# New Concepts in the Vague Graph Structure with an Application in Transportation

Authors: Xiaolong Shi; Saeed Kosari
Journal: Journal of Function Spaces
 (2022)

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## Abstract

Fuzzy graph (FG) models take on the presence being ubiquitous in environmental and fabricated structures by humans, specifically the vibrant processes in physical, biological, and social systems. Owing to the unpredictable and indiscriminate data which are intrinsic in real life and problems being often ambiguous, it is very challenging for an expert to exemplify these problems through applying an FG. Vague graph structure (VGS), belonging to the FG family, has good capabilities when facing with problems that cannot be expressed by FGs. VGS can handle the vagueness connected with the incompatible and determinate information of any real-world problem, where FGs may not succeed to bear satisfactory results. The previous definitions’ restrictions in FGs have led us to propose new definitions in the VGS. Domination is one of the key issues that has many applications in computer science and social networks. Today, many researchers are trying to prove its application in medical sciences and psychology. Therefore, in this paper, different concepts related to domination in VGSs such as Ni-dominating set, vague full dominating set, minimalNi-DS, and strong capacity Ni-dominating set are defined using some examples. Finally, an application of domination in medical sciences has been presented.


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## Body

## 1. Introduction

The FG concept serves as one of the most dominant and extensively employed tools for multiple real-word problem representations, modeling, and analysis. To specify the objects and the relations between them, the graph vertices or nodes and edges or arcs are applied, respectively. Graphs have long been used to describe objects and the relationships between them. Many of the issues and phenomena around us are associated with complexities and ambiguities that make it difficult to express certainty. These difficulties were alleviated by the introduction of fuzzy sets by Zadeh [1]. This concept established well-grounded allocation membership degree to elements of a set. Rosenfeld [2] proposed the idea of the FG in 1975. The existence of a single degree for a true membership could not resolve the ambiguity on uncertain issues, so the need for a degree of membership was felt. Afterward, to overcome the existing ambiguities, Gau and Buehrer [3] introduced false-membership degrees and defined a vague set as the sum of degrees not greater than 1. Kauffman [4] represented FGs based on Zadeh’s fuzzy relation [5, 6]. Samanta and Pal [7] defined fuzzy competition graphs. Vague graph notion was introduced by Ramakrishna in [8]. Borzooei et al. [9–11] investigated new concepts of VGs. Ramakrishnan and Dinesh [12, 13] studied on generalized FGSs. Harinath and Lavanya [14] represented FGSs based on wheel and star graphs. Akram et al. [15–18] defined intuitionistic fuzzy graph structures and m-polar fuzzy graph structures. VGS, belonging to the FG family, has good capabilities when facing with problems that cannot be expressed by fuzzy graphs. A VGS is referred to as a generalized structure of an FG that conveys more exactness, adaptability, and compatibility to a system when coordinated with systems running on FGs. Also, a VGS is able to concentrate on determining the uncertainly coupled with the inconsistent and indeterminate information of any real-world problem, where FGs may not lead to adequate results. Kosari et al. [19] defined the vague graph structure and studied new concepts of irregularity on it.Domination is one of the most important topics that has many applications in social groups and fuzzy social networks. Today, many researchers are trying to prove its application in medical sciences and psychology. For the first time, the concept of DS in graph theory was introduced by Ore [20]. The domination in the FG was defined by Somasundaram [21, 22]. Nagoor Gani et al. [23, 24] investigated several concepts of domination in FGs. Enriquez et al. [25] presented domination in the fuzzy directed graph. Parvathi and Thamizhendhi [26] described the DS in an intuitionistic fuzzy graph. Shi and Kosari [27] examined domination in product vague graphs. Talebi et al. [28] studied new concepts of domination in fuzzy graph structures. Sahoo et al. [29] gave covering and paired domination in the intuitionistic fuzzy graph. In this paper, we introduce several concepts related to domination in VGSs such as Ni-dominating set, vague full dominating set, minimal Ni-DS, and strong capacity Ni-dominating set, with some examples. Finally, an application of domination in transferring patients has been presented.


