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ORIGINAL RESEARCH article

Front. Phys., 18 March 2021
Sec. Statistical and Computational Physics

Equitable Domination in Vague Graphs With Application in Medical Sciences

  • Institute of Computing Science and Technology, Guangzhou University, Guangzhou, China

Considering all physical, biological, and social systems, fuzzy graph (FG) models serve the elemental processes of all natural and artificial structures. As the indeterminate information is an essential real-life problem, which is mostly uncertain, modeling the problems based on FGs is highly demanding for an expert. Vague graphs (VGs) 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. In addition, VGs are a very useful tool to examine many issues such as networking, social systems, geometry, biology, clustering, medical science, and traffic plan. The previous definition restrictions in FGs have made us present new definitions in VGs. A wide range of applications has been attributed to the domination in graph theory for several fields such as facility location problems, school bus routing, modeling biological networks, and coding theory. Concepts from domination also exist in problems involving finding the set of representatives, in monitoring communication and electrical networks, and in land surveying (e.g., minimizing the number of places a surveyor must stand in order to take the height measurement for an entire region). Hence, in this article, we introduce different concepts of dominating, equitable dominating, total equitable dominating, weak (strong) equitable dominating, equitable independent, and perfect dominating sets in VGs and also investigate their properties by some examples. Finally, we present an application in medical sciences to show the importance of domination in VGs.

1 Introduction

Many real-world situations can accessibly be explained by means of a diagram consisting of a set of points together with lines joining certain pairs of these points. Notice that in such diagrams one is mainly interested in whether two given points are joined by a line; the manner in which they are joined is immaterial. A mathematical abstraction of situations of this type gives rise to the concept of a graph. To exemplify the objects and the connection between them, the graph nodes and edges are being employed accordingly. FGs are intended to demonstrate the connection structure among objects so that the concrete object existence (node) and the relationship between two objects (edge) are matters of degree. FG models are advantageous mathematical tools for addressing the combinatorial problems in several fields integrating research, algebra, computing, environmental science, and topology. Owing to the vagueness and ambiguity of natural existence, fuzzy graphical models outperform other graphical models. In 1965, Zadeh [44] proposed fuzzy set (FS) theory as a model for the exemplification of uncertainty and vagueness in real-world systems. FS theory is an exceedingly influential mathematical tool for resolving approximate reasoning-related problems. By defining the VS notion through changing the value of an element in a set with a subinterval of [0,1], Gau and Buehrer [13] introduced the VS theory. More probabilities are illustrated by VSs compared to FSs. A VS is more effective for explaining the false membership degree existence. Many events in the real world provided the incentive for introducing FGs. Kauffman [15] described FGs based on Zadeh’s fuzzy relation [44]. Kosari et al. [16] defined vague graph structure. Fuzzy Graph was introduced by Rosenfeld [32]. Akram et al. [16] proposed new definitions on FGs. Mordeson et al. [1719] studied some results in FGs. Borzooie and Rashmanlou [711] analyzed several concepts of VGs. Samanta et al. [3338] defined fuzzy competition graphs and some bipolar fuzzy graph results. Shao et al. [25, 26, 3941] introduced new results in FGs and intuitionistic fuzzy graphs. Ramakrishna [24] presented VG concepts and examined their properties. Rashmanlou et al. [2731] advanced new concepts in VGs.

A VG is a generalized structure of a FG that provides more exactness, adaptability, and compatibility to a system when matched with systems that run on FGs. In addition, a VG is capable of concentrating on determining the uncertainty coupled with the inconsistent and indeterminate information of any real-world problem, where FGs may not lead to adequate results. There exist an extensive array of applications for domination in graph theory in several fields such as school bus routing, facility location problems, and electrical networks. The domination idea was introduced first in the chessboard problem. In 1962, Ore [22] pioneered to apply the expression “domination” for undirected graphs. Somasundaram [42] presented the domination and independent domination in FGs. Gani and Chandrasekaran [20, 21] investigated the fuzzy-DS and independent-DS notion utilizing strong arcs. Cockayne [12] and Hedetniemi [14] described the independent and irredundance domination number in graphs. The domination concept in intuitionistic fuzzy graphs was examined by Parvathi and Thamizhendhi [23]. Talebi and Rashmanlou [43] studied new applications of domination in VGs. Domination in VGs has several uses in different fields. Hence, this study seeks to consider different concepts of dominating, equitable dominating, total equitable dominating, weak (strong) equitable dominating, equitable independent, and perfect dominating sets in VGs and investigate their properties by some examples.

