# A Study on Vague Graph Structures with an Application

Authors: Zheng Kou; Maryam Akhoundi; Xiang Chen; Saber Omidi
Journal: Advances in Mathematical Physics
 (2022)

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## Abstract

Fuzzy graph (FG) models take on the presence being ubiquitous in environmental and fabricated structures by human, specifically the vibrant processes in physical, biological, and social systems. Owing to the unpredictable and indiscriminate data which are intrinsic in real-life, problems being often ambiguous, so it is very challenging for an expert to exemplify those problems through applying an FG. Vague graph structure (VGS), belonging to FGs family, has good capabilities when facing with problems that cannot be expressed by FGs. VGSs have a wide range of applications in the field of psychological sciences as well as the identification of individuals based on oncological behaviors. Therefore, in this paper, we apply the concept of vague sets (VSs) to GS. We define certain notions, VGS, strong vague graph structure (SVGS), and vagueβi-cycle and describe these notions by several examples. Likewise, we introduce ψ-complement, self-complement (SC), strong self-complement (SSC), and totally strong self-complement (TSSC) in VGS and investigate some of their properties. Finally, an application of VGS is 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]. A GS, proposed by Sampathkumar [2], refers to the generalization of undirected graph being relatively beneficial in investigating some structure including graphs, signed graphs, and graphs in which every edge is labeled or colored. A GS facilitates studying the different relations and the equivalent edges simultaneously. The FS focuses on the membership degree of an object in a particular set. 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 VS as the sum of degrees not greater than 1. Kaufmann [4] represented FGs based on Zadeh’s fuzzy relation [5, 6]. Rosenfeld [7] described the structure of FGs obtaining analogs of several graph theoretical concepts. Harinath and Lavanya [8] studied new concepts in fuzzy graph structures. Bhattacharya [9] gave some remarks on FGs. Several concepts on FGs were introduced by Mordeson and Nair [10]. Dinesh [11] investigated the notion of a FGS and studied some related properties. Ghorai et al. [12, 13] presented fuzzy tolerance graphs and planarity in VGs. Ramakrishna [14] defined VGs. Kosari et al. [15–18] investigated domination set, equitable domination set, and new concepts of domination in vague graphs and vague incidence graphs. Pal and Rashmanlou [19, 20] have given several concepts in FGs. Akram et al. [21, 22] introduced certain fuzzy graph structures. Sunitha and Vijayakumar [23] presented some properties of complement on FGs. Muhiuddin et al. [24–26] investigated new results in cubic graphs.A VGS is concerned with the generalized structure of an FG that expresses more exactness, adaptability, and compatibility to a system when synchronized with systems operating on FGs. Correspondingly, a VGS is capable of focusing on determining the uncertainly combined with the inconsistent and indeterminate information of any real-world problem, in which FGs may not lead to adequate results. Hence, in this paper, we define the certain notions including VGS, SVGS, and vagueβi-cycle and describe these notions by several examples. We study ψ-complement, SC, and SSC in VGSs and investigate some of their properties. Finally, an application of VGS has been presented.


## 2. Preliminaries

A GSG∗=V,R1,R2,…,Rk, consists of a nonempty set V together with relations R1,R2,…,Rk on V, which are mutually disjoint so that each Ri is irreflexive and symmetric. If a,b∈Ri for some i, 1≤i≤k, we name it an Ri-edge and write it as “ab.”A GSG∗=V,R1,R2,…,Rk is CGS, if,(i)
every edgeRi, 1≤i≤k appears at least once in G∗(ii)
among every pair of nodesab in V, ab is an Ei-edge for some i, 1≤i≤kDefinition 1.
(see [1]). A fuzzy subset σ on a set X is a map σ:X⟶0,1. A fuzzy binary relation on X is a fuzzy subset σ on X×X. By a fuzzy relation, we mean a fuzzy binary relation given by σ:X×X⟶0,1.Definition 2.
(see [11]). Let G∗=V,R1,R2,…,Rk be a GS and let ν,λ1,λ2,…,λk be the fuzzy subset of V,R1,R2,…,Rk, respectively, so that 0≤λixy≤σx∧σy, ∀x,y∈V, i=1,2,…,k. Then, G=ν,λ1,λ2,…,λk is an FGS of G∗.Definition 3.
(see [11]). Let G=ν,λ1,λ2,…,λk be an FGS of a GS G∗=V,R1,R2,…,Rk. Then, F=ν,τ1,τ2,…,τk is a PFSSGS of G if G, for i=1,2,…,k.Definition 4.