## 2. Preliminaries

An FG is of the formG=μ,ν which is a pair of mappings μ:V⟶0,1 and ν:V×V⟶0,1 as is defined as νa,b≤μa∧μb, ∀a,b∈V, and ν is a symmetric fuzzy relation on μ, and ∧ denotes minimum.A VSM is a pair tM,fM on set V where tM and fM are taken as real-valued functions which can be defined on V⟶0,1 so that tMa+fMa≤1, for all a∈V.Definition 1 (see [8]).
A pairG=M,N is called to be a VG on a CG G∗, where M=tM,fM is a VS on V and N=tN,fN is a VS on E⊆V×V so that tNab≤mintMa,tMb and fNab≥maxfMa,fMb, ∀ab∈E.Definition 2 (see [12]).
ζ=μ,ν1,ν2,…,νn is named a fuzzy graph structure (FGS) of graph structure ζ∗=V,E1,E2,…,En whenever μ,ν1,ν2,…,νn are FSs of V,E1,E2,…,En, respectively, so that νiab≤μa∧μb, ∀a,b∈V and i=1,2,…,n. Note that if ab∈Suppνi, then ab is called a νi-edge of ζ, for i=1,2,…,n.Definition 3 (see [19]).
ξ=M,N1,N2,…,Nn is called a VGS of a GS ζ∗=V,E1,E2,…,En if M=tM,fM is a VS on V and for every i=1,2,…,n; Ni=tNi,fNi is a VS on Ei so that(1)tNiab≤tMa∧tMb,fNiab≥fMa∨fMb,∀ab∈Ei⊆V×V. Note that tNiab=0=fNiab, ∀ab∈V×V−Ei, and 0≤tNiab≤1 and 0≤fNiab≤1, ∀ab∈Ei, where V and Eii=1,2,…,n are called the underlying vertex set and underlying i-edge set of ξ, respectively.Example 1 (see [19]).
Consider the VGSξ=M,N1,N2 as shown in Figure 1, where M=a,0.2,0.3,b,0.2,0.4,c,0.1,0.2,d,0.2,0.2, N1=ab,0.2,0.4,bc,0.1,0.4, and N2=cd,0.1,0.2,ad,0.1,0.3. Clearly, ξ is a VGS.Figure 1
Vague graph structureξ.Definition 4 (see [19]).
A VGSξ=M,N1,N2,…,Nn is called SVGS whenever tNiab=tMa∧tMb and fNiab=fMa∨fMb, ∀ab∈Ni, i=1,2,…,n.Definition 5 (see [19]).
A VGSξ=M,N1,N2,…,Nn is called the CVGS if(i)
ξ is a SVGS(ii)
SuppNi≠∅, ∀i=1,2,…,n(iii)
For every pair of nodesab∈V, ab is an Ni-edge for some iDefinition 6 (see [19]).
Suppose thatξ=M,N1,N2,…,Nn is a VGS. Then,(2)p=M=∑a∈VtMa,∑a∈VfMa,qi=Ni=∑ab∈EitMab,∑ab∈EifMab,q=∑i=1nqi=∑i=1nNi.Definition 7 (see [19]).
A VGSξ=M,N1,N2,…,Nn is named CBVGS if(i)
V can be partitioned into two subsets V1 and V2 so that each Ni-edge joins one node of V1 to one node of V2, for some i(ii)
ξ is a SVGS(iii)
SuppNi≠∅, ∀i,1≤i≤nDefinition 8.
Letξ=M,N1,N2,…,Nn be a VGS; then, the VC of T⊆V is defined as(3)T=∑a∈T1+tMa−fMa2.Some of the basic notations are listed in Table1.Table 1
Some basic notations.
NotationMeaningFGFuzzy graphVGVague graphVCVertex cardinalityDNDomination numberMI-CMinimal cardinalityFEFull edgeCGSComplete graph structureFGSFuzzy graph structureCNClosed neighborhoodNi-IVNi-isolated vertexNiMI-DSNi minimal-dominating setCVGSComplete vague graph structureVFDSVague full dominating setCBVGSComplete bipartite vague graph structureVSVague setVGSVague graph structureDSDominating setCGCrisp graphFDFully dominatedGSGraph structureSCStrong capacityFNFull neighborhoodNDNeighborhood degreeFIVFull isolated vertexNi-NDNi-neighborhood degreeSVGSStrong vague graph structureWCWeak capacity