Previously, many emergency patients died due to delays in transportation to the hospital; therefore, we introduce an application in the transportation system to show the importance of domination in VGs.

2 Preliminaries

In this section, to consider the stage for our analysis and to facilitate the following of our discussion, a brief overview of some of the basic definitions is introduced. A graph denotes a pair G*=(V,E) satisfying EV×V. The elements of V and E are the nodes and edges of the graph G*, correspondingly.

An FG has the form of ξ=(γ,ν), where γ:V[0,1] and ν:V×V[0,1] are defined as ν(ab)γ(a)γ(b), a,bV, and ν is a symmetric fuzzy relation on γ and denotes the minimum.

Definition 2.1. [13] A VS A 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 tA(a)+fA(a)1, for all a belongs V. The interval [tA(a),1fA(a)]is considered as the vague value of a in A.G* will be a crisp graph (V,E) and ζ a VG (A,B) throughout this article.

Definition 2.2. [13] The support of a vague set A=(tA,fA), denoted by supp(A), is defined as supp(A)=suppt(A)suppf(A), where suppt(A)={a|tA(a)>0}, suppf(A)={a|fA(a)>0}.

Definition 2.3. [24] A pair ζ=(A,B)is called a VG on a crisp graph G* that A=(tA,fA)is a VS on V and B=(tB,fB) is a VS on EV×Vsuch that tB(ab)min(tA(a),tA(b)) and fB(ab)max(fA(a),fA(b)), for each edge abE.

Definition 2.4. [8] Let ζ=(A,B)be a VG. Then, (i) the vertex cardinality of ζ is described by |A| and defined as |A|=aiV(tA(ai),fA(ai)). and (ii) the edge cardinality of ζ is described by |B| and defined as

|B|=ai,ajV(tB(aiaj),fB(aiaj)).

Definition 2.5. [8] Let ζ=(A,B) be a VG. If ai,ajV, then the t-strength of connectedness between ai and aj is defined as tB(ai,aj)=sup{tBk(ai,aj)|k=1,2,,n} and f-strength of connectedness is as fB(ai,aj)=inf{fBk(ai,aj)|k=1,2,,n}. In addition, we have

tBk(ab)=sup{tB(a,b1)tB(b1,b2)tB(b2,b3)tB(bk1,b)|(a,b1,b2,,bk1,b)V}.

and

fBk(ab)()=inf{fB(a,b1)fB(b1,b2)fB(b2,b3)fB(bk1,b)|(a,b1,b2,,bk1,b)V}.

Definition 2.6. [8] An edge abin a VG ζ=(A,B)is called strong edge if tB(ab)(tB)(ab)and fB(ab)(fB)(ab).

Definition 2.7. [8] Two nodes ai and aj in a VG ζ=(A,B) are called to be adjacent if either one of the following conditions holds. (i)tB(aiaj)>0 and fB(aiaj)0. (ii)tB(aiaj)0 and fB(aiaj)>0, ai,ajV. (iii) A node a in a VG ζ is called an isolated node if tB(ab)=0and fB(ab)=0, bV, ab. That is, N(a)=.

Definition 2.8. [9] The degree of a node a in a VG ζ is defined as the sum of weights of edges incident to a. It is defined by dζ(a)=(degt(a),degf(a)). The minimum degree of ζ is δ(ζ)=min{dζ(a)|aV}. The maximum degree of ζ is Δ(ζ)=max{dζ(a)|aV}.

Definition 2.9. [8] Let ζ=(A,B)be a VG. Suppose that a,bV; then, a dominates b in ζ if a strong edge between a and b.

Definition 2.10. [8] A subset S of V is called a DS in ζ if for each aVS, bSso that a dominates b. A DS S of a VG ζ is referred to as a Minimal DS if no proper subset of S is a DS.

Definition 2.11. [8] If ζ is a VG, then the vertex cardinality of SV is defined as follows:

|S|=|aS1+tA(a)fA(a)2|.