(see [11]) The strength of a λi-path a0a1,⋯an of a FGS G is ∧j=1nλiaj−1aj, for i=1,2,…,k.Definition 5.
(see [11]) In a FGS G, λi2ab=λi°λiab=∨cλiac∧λibc, λijab=λij−1°λiab=∨cλij−1ac∧λibc, j=1,2,…,n, for any n≥2. Also, λi∞ab=∨λijab,j=1,2,….Definition 6.
Reference [11] G=ν,λ1,λ2,…,λk is a λi-cycle if and only if Suppν,Suppλ1,Suppλ2,…,Suppλk is a Ri-cycle.Definition 7.
Reference [11] G=ν,λ1,λ2,…,λk is a fuzzy λi-cycle if and only if Suppν,Suppλ1,Suppλ2,…,Suppλk is a Ei-cycle and ∃ no unique ab in Suppλi so that(1)λiab=∧λicd|cd∈Suppλi.Definition 8.
(see [3]) A VS A is a pair tA,fA on set X where tA and fA are taken as real valued functions which can be defined on V⟶0,1 so that tAa+fAb≤1, ∀a∈X.Definition 9.
(see [14]) A VG is a pair G=A,B, where A=tA,fA is a VS on V and B=tB,fB is a VS on E⊆V×V so that tBab≤mintAa,tAb and fBab≥maxfAa,fAb, for ab∈E.
All the basic notations are shown in Table1.Table 1
Some basic notations.
NotationMeaningFGFuzzy graphFSFuzzy setVSVague setGSGraph structureCGSComplete graph structureVGSVague graph structureVSGSVague subgraph structureSGSSubgraph structureFGSFuzzy graph structureSVGSStrong vague graph structureUGUnderlying graphENEnd nodeSCSelf-complementarySSCStrong self-complementaryTSCTotally self-complementaryTSSCTotally strong self-complementPFSSGSPartial fuzzy spanning subgraph structureUVSUnderlying vertex setTSSCTotally strong self-complementary


## 3. New Concepts in Vague Graph Structure

Definition 10.
G^v=A,B1,B2,…,Bn is said to be a VGS of a GS G∗=V,R1,R2,…,Rn, if A=tA,fA is a VS on V and for every i=1,2,…,n; Bi=tBi,fBi is a VS on Ri so that(2)tBiab≤tAa∧tAb,fBiab≥fAa∨fAb.
∀ab∈Ri⊆V×V. Note that tBiab=0=fBiab, ∀ab∈V×V−Ri and 0≤tBiab≤1, 0≤fBiab≤1, ∀ab∈Ri, where V and Ri,i=1,2,…,n are named UVS and underlying i-edge set of G^v, respectively.Example 1.
LetG∗=V,R1,R2 be a GS so that V=a1,a2,a3,a4, R1=a1a2,a2a3, and R2=a3a4,a1a4. Let A, B1, and B2 be vague subsets of V, R1, and R2, respectively, so that(3)A=a1,0.4,0.5,a2,0.4,0.6,a3,0.3,0.4,a4,0.4,0.4,B1=a1a2,0.4,0.6,a2a3,0.3,0.6,B2=a3a4,0.3,0.4,a1a4,0.3,0.5.
Then,G^v=A,B1,B2 is a VGS of G∗ drawn in Figure 1.Figure 1
VGSG^v=A,B1,B2.Definition 11.
(i)
AVGSH⌣v=C,D1,D2,…,Dn is said to be a VSGS of a VGS G⌣v=A,B1,B2,…,Bn with UVS V, if C⊆A and Di⊆Ci, ∀i, that is,(4)tCa≤tAa,fCa≥fAa;for alla∈V,and fori=1,2,…,n,(5)tDiab≤tBiab,fDiab≥fBiab;for allab∈V×V.(ii)
H⌣v is named a VSSGS of a VGS Gˇv, if C=A.(iii)
H⌣v is named a VPSSGS of a VGS Gˇv, if it excludes some edges of Gˇv.Example 2.
Consider a VGSG⌣v=A,B1,B2 as shown in Figure 1. Let(6)C=a1,0.3,0.6,a2,0.0,0.7,a3,0.3,0.4,a4,0.3,0.5,D1=a1a2,0,0.7,a2a3,0,0.7,D2=a3a4,0.2,0.5,a1a4,0.3,0.6,C1=a1a2,0.2,0.6,a2a3,0.3,0.6,C2=a3a4,0.2,0.4,a1a4,0.3,0.7,F1=a1a2,0.4,0.6,a2a3,0.3,0.6,F2=a1a4,0.3,0.6.