## 3. Domination in Vague Graph Structures

Definition 9.
Suppose thatξ=M,N1,N2,…,Nn is a VGS of ζ∗=V,E1,E2,…,En. We say that aNi-dominates b, and conversely, ab is an Ni-edge in ξ. K⊆V is named the Ni-DS if for each b∈V−K, ∃ some a∈K so that aNi-dominates b. The minimum cardinality of Ni-DSs is called Ni-DN and is described by γiξ or γi.Definition 10.
K⊆V is called the vague Ni1,i2,…,it-dominating set (Ni1,i2,…,it-DS) in ξ if for every b∈V−K, there exists a∈K so that a dominated b for one of the edges Ni1,Ni2,…,Nit. The Ni1,i2,…,it-dominating number γi1,i2,…,itξ or γi1,i2,…,it is the MI-C of a Ni1,i2,…,it-DS. It is obvious that γi1,i2,…,it≤P.Definition 11.
T⊆V is called the VFDS if, for each b∈V−T, there is some a∈T so that a dominates b for one of the edges N1,N2,…,Nn. The full dominating number on VGS ξ=M,N1,N2,…,Nn is the MI-C of VDSs in ξ and is described by γfξ or γf. A VFDS is a N1,2,…,k-DS.Example 2.
Consider the VGSξ=M,N1,N2,N3 as shown in Figure 2, where(4)M=a,0.1,0.2,b,0.3,0.5,c,0.2,0.4,d,0.1,0.3,e,0.3,0.4,f,0.2,0.3,N1=af,0.1,0.3,ac,0.1,0.5,be,0.2,0.5,cd,0.1,0.4,N2=ab,0.1,0.6,df,0.1,0.3,ce,0.2,0.6,N3=bf,0.2,0.7,de,0.1,0.4.
TheN1-DSs of ξ are as follows:(5)K1=a,b,d,K2=b,c,f,K3=b,d,f,K4=a,d,e,K5=c,e,f,K6=d,e,f.
With a simple calculation,γ1=1.25. The N2-DSs of ξ are as follows:(6)K1=a,c,f,K2=a,c,d,K3=a,e,f,K4=b,e,f,K5=a,c,d,K6=b,c,f,K7=a,d,e,K8=b,d,e,in which γ2=1.2. Accordingly, the N3-DSs are as follows:(7)K1=a,b,c,e,K2=a,c,e,f,K3=a,b,c,d,K4=a,c,d,f,where γ3=1.65. The N1,2-DSs of ξ are as follows:(8)K1=a,d,e,K2=b,c,f,K3=b,d,K4=e,f,where γ1,2=0.8. Similarly, γ1,3=0.8, and γ2,3=0.85. The VFDSs of ξ are as follows:(9)T1=b,c,T2=b,d,T3=a,d,T4=e,f,T5=a,e,T6=c,f,where γf=0.8.Figure 2
VGSξ=M,N1,N2,N3.Theorem 1.
A VGSξ=M,N1,…,Nn is a CVGS if and only if every node is a VFDS.Proof.
Letξ=M,N1,…,Nn be a CVGS. Consider an arbitrary node a∈V. Since a is incident with each node of b∈V−a by some Ni-edge, a is FD by b. Therefore, T=a is a VFDS, ∀a∈V.
Conversely, assume thatT=a is a VFDS, ∀a∈V. So, a FD b, ∀b∈V−T. In other words, there exists a FE from a to b, ∀b∈V−T. Hence, tNab=tMa∧tMb and fNab=fMa∨fMb, for all ab∈Ei, i=1,2,…,n and a,b∈V.Remark 1.
Ifξ=M,N1,N2,…,Nn is a CBVGS with V=V1∪V2, then(i)
γ1=γ2=⋯=γn(ii)
γf=minV1,V2Example 3.