All the basic notations are shown in Table 1.

TABLE 1
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TABLE 1. Some basic notations.

3 Domination in VGs

Definition 3.1. Let ζ=(A,B)be a VG. The equitable neighborhood (EN) of a node aV, described by EN(a) and defined as EN(a)=(ENt(a),ENf(a)), where ENt(a)={bV|bN(a),|degt(b)degt(a)|1}, tB(ab)=min{tA(b),tA(a)}, ENf(a)={bV|bN(a),|degf(b)degf(a)|1}, fB(ab)=max{fA(b),fA(a)}.

Definition 3.2. The END of a node aV, denoted by degEN(a), is defined as degEN(a)=(degENt(a),degENf(a)), where degENt(a)=bEN(a)tB(ab)and degENf(a)=bEN(a)fB(ab).The minimum END, denoted by δEN(ζ), is defined as δEN(ζ)=(δENt(ζ),δENf(ζ)), where δENt(ζ)=min{degENt(a)|aV} and δENf(ζ)=min{degENf(a)|aV}.The maximum END, denoted by ΔEN(ζ), is defined as ΔEN(ζ)=(ΔENt(ζ),ΔENf(ζ)), where ΔENt(ζ)=max{degENt(a)|aV} and ΔENf(ζ)=max{degENf(a)|aV}.

Example 3.3. Let ζ=(A,B) be a VG on V={a,b,c,d,e,f} so that A=(tA,fA) is a vague subset of V, in Table 2, and B=(tB,fB) is a vague subset of V×V defined in Table 3. The VG is shown in Figure 1. By simple calculation, we have EN(a)={b}, EN(b)={a,f,d}, EN(c)={e}, EN(d)={b}, EN(e)={c}, and EN(f)={b}.The ENDs of nodes are calculated as degEN(a)=(0.2,0.5), degEN(b)=(0.8,1.6), degEN(c)=(0.1,0.6), degEN(d)=(0.3,0.5), degEN(e)=(0.1,0.6), and degEN(f)=(0.3,0.6). The minimum END of VG ζ is δEN(ζ)=(0.1,0.5) and the maximum END of a VG ζ is ΔEN(ζ)=(0.8,1.6).

TABLE 2
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TABLE 2. Vague set A on set V.

TABLE 3
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TABLE 3. Vague set B in V×V.

FIGURE 1
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FIGURE 1. EN degree of nodes in ζ.

Definition 3.4. Let ζ=(A,B) be a VG. A node bV is called an EIN in ζ if, for each aV, |degt(a)degt(b)|>1, tB(ab)<min{tA(a),tA(b)}, and |degf(a)degf(b)|<1, fB(ab)>max{fA(a),fA(b)}, i.e., EN(b)=.

Example 3.5.Consider a VG ζ=(A,B) on V={a,b,c,d,e} which is shown in Figure 2.From Figure 2, we have deg(a)=(1.3,1.3), deg(b)=(1.2,0.8), deg(c)=(1.4,1.3), deg(d)=(0.2,0.6), and deg(e)=(0.1,0.6). Since |degt(a)degt(e)|>1, tB(ae)<min{tA(a),tA(e)}, and |degf(a)degf(e)|<1, fB(ae)>max{fA(a),fA(e)}, i.e., EN(e)=. Also, |degt(c)degt(d)|>1, tB(cd)<min{tA(c),tA(d)}, and |degf(c)degf(d)|<1, fB(cd)>max{fA(c),fA(d)}, i.e., EN(d)=. Hence, e and d are isolated nodes in ζ.

FIGURE 2
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FIGURE 2. Equitable isolated node.

Definition 3.6.Let ζ be a VG. A subset SV is called an EDS of ζ if for each node bVS, a node aS so that abE, |degt(a)degt(b)|1, tB(ab)=min{tA(a),tA(b)}, and |degf(a)degf(b)|1, fB(ab)=max{fA(a),fA(b)}. The EDN of ζ, denoted by γe(ζ), is defined as the minimum cardinality of an EDS of S.

Definition 3.7.An EDS S of a VG ζ is called a MI-EDS of ζ if for each node bS, the set S{b} is not an EDS; i.e., no proper subset of S is an EDS of ζ.