It is easy to see thatH⌣v=C,D1,D2, J⌣v=A,C1,C2, and K⌣v=A,F1,F2 are the VSGS, VSSGS, and VPSSGS of Gˇv, respectively.
Their corresponding graphs are drawn in Figures2–4.Figure 2
Vague subgraph structureH⌣v=C,D1,D2.Figure 3
Vague spanning subgraph structureJ⌣v=A,C1,C2.Figure 4
Vague partial spanning subgraph structureK⌣v=A,F1,F2.Definition 12.
LetG⌣v=A,B1,B2,…,Bn be a VGS with UVS V. Then, there exists a Bi-edge among two nodes a and b of V so that we have(i)
tBiab>0, fBiab>0, or(ii)
tBiab>0, fBiab=0, or(iii)
tBiab=0, fBiab>0, for some i.Definition 13.
For a VGSG⌣v=A,B1,B2,…,Bn, the support of Bi is described as(7)SuppBi=ab∈V×V:tBiab≠0orfBiab≠0,i=1,2,…,n.Definition 14.
Bi-path of a VGS Gˇv with UVS V is a sequence of distinct nodes v1,v2,…,vm∈U (except the choice vm=v1) so that vj−1vj is a Bi-edge, ∀j=2,3,…,m.Definition 15.
In a VGSG⌣v=A,B1,B2,…,Bn with UVS V, two nodes a and b called Bi-connected, if they are connected by a Bi-path, for some i∈1,2,3,…,n.Definition 16.
A VGSG⌣v=A,B1,B2,…,Bn with UVS V is called to be Bi-strong, if ∀Bi-edges ab,(8)tBiab=tAa∧tAb,fBiab=fAa∨fAb,for some i∈1,2,3,…,n.Example 3.
Consider the VGSG⌣v=A,B1,B2, drawn in Figure 1. Then,(i)
a1a2,a2a3 are B1-edges and a3a4,a1a4 are B2-edges(ii)
a1a2a3 and a3a4a1 are B1 and B2 paths, respectively(iii)
a1 and a3 are B1-connected nodes of U(iv)
G⌣v is a B1-strong since SuppB1=a1a2,a2a3 and we have(9)tB1a1a2=0.4=tAa1∧tAa2,fB1a1a2=0.6=fAa1∨fAa2,tB1a2a3=0.3=tAa2∧tAa3,fB1a2a3=0.6=fAa2∨fAa3.Definition 17.
A VGSG⌣v=A,B1,B2,…,Bn is named to be strong, if it is Bi-strong, ∀i∈1,2,3,…,n.Definition 18.
A VGSG⌣v=A,B1,B2,…,Bn with UVS V, is named complete or B1,B2,…,Bn-complete if(i)
G⌣v is a SVGS(ii)
SuppBi≠∅, ∀i=1,2,3,…,n(iii)
For each nodesa,b∈V, ab should be a Bi-edgeExample 4.
SupposeG⌣v=A,B1,B2 drawn in Figure 5, be VGS of the GS G∗=V,R1,R2 where V=a1,a2,a3,a4, R1=a1a2,a2a3,a1a3, and R2=a1a4,a3a4,a2a4. Then, G⌣v is a SVGS, since it is both B1-strong and B2-strong. Moreover, SuppB1≠∅, SuppB2≠∅, and each nodes in V, is either a B1-edge or a B2-edge, so G⌣v is a complete or B1B2-complete-VGS as well.Figure 5
VGSG⌣v=A,B1,B2.Definition 19.
In a VGSG⌣v=A,B1,B2,…,Bn with UVS V, tBi and fBi-strength of a Bi-path “PBi=v1v2,…,vm” are denoted by δ·PBi and Δ·PBi, respectively, so that δ·PBi=∧j=2mtBiaj−1aj and Δ·PBi=∨j=2mfBiaj−1aj.
Then, we write strength of the pathPBi=δ·PBi,Δ·PBi.Example 5.
InG⌣v=A,B1,B2 shown in Figure 5, P1=a1a2a3a1 is a B1-path, and a1a4a3 is a B2-path, we have(10)δ·P1=tB1a1a2∧tB1a2a3∧tB1a3a1=0.2∧0.3∧0.2=0.2,Δ·P1=fB1a1a2∨fB1a2a3∨fB1a3a1=0.3∨0.5∨0.5=0.5,δ·P2=tB2a1a4∧tB2a4a3=0.2∧0.4=0.2,Δ·P2=fB2a1a4∨fB2a4a3=0.6∨0.6=0.6.