Consider a CBVGSξ=M,N1,N2,N3 as shown in Figure 3, where(10)M=a,0.2,0.3,b,0.3,0.4,c,0.1,0.2,d,0.2,0.4,e,0.3,0.5,N1=ac,0.1,0.3,bc,0.1,0.4,N2=ad,0.2,0.4,bd,0.2,0.4,N3=ae,0.2,0.5,be,0.3,0.5.
The DSs and DN for Figure3 are as follows:(11)c,d,e=1.25,a,b,d,e=1.7,γ1=1.25,c,d,e=1.25,a,b,c,e=1.75,γ2=1.25,e,c,d=1.25,a,b,c,d=1.75,γ3=1.25.
So,γ1=γ2=γ3=1.25. Furthermore, a,b=0.9, c,d,e=1.25, and γf=0.9. It is clear that γf=minV1,V2 so that V1=a,b and V2=c,d,e.Figure 3
A complete bipartite VGSξ.Definition 12.
Letξ=M,N1,N2,…,Nn be a VGS. The Ni-neighborhood of a is described by(12)Nia=b∈V|abis anNi−edge.
Likewise,Ni1,i2,…,it-neighborhood and FN of a are described as(13)Ni1,i2,…,ita=b∈V|abisanNi1,i2,…,it−edge,Nfa=b∈V|abisafullμ−edge.
TheNi-CN of a is described as(14)Nia=Nia∪a.
In the same way,Ni1,i2,…,itx=Ni1,i2,…,ita∪a and Nfa=Nfa∪a. For nonempty set B⊆V, we have(15)NiB=Nia|a∈B,NiB=NiB∪B,NfB=Nfa|a∈B,NfB=NfB∪B.Definition 13.
In a VGSξ=M,N1,N2,…,Nn, the Ni-ND of node a is described as(16)dNia=∑b∈NiaMAb,∑b∈NiaNAb.
Similarly,(17)dNi1,i2,…,ita=∑b∈Ni1,i2,…,itaMAb,∑b∈Ni1,i2,…,itaNAb,(18)dNfa=∑b∈NfaMAb,∑b∈NfaNAb.
The minimum and maximumNi-ND of ξ are denoted by δNi and ΔNi, respectively. In the same way, for the FND, they are shown by δNf and ΔNf.Example 4.
Consider the VGSΓ=M,N1,N2,N3 in Figure 4. We have(19)N1a=b,f,N1c=∅,N1b=a,d,f,dN1a=0.3,0.5,dN1b=0.6,1.1,dN1c=0.
Obviously,δN1=0, ΔN1=0.6,1.1, N2,3a=f,c, N2,3b=d, N2,3d=b,e, N2,3c=a, N2,3e=d, N2,3f=a, dN2,3a=0.5,0.7, dN2,3b=0.2,0.5, dN2,3c=0.2,0.3, dN2,3d=0.2,0.6, dN2,3e=0.2,0.5, and dN2,3f=0.2,0.3. Hence, δN2,3=0.2,0.3 and ΔN2,3=0.5,0.7.
For full neighborhood, we haveNfa=f, Nfb=d, Nfc=∅, Nfd=b, Nfe=∅, Nff=a, δNf=0, and ΔNf=0.2,0.5.Figure 4
VGSξ=M,N1,N2,N3.Definition 14.
For VGSξ=M,N1,N2,…,Nn, a∈V is named an Ni-IV whenever Nia=∅, ∀i=1,…,n. In fact, for every node b∈V, where b≠a, ab is not an Ni-edge. Also, a is FIV if Nfa=∅. For the set B⊆V, a∈B is called an Ni-IV in B if Nia⊆V−B.Example 5.
For VGSξ=M,N1,N2,N3 shown in Figure 2, nodes a and c are N3-IV. N3a=N3c=∅. Note that a is N1-IV in B=a,b,d too because its N1-neighborhood is not in B.(20)N1a=f,c⊂c,e,f=V−B.Remark 2.
γf=γi=p iff every node in ξ is Ni-IV, ∀i=1,2,…,n. In addition, if ∀i=1,2,…,n, a is an Ni-IV, then a is a FIV.Theorem 2.
IfT is a MI-VFDS of a VGS ξ=M,N1,N2,…,Nn without FIVs, then V−T is a VFDS of ξ.Proof.