Example 3.8.Consider a VG ζ=(A,B), as shown in Figure 3.It is easy to show that the MI-EDS of VG ζ is S={a}. The EDN of ζ is γe(ζ)=0.45.

FIGURE 3
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FIGURE 3. Equitable dominating set of ζ.

Definition 3.9.Let ζ=(A,B) be a VG. ζ is called a DEVG if, for each bV, a node aV so that absupp(B), |degt(a)degt(b)|1, tB(ab)=min{tA(a),tA(b)}, and |degf(a)degf(b)|1, fB(ab)=max{fA(a),fA(b)}.

Example 3.10.Consider a VG ζ, as shown in Figure 4. Simple calculations show that ζ is a degree equitable VG.

FIGURE 4
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FIGURE 4. Degree equitable VG.

Definition 3.11.A subset IV is called an EIS of a VG ζ=(A,B) if |degt(a)degt(b)|>1, tB(ab)<min{tA(a),tA(b)}, and |degf(a)degf(b)|<1, fB(ab)>max{fA(a),fA(b)}, for all a,bI. The EIN of ζ, denoted by γie(ζ), is defined as the minimum cardinality of an EIS of ζ.

Definition 3.12.An EIS I is called a MA-EIS of ζ if, for each node aVI, the set I{a} is not an EIS.

Example 3.13. Consider a VG ζ=(A,B) given in Figure 5. It is clear that S={a,d} is maximal EIS. The EIN is γie(ζ)=1.25.

FIGURE 5
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FIGURE 5. EIS of ζ.

Definition 3.14.Let ζ=(A,B) be a VG. For any two nodes a,bV, a strongly dominates b in ζ if tB(ab)=min{tA(a),tA(b)}, fB(ab)=max{fA(a),fA(b)}, and degζt(a)degζt(b), degζf(a)degζf(b). Similarly, a weakly dominates b if tB(ab)=min{tA(a),tA(b)}, fB(ab)=max{fA(a),fA(b)}, degζt(b)degζt(a), and degζf(b)degζf(a).

Definition 3.15.An EDS SV is called a weak (strong) EDS of ζ if, for each node bVS, at least one node aS so that a weakly (strongly) dominates b. The weak (strong) EDN of ζ, denoted by γwe(ζ)(γse(ζ)), is called as the minimum cardinality of a weak (strong) EDS of ζ.

Example 3.16.Consider the VG ζ=(A,B) given in Figure 6. It is easy to see that the strong EDSs of ζ are S1={a,b} and S2={a,d}. The strong EDN of ζ is γse(ζ)=0.8.

FIGURE 6
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FIGURE 6. Strong EDS of ζ.

Theorem 3.17.Let ζ=(A,B) be a VG. An EDS S of ζ is a minimal EDS if and only if, for every bS, one of the following conditions holds: (i) b is an isolated node in S, (ii)EN(b)(VS), or (iii) a node aVS so that EN(a)S={b}.

Proof. Let ζ be a VG with minimal EDS S; then, for each node bS, the set S'=S{b} is not an EDS. Hence, at least one node aVS' so that a is not dominated by any node in S'. So, we have two cases.

If a=b, then b is an isolated node in S; i.e., b is not neighbor to any node cS so that tB(bc)=0=fB(bc). Thus, EN(b)(VS); that is, every node in S has a neighbor in VS.

If ab, i.e., aVS, then a is dominated by some node of S but not dominated by any node in S'. Hence, a is neighbor only to one node bS, so EN(a)S={b}.

Conversely, suppose that S is an EDS of a VG ζ and, for every node bS, one of the given conditions holds. Assume that S is not a MI-EDS, then clearly a node bS so that S{b} is an EDS of ζ. Therefore, b is neighbor to at least one node of set S{b}; i.e., b is not an isolated node in S, and thus condition (i) is false. In addition, if we get S'=S{b} as an EDS of ζ, then each node of VS' is neighbor to at least one node in S'. Hence, conditions (ii) and (iii) are also false which is a contradiction. ∎

Theorem 3.18.Let ζ=(A,B) be a VG with order o(ζ), then: (i)γe(ζ)γse(ζ)o(G*)ΔENt(ζ), (ii)γe(ζ)γwe(ζ)o(G*)δENt(ζ).