Thus, strength ofB1-path is P1=0.2,0.5, and strength of P2-path is P2=0.2,0.6.Definition 20.
In a VGSG⌣v=A,B1,B2,…,Bn with UVS V,(i)
tBi-strength of connectedness between a and b is defined by tBi∞ab=∨j≥1tBijab, where tBijab=tBij−1°tBiab, for j≥2 and tBi2ab=tBi°tBiab=∨ztBiac∧tBibc;(ii)
fBi-strength of connectedness among a and b is described by fBi∞ab=∧j≥1fBijab, where fBijab=fBij−1°fBiab, for j≥2 and fBi2ab=fBi°fBiab=∨zfBiac∨fBibc.Example 6.
LetG⌣v=A,B1,B2 be a VGS of GS G∗=V,R1,R2 as shown in Figure 6 so that V=a1,a2,a3, R1=a1a2,a1a3, and R2=a2a3. Since tB1a1a2=0.2, tB1a1a3=0.2, and tB1a2a3=0, therefore,(11)tB12a1a2=tB1°tB1a1a2=tB1a1a3∧tB1a3a2=0.2∧0.3=0.2,tB12a2a3=tB1°tB1a2a3=tB1a2a1∧tB1a1a3=0.2∧0.2=0.2,tB12a1a3=tB1°tB1a1a3=tB1a1a2∧tB1a2a3=0.2∧0.3=0.2,tB13a1a2=tB12°tB1a1a2=tB12a1a3∧tB1a3a2=0.2∧0.3=0.2,tB13a2a3=tB12°tB1a2a3=tB12a2a1∧tB1a1a3=0.2∧0.2=0.2,tB13a1a3=tB12°tB1a1a3=tB12a1a2∧tB1a2a3=0.2∧0.3=0.2.
Thus, we have,(12)tB1∞a1a2=∨0.2,0.2,0.2=0.2,tB1∞a2a3=∨0.2,0.2,0.2=0.2,tB1∞a1a3=∨0.2,0.2,0.2=0.2.
SincefB1a1a2=0.6, fB1a1a3=0.7, and fB1a2a3=0, therefore,(13)fB12a1a2=fB1°fB1a1a2=fB1a1a3∨fB1a3a2=0.7∨0.7=0.7,fB12a2a3=fB1°fB1a2a3=fB1a2a1∨fB1a1a3=0.6∨0.7=0.7,fB12a1a3=fB1°fB1a1a3=fB1a1a2∨fB1a2a3=0.6∨0=0.6,fB13a1a2=fB12°fB1a1a2=fB12a1a3∨fB1a3a2=0.6∨0.7=0.7,fB13a2a3=fB12°fB1a2a3=fB12a2a1∨fB1a1a3=0.7∨0.7=0.7,fB13a1a3=fB12°fB1a1a3=fB12a1a2∨fB1a2a3=0.7∨0.7=0.7,and(14)fB14a1a2=fB13°fB1a1a2=fB13a1a3∨fB1a3a2=0.7∨0.7=0.7,fB14a2a3=fB13°fB1a2a3=fB13a2a1∨fB1a1a3=0.7∨0.7=0.7,fB14a1a3=fB13°fB1a1a3=fB12a1a2∨fB1a2a3=0.7∨0.7=0.7.
Thus, we have(15)fB1∞a1a2=∨0.6,0.7,0.7,0.7=0.7,fB1∞a2a3=∨0.0,0.7,0.7,0.7=0.7,fB1∞a1a3=∨0.7,0.6,0.7,0.7=0.7.
By similarity way, we can calculatetB2∞a1a2, tB2∞a2a3, and tB2∞a1a3.Figure 6
VGSG⌣v=A,B1,B2.Definition 21.
A VGSG⌣v=A,B1,B2,…,Bn of a GS G∗=V,R1,R2,…,Rn is a Bi-cycle, if G∗ is an Ri-cycle.Definition 22.
A VGSG⌣v=A,B1,B2,…,Bn of a GS G∗=V,R1,R2,…,Rn is a vague Bi-cycle, for some i, if we have(i)
G is a Bi-cycle;(ii)
There is no uniqueBi-edge ab in G⌣v so that tBiab=mintBizw:zw∈Ri=SuppBi or fBiab=maxfBizw:zw∈Ri=SuppBi.Example 7.
VGSG⌣v=A,B1,B2 in Figure 7 is a B1-cycle as well as vague B1-cycle, since SuppA,SuppB1,SuppB2 is an R1-cycle and there are two B1-edges with minimum degree of membership and two B1-edges with maximum degree of membership of all B1-edges.Figure 7
VGSG⌣v=A,B1,B2.Definition 23.