Letξ=M,N1,N2,…,Nn be a VGS without FIVs and T be a MI-VFDS of ξ. Consider an optional node a∈T. Since ξ has no FIVs, ∃ a node b∈Nfa so that ab is an Ni-edge for some i=1,2,…,n. Hence, b∈V−T and is FD by a. Again, b∈V−T is incident with some element of T. So, each node of V−T is FD by some node of T. Therefore, V−T is a VFDS of ξ.Theorem 3.
AnNi-DS K in VGS ξ=M,N1,N2,…,Nn is an Ni-MI-DS if and only if for each a∈K, one of the two following conditions holds:(i)
a is an Ni-IV of K(ii)
There is a nodeb∈V−K so that Nib∩D=aProof.
Suppose thatK is the Ni MI-DS and a∈D. Since K is MI-DS, K′=K−a is not a Ni-DS. Hence, ∃ some b∈V−K′=V−K−a=V−K∪a so that b is not Ni-D by each element of K−a. If b=a, then a is an Ni-IV of K. If b∈V−K, then b is Ni-D by a∈K since b is not Ni-D by K−a, but it is Ni-D by K. Hence, Nib∩K=a.
Conversely, assume thatK is an Ni-DS, and for each node a∈K, one of the two conditions holds. We show that K is an Ni MI-DS. Assume K is not an Ni MI-DS. Hence, ∃ a node a∈K so that K−a is an Ni-DS. Therefore, a is Ni-D by at least one node in K−a. So, condition (i) does not hold. Also, if K−a is an Ni-DS, then each node in V−K is Ni-D by at least one node in K−a. Thus, condition (ii) does not hold, and this contradicts our assumption.Proposition 1.
Ifξ=M,N1,N2,…,Nn is a VGS without Ni-IV, then(21)γi≤p2.Proof.
Suppose thatK is an Ni MI-DS of ξ. Since V−K is an Ni-DS of ξ, we have K|≤|V−K. So, 2|K|≤|V| and K≤V/2. Therefore, γi≤p/2.Theorem 4.
Ifξ=M,N1,N2,…,Nn is a VGS without Ni-IV, then(22)γi≤p−ΔNi≤p−δNi.Proof.
Letξ=M,N1,N2,…,Nn be a VGS and a∈V be a node with the maximum Ni-ND of ξ. Clearly, V−Nia is an Ni-DS of ξ. If K is a minimum Ni-DS of ξ, then K≤V−Nia, γi≤p−ΔNi. Since ΔNi≥δNi, γi≤p−ΔNi≤p−δNi.Corollary 1.
In a VGSξ=M,N1,N2,…,Nn without Ni-IV,(23)γf≤p−ΔNf≤p−δNf.Definition 15.
Supposeξ=M,N1,N2,…,Nn is a VGS and a∈V. The Ni-degree of a is shown as dia in that we have(24)dia=∑ab∈NitNab,∑ab∈NifNab.
The minimumNi-degree and the maximum Ni-degree of nodes in ξ are defined as(25)δiξ=mindia|a∈V,Δiξ=maxdia|a∈V.
TheNi1,i2,…,it-degree and full degree of a are defined by(26)di1,i2,…,ita=∑m=1ab∈NimttNimab,∑m=1ab∈NimtfNimab,dfa=∑i=1ab∈NintNiab,∑i=1ab∈NinfNiab,respectively. Also, δfξ=mindfa|a∈V and Δfξ=maxdfa|a∈V. For nonempty set B⊆V, the degree is described as follows:(27)diB=∑a∈Bdia,dfB=∑a∈Bdfa,di1,i2,…,itB=∑a∈Bdi1,i2,…,ita.Example 6.