Proof.According to definition, every weak (strong) EDS of a VG ζ is an EDS of ζ, γe(ζ)γse(ζ) and γe(ζ)γwe(ζ). Let a and b be two arbitrary nodes of ζ. If degENt(b)=ΔENt(ζ) and degENt(a)=δENt(ζ), then VEN(b) is a strong EDS of ζ and VEN(a) is a weak EDS of ζ. Hence, γse(ζ)|VEN(b)| and γwet(ζ)|VEN(a)|, i.e.,

γse(ζ)o(G*)ΔENt(ζ)

and

γwe(ζ)o(G*)δENt(ζ).

Theorem 3.19.Let ζ be a VG without single nodes and S be a MI-EDS of ζ; then VSis an EDS of ζ.

Proof.Let ζ be a VG with MI-EDS S; then, for each node bS, there is at least one node aVS so that |degt(a)degt(b)|1, tB(ab)=min{tA(a),tA(b)} and |degf(a)degf(b)|1, fB(ab)=max{fA(a),fA(b)}. Hence, VS dominates each element of S. So, VS is an EDS of ζ. ∎

Theorem 3.20. Let ζ be a VG with EIDS I; then I is both a MI-EDS and a MA-EIS of ζ. Conversely, any MA-EIS I of a VG ζ is an EIDS of ζ.

Proof. Let ζ be a VG with EIDS K; then, for each node bVK, the set K{b} is not an EIS and the set K{b}is not an EDS of ζ. So, K is both a MI-EDS and a MA-EIS of ζ.Conversely, assume that K is a MA-EIS of ζ; then, for each node bVK, the set K{b} is not an EIS of ζ. Hence, the set K dominates each node aVK and so K is an EDS of ζ. Therefore, K is an EIDS of ζ. ∎

Theorem 3.21.A subset IVis an EIS and EDS of a VG ζ if and only if I is a MA-EIS of ζ.

Proof.Assume that K is both an EDS and an EIS of a VG ζ. Suppose that K is not a MA-EIS of ζ, then clearly there exists a node aVK so that K{a} is an EIS; namely, a is not dominated by any node bK that shows K is not an EDS of ζ, a contradiction, so K is a MA-EIS of ζ. Conversely, let K be a MA-EIS of ζ; then, for each node aVK, the set K{a} is not an EIS of ζ. Hence, the set K dominates each node aVK; that is, K is an EDS of ζ. So, K is both an EDS and an EIS of ζ. ∎

Definition 3.22.A total-EDS (TEDS) of a VG ζ=(A,B) is a subset SVif for each node bV, at least one node aS so that abE(ζ), |degt(a)degt(b)|1, tB(ab)=min{tA(a),tA(b)}, and |degf(a)degf(b)|1, fB(ab)=max{fA(a),fA(b)}. The TEDN of ζ, denoted by γte(ζ), is defined as the minimum cardinality of a TEDN S.

Definition 3.23.A TEDS S of a VG ζ is called a minimal TEDS if, for each node bS, the set S{b}is not a TEDS; i.e., no proper subset of S is a TEDS of ζ.

Example 3.24.Consider a VG ζ=(A,B), as shown in Figure 7. It is clear that {a,b} and {b,c} are TEDSs of ζ.

FIGURE 7
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FIGURE 7. TEDS of ζ.

Theorem 3.25.Let ζ be a VG with no isolated nodes; then γt(ζ)γte(ζ).

Proof.As each TEDS of a VG ζ is a total dominating set, so γt(ζ)γte(ζ). ∎

Definition 3.26.Let ζ=(A,B) be a VG. A subset SV is called a PDS of ζ if, for each node bVS, there exists exactly one vertex aSso that a dominates b.

Definition 3.27.A PDS S of a VG ζ is called a minimal PDS if, for each aS, the set S{a} is not a PDS in ζ. The minimum cardinality between all MI-PDSs is called the PDN of ζ and it is denoted by γpif(ζ) or simply γpif.

Example 3.28.Consider a VG ζ=(A,B) given in Figure 8. By simple computation, it is clear that S={a,d}is a MI-PDS. The PDN of ζ is γpif=0.9.