A VGSG⌣v1=A1,B11,B12,…,B1n of GS G1∗=V1,R11,R12,…,R1n is isomorphic to a VGS G⌣v2=A2,B21,B22,…,B2n of G2∗=V2,R21,R22,…,R2nG⌣v1≅G⌣v2 if ∃ a bijective g:V1⟶V2 and a permutation ψ on the set 1,2,…,n so that(16)tA1u1=tA2gu1,fA1u1=fA2gu1,∀u1∈V1,and forψi=j,tB1iu1u2=tB2jgu1gu2, fB1iu1u2=fB2jgu1gu2, for all u1u2∈R1i, i=1,2,…,n.Definition 24.
A VGSG⌣v1=A1,B11,B12,…,B1n of GS G1∗=V,R11,R12,…,R1n is identical to a VGS G⌣v2=A2,B21,B22,…,B2n of G2∗=V,R21,R22,…,R2n if ∃ a bijection g:V⟶V so that(17)tA1u=tA2gu,fA1u=fA2gu,for allu∈V,and(18)tB1iu1u2=tB2igu1gu2,fB1iu1u2=fB2igu1gu2,for allu1u2∈R1i,i=1,2,…,n.Example 8.
LetG⌣v1 and G⌣v2 be two VGSs of GS G1∗=V1,R1,R2 and G2∗=V2,R1′,R2′, respectively, drawn in Figure 8. Here, G⌣v1 is isomorphic (not identical) to G⌣v2 under the mapping g:V1⟶V2, defined by ga1=b1, ga2=b2, and ga3=b3 and a permutation ψ given by ψ1=2, ψ2=1 so that(19)tA1ai=tA2gai,fA1ai=fA2gai,∀ai∈V,tBkaiaj=tBψkgaigaj,fBkaiaj=fBψkgaigaj,∀aiaj∈Rk,k=1,2.Figure 8
Isomorphic vague graph structures.Example 9.
LetG⌣v1=A,B1,B2 and G⌣v2=A′,B1′,B2′ be two VGSs of GSs G1∗=V,R1,R2 and G2∗=V′,R1′,R2′, respectively, as shown in Figure 9. Here, G⌣v1 is identical with G⌣v2 under the mapping g:V⟶V′, defined by ga1=b6, ga2=b2, ga3=b4, ga4=b5, ga5=b1, and ga6=b3 so that(20)tAai=tA′gai,fAai=fA′gai,∀ai∈V,tBkaiaj=tBk′gaigaj,fBkaiaj=fBk′gaigaj,∀aiaj∈Rk,k=1,2.Figure 9
Identical vague graph structures.Definition 25.
LetG⌣v be a VGS of a GS G∗. Let ψ denote a permutation on the set R1,R2,…,Rn and the corresponding permutation on B1,B2,…,Bn, i.e., ψBi=Bj if ψRi=Ej, ∀i. If ab∈Br for some r and tBiψab=tAa∧tAb−∨j≠itψBjab, fBiψab=fAa∨fAb−∨j≠ifψBjab, i=1,2,…,n, then, ab∈Bmψ while m is chosen so that tBmψab≥tBiψab and fBmψab≥fBiψab, ∀i. Then, VGS A,B1ψ,B2ψ,…,Bnψ denoted by Gvψc, is named the ψ-complement of VGS.Example 10.
Consider VGSG⌣v=A,B1,B2 shown in Figure 10 and let ψ be a permutation on the set B1,B2 so that ψB1=B2 and ψB2=B1. Now, for a1a2∈B1,(21)tB1ψa1a2=tAa1∧tAa2−∨j≠1tψBja1a2=0.2−0.2=0,fB1ψa1a2=fAa1∨fAa2−∨j≠1fψBja1a2=0.5−0.5=0,tB2ψa1a2=tAa1∧tAa2−∨j≠2tψBja1a2=0.2−0=0.2,fB2ψa1a2=fAa1∨fAa2−∨j≠2fψBja1a2=0.5−0=0.5.
Clearly,tB2ψa1a2=0.2>0=tB1ψa1a2 and fB2ψa1a2=0.5>0=fB1ψa1a2, so a1a2∈B2ψ.(22)tB1ψa1a3=tAa1∧tAa3−∨tψB2a1a3=0.2−0.2=0,fB1ψa1a3=fAa1∨fAa3−∨fψB2a1a3=0.4−0.4=0,tB2ψa1a3=tAa1∧tAa3−∨tψB1a1a3=0.2−0=0.2,fB2ψa1a3=fAa1∨fAa3−∨fψB1a1a3=0.4−0=0.4.