Consider the VGSξ=M,N1,N2,N3 as shown in Figure 5, where(28)M=a,0.1,0.2,b,0.2,0.4,c,0.3,0.5,d,0.2,0.3,e,0.2,0.5,N1=ac,0.1,0.6,be,0.1,0.5,de,0.1,0.5,N2=ab,0.1,0.4,ce,0.1,0.6,N3=ae,0.1,0.7,bc,0.2,0.5,cd,0.2,0.6.
TheN3-degree of B=a,d,e is as follows: d3a=0.1,0.5, d3d=0.2,0.6, and d3e=0.1,0.5. Hence, d3B=0.4,1.6. Also, d3ξ=0.1,0.5 and Δ3ξ=0.4,1.1. The N1,2-degree of B is as follows:(29)N1,2a=0.2,1,N1,2d=0.1,0.5,N1,2e=0.3,1.6.
Hence,N1,2B=0.6,3.1. The full degree of B is as follows: dfa=0.3,1.5, dfd=0.3,1.1, and dfe=0.4,2.1. So, dfB=1,4.7.Figure 5
VGSξ=M,N1,N2,N3.Proposition 2.
Ifξ=M,N1,N2,…,Nn is a VGS, then(i)
γi≤p−Δiξ(ii)
γf≤p−dfT, T is a γf-setProof.
Assume thatΔiξ is the maximum Ni-degree of nodes in ξ and K is a γi-set. Then, V−K is an Ni-DS. So, K≤V−K and γi≤p−K≤p−Δiξ. Hence, (i) holds. Suppose T is a set of nodes with MI-C such that every Ni-edge, 1≤i≤k, is incident with some nodes in T. Then, dfT≤Δiξ, and we have γf≤γi≤p−Δiξ≤p−dfT. So, γf≤p−dfT.Definition 16.
The capacity of a nodea in the VGS ξ=M,N1,N2,…,Nn is the sum membership of different Ni-edges in a, and it is shown by Ca. δCa and ΔCa are MI-C and MA-C of a for different Ni-edges in a. Clearly, δCa≤Ca≤ΔCa≤n and Ca≤dfa. Likewise, δCξ=minδCa|a∈V, and ΔCξ=maxΔCa|a∈V. For B⊆V, the capacity of B is described by(30)CB=∑a∈BCa.Example 7.
Consider the VGSξ=M,N1,N2,N3 as in Figure 5. We have(31)Ca=0.3,1.5,Cb=0.4,1.4,Cd=0.3,1.1,δCc=0.4,1.7,ΔCc=0.4,1.8,δCe=0.3,1.6=ΔCe.Definition 17.
LetK be an Ni-DS in the VGS ξ=M,N1,N2,…,Nn. Then, K is named the Ni-SCDS if for each node b∈V−K, ∃ a node a∈D neighbor to b so that Ca≥Cb.
The MI-C ofNi-SCDSs is named SC Ni-DN γSCi, and the MI-C of Ni-WCDSs is called WC Ni-DN γWCi. Similarly, the MI-C of SC and WC of VFDSs are denoted by γSCf and γWCf, respectively.Example 8.
For VGSξ=M,N1,N2,N3 as shown in Figure 2, K1=a,b,c,e is a SC N3-DS among other N3-DSs, and K4=d,f,a,c is a WC N3-DS.Theorem 5.
Ifξ=M,N1,N2,…,Nn is a VGS, then(i)
γSCf≤p−ΔCξ(ii)
γWCi≤p−δCξProof.
(i) Leta be a node with capacity ΔCξ and K be a set of nodes neighbor to a. In this case, each node b∈K is a neighbor to a∈V−K, and Cb≤Ca. Hence, V−K is a SC-VFDS. Therefore,(32)γSCf≤V−K≤p−ΔCξ.
(ii) Leta be a node of capacity δCξ and K′ be the set of nodes neighbor to a by Ni-edge. Then, each node b∈K′ is a neighbor to a∈V−K′, and Cb≥Ca. So, V−K′ is an Ni-WCDS. Hence, γWCi≤V−K′≤p−δCξ.