FIGURE 8
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FIGURE 8. VG ξ.

4 The Application of VDS in Medical Sciences

In the past, many emergency patients died due to the delays in transportation to the hospital, but today the number has dropped dramatically. Traffic problems in cities are one of the factors influencing this delay. In addition, the specialization of hospitals has meant that each patient must be transferred to the relevant hospital based on the main complaint, even though this specialized hospital is further away than other available hospitals. Therefore, in this study, we have tried to identify the nearest hospital based on distance, traffic load, and patient complaints. For this purpose, we consider four hospitals located in one city. We show hospitals as B, C, D, and E. In this vague graph, one vertex represents the patient’s home and other vertices are related to the hospitals in the city. The edges indicate the accumulation of cars in the city. (See Figure 9).

FIGURE 9
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FIGURE 9. VG ζ.

The node B(0.3,0.5) means that it has 30% of the necessary facilities for treating the patient and unfortunately lacks 50% of the necessary equipment.

The edge AB shows that only 30% of the patient’s transport route to the hospital by ambulance has a low traffic load, and unfortunately 70% of the route between these two points has a heavy traffic load during most hours of the day. The EDSs for Figure 8 are as follows:

S1={A,B},
S2={A,D},
S3={A,B,C},
S4={A,B,D},
S5={A,B,E},
S6={A,C,D},
S7={A,D,E},
S8={B,C,E},
S9={C,D,E},
S10={A,B,D,E},
S11={A,C,D,E},
S12={B,C,D,E},
S13={A,B,C,E},
S14={A,B,C,D}.

After calculating the cardinality of S1,,S14, we obtain

|S1|=0.7,
|S2|=0.8,
|S3|=0.9,
|S4|=1.2,
|S5|=1.1,
|S6|=1,
|S7|=1.3,
|S8|=1,
|S9|=1.1,
|S10|=1.6,
|S11|=1.4,
|S12|=1.5,
|S13|=1.3,
|S14|=1.4.

It is obvious that S1 has the smallest size between other DSs; hence, we conclude that it can be the best choice because first there is more free space for the ambulance from the patient’s home to hospital B, so that it can get the patient to the desired location faster, saving time and money. Second, hospital B has more medical services compared to other hospitals. So, the government should invest more on widening roads and controlling traffic between cities so that ambulances can transport patients to the relevant specialized hospitals faster.

5 Conclusion

Considering the precision, elasticity, and compatibility in a system, vague models outweigh the other FGs. The VG concept generally has a large variety of applications in different areas such as computer science, operation research, topology, and natural networks. Domination in graph theory has a wide range of applications in several fields such as facility location problems, school bus routing, and coding theory. Therefore, in this research, we described several concepts of dominating sets, ED, TED, weak (strong) ED, EISs, and PDS, in VGs and also studied their properties incorporating some basic examples. Finally, we introduced an application of domination in the transportation system. Future research will hold the investigation of new concepts of vague planer graphs, vague bridges, vague cycles, and vague competition graphs and represent their applications in medical sciences and social networks.

Data Availability Statement

The original contributions presented in the study are included in the article/Supplementary Material; further inquiries can be directed to the corresponding author.

Author Contributions

All authors have contributed equally to this work. All authors have read and agreed to the possible publication of the manuscript.

Funding

This work was supported by the National Key R&D Program of China (No. 2018YFB1005100).

Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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Keywords: vague set, vague graph, equitable dominating set, equitable neighborhood, medical science, Mathematics Subject Classification: 05C99, 03E72

Citation: Rao Y, Kosari S, Shao Z, Qiang X, Akhoundi M and Zhang X (2021) Equitable Domination in Vague Graphs With Application in Medical Sciences. Front. Phys. 9:635642. doi: 10.3389/fphy.2021.635642

Received: 30 November 2020; Accepted: 15 January 2021;
Published: 18 March 2021.

Edited by:

Jinjin Li, Shanghai Jiao Tong University, China

Reviewed by:

Veena Mathad, University of Mysore, India
Hossein Rashmanlou, University of Mazandaran, Iran

Copyright © 2021 Rao, Kosari, Shao, Qiang, Akhoundi and Zhang. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Saeed Kosari, saeedkosari38@gzhu.edu.cn

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