So,a1a3∈B2ψ. Also, for a2a3, we have(23)tB1ψa2a3=tAa2∧tAa3−∨tψB2a2a3=0.2−0=0.2,fB1ψa2a3=fAa2∨fAa3−∨fψB2a2a3=0.5−0=0.5,tB2ψa2a3=tAa2∧tAa3−∨tψB1a2a3=0.2−0.2=0,fB2ψa2a3=fAa2∨fAa3−∨fψB1a2a3=0.5−0.5=0.
So,a2a3∈B1ψ.Figure 10
Vague graph structuresG⌣v and G⌣vψc.Theorem 1.
Aψ-complement of a VGS is a SVGS. In addition, if ψi=r, for r,i∈1,2,…,n, then all Br-edges in VGS G⌣v=A,B1,B2,…,Bn become Biψ-edges in G⌣vψc=A,B1ψ,B2ψ,…,Bnψ.Proof.
The proof of the first part is clear according to the definition of theψ-complement G⌣vψc of VGS Gˇv, since for any Biψ-edge ab, tBiψab and fBiψab, respectively, have the maximum values of(24)tAa∧tAb−∨j≠itψBjabfAa∨fAb−∨j≠ifψBjab.
That is,(25)tBiψab=tAa∧tAb,fBiψab=fAa∨fAb,for all edges ab in G⌣vψc; hence, G⌣vψc is a SVGS. Now assume on contrary that ψi=r, but ab is Bs-edge in G⌣v with s≠r that shows ψBi≠Bs.
By comparing equations (24) and (25), we have(26)∨j≠itψBjxy=0,∨j≠ifψBjxy=0,that it is not true because Bs=ψBj, for some j∈1,2,…,i−1,i+1,…,n. Therefore, our assumption is wrong and ab should be a Br-edge. Thus, we get if ψi=r, then, all Br-edges in VGS G⌣v=A,B1,B2,…,Bn become Biψ-edges in G⌣vψc=A,B1ψ,B2ψ,…,Bnψ, for r,i∈1,2,…,n.Definition 26.
LetG⌣v=A,B1,B2,…,Bn be a VGS and ψ be a permutation on the set 1,2,…,n. Then,(i)
G⌣v is SC, if G⌣v≅G⌣vψc,(ii)
G⌣v is SSC, if it is identical to G⌣vψc.(iii)
G⌣v is TSC if G⌣v≅G⌣vψc, ∀ permutations ψ on the set 1,2,…,n.(iv)
G⌣v is TSSC, if it is identical to G⌣vψc, ∀ permutations ψ on the set 1,2,…,n.Theorem 2.
A VGSG⌣v is strong if G⌣v is TSC.Proof.
AssumeGˇv be a SVGS and ψ be a permutation on the set 1,2,…,n. If ψ−1i=j, then, by Theorem 1, all Bi-edges in G⌣v=A,B1,B2,…,Bn become Bjψ-edges in G⌣vψc=A,B1ψ,B2ψ,…,Bnψ. Also, G⌣vψc is strong, so,(27)tBia1a2=tAa1∧tAa2=tBjψa1a2,fBia1a2=fAa1∨fAa2=fBjψa1a2.
Then,G⌣v is isomorphic to G⌣vψc, under the identity mapping g:V⟶V and a permutation ψ, so that(28)tAa=tAga,fAa=fAga,for alla∈V,and(29)tBia1a2=tBjψa1a2=tBjψga1ga2,fBia1a2=fBjψa1a2=fBjψga1ga2,for all a1a2∈Ri. This holds for all the permutation on the set 1,2,…,n. Thus, G⌣v is TSC. Conversely, let ψ be any permutation on the set 1,2,…,n and G⌣v and G⌣vψc be isomorphic. According to the definitions of ψ-complement and isomorphism, we get(30)tBia1a2=tBjψga1ga2=tAga1∧tAga2=tAa1∧tAa2,fBia1a2=fBjψga1ga2=fAga1∨fAga2=fAa1∨fAa2,for all a1a2∈Ei, i=1,2,…,n. Hence, G⌣v is a SVGS.Theorem 3.
If GSG∗=V,R1,R2,…,Rn is TSSC and A is a VS of V with constant fuzzy mapping tA and fA, then, a SVGS G⌣v=A,B1,B2,…,Bn of G∗ is TSSC.Proof.
