that exist between this country and other countries. In this vague graph, the nodes representing the countries and edges indicate the extent of political relations and friendship between the two countries.The vertex of China0.6,0.3 shows that the efficiency and effectiveness of vaccines in this country are 60%, and, unfortunately, it is as harmful as 30%. The edge Iran-India shows that only 20% on the friendship is established between the two countries and there is 70% of the political conflict and tension between them. The restrained dominating sets (RDSs) for Figure 8 are as follows:(8)S1=a,b,S2=a,d,S3=b,c,S4=b,e,S5=c,d,S6=d,e,S7=a,b,c,S8=a,b,e,S9=b,c,e,S10=c,d,e.Figure 8
VGξ.After calculating the cardinality ofS1,S2,…,S10, we obtain(9)S1=1,S2=1.05,S3=1.3,S4=1.1,S5=1.35,S6=1.15,S7=1.65,S8=1.45,S9=1.75,S10=1.8.It is clear thatS1 has the smallest size among other RDSs, so we conclude that it can be the best choice because, first, China has the most effective vaccine in terms of susceptibility to the virus, and, second, there is a relatively good friendship between Iran and China. Therefore, governments must provide the necessary facilities for the delivery of efficient and useful vaccines to deprived countries in order to prevent the transmission of this deadly virus to the rest of the people as soon as possible.


## 5. Conclusion

Domination in FGs theory is one of the most widely discussed topics in other sciences including psychology, computer science, nervous systems, artificial intelligence, and combinations. They have also been utilized in summarizing document and in designing secure systems for electrical grids. Hence, in this paper, we introduced several concepts of domination, such as RDS, PDS, GRDS, EDS, and total K-dominating set in VGs and also investigated their properties by some examples. Finally, we described an application of domination in the field of medical sciences and discussed a topic in today’s world, namely, the coronavirus. In our future work, we will introduce vague incidence graphs and study the concepts of connected perfect dominating set, regular perfect dominating set, inverse perfect dominating set, and independent perfect dominating set on vague incidence graph.



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