## 4. Application of Dominating Sets in Transferring Patients

Nowadays, many hospitals in cities have to transfer patients to neighboring cities that have the necessary facilities owing to the lack of facilities and equipment for treatment, but there are many factors that can be important in deciding which hospital to choose. One of them is the suitability of the roads in terms of smoothness and having the necessary traffic signs for ambulance drivers because the unsuitability of the roads can be dangerous and stressful for both the patient and the vehicle. Another factor that can be very useful is the amount of congestion and traffic between roads because if this congestion is less, then the patient will be transported to the hospital faster and easier and will receive the required treatment. As we see today, many patients lose their lives because of not arriving in hospitals on time and not receiving immediate care. Therefore, the amount of traffic and congestion can be very vital factors in this case. Additionally, another factor that can be effective in this decision is the patients’ admission rate in the desired hospital because the more patients a hospital accepts, the easier it is for the patients to be treated more effortlessly and quickly. Note that this issue plays an important role in a patient’s treatment process. Therefore, in this paper, we intend to express the use of the vague full dominating set in transferring a patient to the most effective hospital in the shortest possible time. For this purpose, we consider four hospitals in Sari, Iran. Suppose a patient lives in placeA (Ghaemshahr) and has to go to one of the medical centers in Sari for treatment called Ibn Sina B, Amir Mazandarani C, Imam Khomeini D, and Fatemeh Zahra E. Hospitals are shown in the graph with the symbols B, C, D, and E.In this vague graph structure, the vertices represent the hospitals, and the edges indicate the level of smoothness and quality of roads, road traffic, and the patients’ admission rate in the desired hospitals, respectively. The weight of vertices and edges is shown in Tables2 and 3. The location of the hospitals is shown in Figure6.Table 2
Weight of vertices.
ABCDEMA0.50.20.50.30.3NA0.30.40.20.20.3Table 3
Weight of edges.
ξN1ABN2ABN1BCN2BCtN,fN0.4,0.60.4,0.20.1,0.50.2,0.4ξN1ACN2ACN3ACN2AEtN,fN0.5,0.30.2,0.60.4,0.30.3,0.4ξN1ADN3ADN1DEN2DEtN,fN0.2,0.50.3,0.50.2,0.60.3,0.4ξN3DEtN,fN0.2,0.6Figure 6
Location of hospitals in Sari city.The vertexC0.5,0.2 shows that Amir Mazandarani hospital has 50% of medical devices and equipment to treat a patient, but unfortunately, it does not have 20% of the necessary facilities for treatment. The edge N1AC shows that the ACroute meets only 50% of the global transport standards in terms of quality and construction and has 30% breakdowns. The edge N3AD indicates that 30% of patients can be admitted to Imam Khomeini hospital, but the hospital is not able to accept another 50% of patients. The edge N2AB shows this route has 40% of city traffic for most hours of the day, and 60% of it is free of traffic. The vague full dominating sets for Figure 7 are as follows:(33)T1=A,B,T2=A,C,T3=A,D,T4=B,D,T5=B,E,T6=A,E,T7=C,E,T8=C,D.Figure 7
VGSξ.After calculating the cardinality ofT1,…,Tn, we obtain(34)T1=1,T2=1.25,T3=1.15,T4=0.95,T5=0.9,T6=1.1,T7=1.15,T8=1.2.Clearly,T2 has the largest cardinal wards among other vague full dominating sets, so it turns out that it can be the best choice because firstly, the path from A to C has the smoothest and best communication path, and secondly, only 20% of this path has traffic, and congestion is caused by cars, and thirdly, hospital C has 40% admission for patients, which is more than other hospitals. Note that hospital C has 50% of the necessary medical equipment, which is significant compared with other hospitals. Therefore, it is concluded that the government should provide the necessary facilities for hospitals so that patients do not have to go to hospitals far away for treatment. Also, in terms of transportation, the roads must be of good quality so that ambulances can transport patients for treatment as soon as possible.


## 5. Conclusion

Fuzzy graph has various applications in modern science and technology, especially in the fields of neural networks, computer science, operation research, and decision-making. Vague graph structures have more precision, flexibility, and compatibility, as compared to the fuzzy graphs. Today, VGSs play an important role in social networks and allow users to find the most effective person in a group or organization. One of the most important features of VGSs that has many applications in real problems is the concept of domination. Domination has many applications in psychology, medical science, social groups, and computer networks. Therefore, in this research, we defined different concepts related to domination in VGSs such asNi-dominating set, vague full dominating set, and minimal Ni-dominating set, with several examples. Finally, an application of domination in medical sciences has been presented. In our future work, we will introduce ψ-complement, self-complement, strong self-complement, and totally strong self-complement in VGSs and investigate some of their properties.



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*Source: 1504397-2022-02-08.xml*