Consider a SVGSG⌣v=A,B1,B2,…,Bn of a GS G∗=V,R1,R2,…,Rn. Assume that G∗ is TSSC and that for some constants s,t∈0,1, A=tA,fA is a VS of u so that tAu=s, fAu=t, ∀u∈V. Then, we have to prove that G⌣v is TSSC. Let ψ be an arbitrary permutation on the set 1,2,…,n and ψ−1j=i. Since G∗ is TSSC, so ∃ a bijection g:V⟶V so that for each Ri-edge a1a2 in G∗, ga1ga2 is an Ri-edge in G∗ψ−1c. Consequently, for every Bi-edge a1a2 in G⌣v, ga1ga2 is a Biψ-edge in G⌣vψc. From the definition of A and the definition of SVGS G⌣v,(31)tAa=s=tAga,fAa=t=fAga,for alla,ga∈V,and(32)tBia1a2=tAa1∧tAa2=tAga1∧tAga2=tBiψga1ga2,fBia1a2=fAa1∨fAa2=fAga1∨fAga2=fBiψga1ga2,for all a1a2∈Bi, i=1,2,…,n, which shows G⌣v is SSC. Hence, G⌣v is TSSC, since ψ is arbitrary.Example 11.
A VGSG⌣v=A,B1,B2,B3 of GS G∗=V,R1,R2,R3 as shown in Figure 11 is TSSC.Figure 11
Totally strong self-complementary.


## 4. Application

Providing services to the people in the shortest possible time is one of the most important issues that governments strive to do in the best possible way. Unfortunately, in the past, due to the lack of medical and welfare services as well as the lack of educational centers, people had to travel to distant cities to provide these necessary facilities and equipment, and in this way, unfortunately, they incurred a lot of costs. But today, with the help of governments as well as the equipping of medical and welfare centers, people can move to their nearest neighboring city to benefit from these facilities. A vague graph structure helps us to understand which route is more appropriate for a particular service. It also shows us which service is increasing so that governments can strengthen it in the same way and also try to increase the rest of the services so that people can benefit from them. Most importantly, a VGS can also be useful to the mayor of the area, as it helps the mayor make appropriate policies for the city to develop appropriate services. For example, if one of the welfare services in that area is more than other services (the number of people referring to that welfare center is higher), then, the VGS shows this issues and as a result, the mayor takes the necessary measures to maintain and increase this opportunity. SupposeA is a set consisting of 7 cities as A=Neka,Behshahr,Sari,Ghaemshahr,Babol,Bahnamir,Babolsar, and H is a VS on A as Table 2.Table 2
Vague setH on A.
CityMembership valueNeka0.3,0.4Behshahr0.6,0.3Sari0.8,0.2Ghaemshahr0.5,0.3Babol0.5,0.2Bahnamir0.3,0.2Babolsar0.6,0.3In Table2, true membership value indicates the amount of welfare, educational, medical, and peace facilities in that area, and the false-membership value reveals the lack of necessary facilities in each of the medical services. In Tables 3–9, we show the membership rate of various welfare services on the route between each pair of cities. The relations that considered on set A are as follows:Table 3
Vague set of services on roads connecting Neka and other cities.
Services(Neka, Behshahr)(Neka, Sari)(Neka, Bahnamir)(Neka, Babolsar)Grocery store0.6,0.20.5,0.20.5,0.20.7,0.2Banking services0.6,0.20.6,0.20.4,0.30.6,0.3Car dealership0.5,0.30.5,0.20.3,0.20.5,0.4Hospitals0.6,0.20.5,0.30.4,0.20.6,0.3University and schools0.5,0.40.5,0.30.6,0.20.5,0.2Table 4
Vague set of services on roads connecting Behshahr and other cities.
Services(Behshahr, Neka)(Behshahr, Babol)(Behshahr, Sari)Grocery store0.7,0.20.6,0.20.7,0.2Banking services0.8,0.10.7,0.20.4,0.3Car dealership0.6,0.20.6,0.30.8,0.2Hospitals0.8,0.20.7,0.20.6,0.2University and schools0.7,0.20.5,0.30.6,0.3Table 5
Vague set of services on roads connecting Sari and other cities.
Services(Sari, Behshahr)(Sari, Neka)(Sari, Ghaemshahr)Grocery store0.7,0.20.5,0.20.6,0.2Banking services0.5,0.30.4,0.20.4,0.3Car dealership0.6,0.20.5,0.30.6,0.2Hospitals0.6,0.40.4,0.30.7,0.3University and schools0.7,0.20.5,0.20.6,0.2Table 6
Vague set of services on roads connecting Babol and other cities.
Services(Babol, Behshahr)(Babol, Babolsar)(Babol, Ghaemshahr)Grocery store0.5,0.20.6,0.20.4,0.3Banking services0.6,0.30.6,0.10.4,0.2Car dealership0.7,0.20.5,0.30.4,0.3Hospitals0.6,0.40.6,0.20.4,0.2University and schools0.6,0.20.7,0.10.5,0.1Table 7
Vague set of services on roads connecting Ghaemshahr and other cities.
Services(Ghaemshahr, Babol)(Ghaemshahr, Sari)(Ghaemshahr, Bahnamir)Grocery store0.7,0.10.7,0.20.6,0.2Banking services0.6,0.30.8,0.10.4,0.3Car dealership0.6,0.20.7,0.30.7,0.2Hospitals0.5,0.40.6,0.20.4,0.2University and schools0.6,0.20.5,0.40.6,0.3Table 8
Vague set of services on roads connecting Babolsar and other cities.
Services(Babolsar, Babol)(Babolsar, Sari)(Babolsar, Bahnamir)Grocery store0.7,0.10.5,0.20.6,0.2Banking services0.5,0.30.4,0.30.4,0.3Car dealership0.8,0.20.6,0.30.5,0.2Hospitals0.8,0.20.5,0.30.4,0.3University and schools0.7,0.20.5,0.30.5,0.3Table 9
Vague set of services on roads connecting Bahnamir and other cities.
Services(Bahnamir, Neka)(Bahnamir, Sari)(Bahnamir, Babolsar)Grocery store0.5,0.20.5,0.30.5,0.3Banking services0.6,0.30.6,0.20.6,0.2Car dealership0.6,0.30.5,0.20.7,0.2Hospitals0.7,0.10.6,0.20.5,0.3University and schools0.6,0.20.5,0.30.4,0.3R1=Grocery stores, R2=Banking services, R3=Car daelership, R4=Hospitals, and R5=University and Schools so that A,R1,R2,R3,R4,R5 is a graph structure. Each object in a particular relation indicates the amount of services provided between the two cities. In these relations, each element is determined by its corresponding membership rate, and the corresponding sets are FSs on R1, R2,R3, R4, and R5, respectively, which are shown by B1, B2, B3, B4, and B5, respectively.Suppose that(33)R1=Neka,Sari,Sari,Behshahr,R2=Behshahr,Babol,Babolsar,Babol,R3=Babol,Ghaemshar,Sari,Ghaemshahr,Bahnamir,Babolsar,R4=Neka,Bahnamir,Sari,Bahnamir,R5=Ghaemshahr,Bahnamir,Babolsar,Sari.Now, corresponding vague sets are(34)B1=Neka,Sari,0.6,0.2,Sari,Behshahr,0.7,0.2,B2=Behshahr,Babol,0.7,0.2,Babolsar,Babol,0.8,0.2,B3=Babol,Ghaemshahr,0.5,0.1,Sari,Ghaemshahr,0.7,0.3,Bahnamir,Babolsar,0.7,0.2,B4=Neka,Bahnamir,0.7,0.1,Sari,Bahnamir,0.6,0.2,B5=Ghaemshahr,Bahnamir,0.7,0.2,Babolsar,Sari,0.6,0.3.It is obvious thatH,B1,B2,B3,B4,B5 is a VGS and is denoted in Figure 12. In the VGS shown in Figure 12, each edge represents the most services that occur on the connecting route between the two cities. For example, most services on the way from Babolsar to Bahnamir are grocery services, and their membership value are 0.7,0.2. It is clear that the city of Sari has the highest membership in terms of banking services, which means that this city is the most sensitive city for providing banking services. According to the VGS in Figure 12, the most common services is car dealership. Therefore, we conclude that government should identify places that are suitable for providing services between cities so that people can benefit from these services at the lowest cost and in the fastest possible time.Figure 12
VGS showing the most important services along the roads connecting the two cities.


## 5. Conclusion

Fuzzy graph has various uses in modern science and technology, especially in the fields of neural networks, computer science, operation research, and decision making. VGS 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. So, in this paper, we presented certain notions, including VGS, SVGS, and vagueBi-cycle and illustrated these notions by several examples. We investigated ψ-complement, SC, SSC, and TSSC in VGS and studied some of their properties. Finally, an application of VGS has been introduced. In our future work, we will introduce neighborly irregular, highly irregular, strongly irregular, and edge irregularity in VGSs and investigate some of their properties.



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