BipolarNeutrosophicPlanarGraphs
MuhammadAkrama andK.P.Shumb
a.DepartmentofMathematics,UniversityofthePunjab,NewCampus,Lahore,Pakistan. E-mail:makrammath@yahoo.com,m.akram@pucit.edu.pk b.InstituteofMathematics,YunnanUniversity,China E-mail:kpshum@ynu.edu.cn Abstract
Fuzzygraphtheoryisusedforsolvingreal-worldproblemsindifferentfields,includingtheoreticalcomputerscience,engineering,physics,combinatoricsandmedicalsciences.Inthispaper,wepresentconepts ofbipolarneutrosophicmultigraphs,bipolarneutrosophicplanargraphs,bipolarneutrosophicdualgraphs, andstudysomeoftheirrelatedproperties.Wealsodescribe applicationsofbipolarneutrosophicgraphsin roadnetworkandelectricalconnections.
Keywords: Bipolarneutrosophicplanargraphs,bipolarneutrosophic dualgraphs. 2000MathematicsSubjectClassification:03E72,68R10,68R05
1Introduction
Fuzzygraphtheoryhasanumberofapplicationsinmodelingrealtime systemswherethelevelofinformation inherentinthesystemvarieswithdifferentlevelsofprecision.Fuzzy modelsarebecomingusefulbecauseof theiraiminreducingthedifferencesbetweenthetraditionalnumericalmodelsusedinengineeringandsciences andthesymbolicmodelsusedinexpertsystems.Kaufmanndefinedfirstfuzzygraph[11],thenRosenfeld [14]discussedseveralbasicgraph-theoreticconcepts,including bridges,cut-nodes,connectedness,treesand cycles.Bhattacharya[8]gavesomeremarksonfuzzygraphs,andSunithaandVijayakumar[10]characterized fuzzytrees.Abdul-jabbar etal. [1]introducedtheconceptofafuzzydualgraphanddiscussedsomeofits interestingproperties.SamantaandPal[15,16]introducedandinvestigatedtheconceptoffuzzyplanargraphs andstudiedseveralproperties.Onotherhand,AlshehriandAkram[6]introducedtheconceptofintuitionistic fuzzyplanargraphs.Akrametal.[5]discussedtheconceptofbipolarfuzzyplanargraphs.Dhavaseelanetal. [10]definedstrongneutrosophicgraphs.AkramandShahzadi[2] introducedthenotionsofneutrosophicgraphs andneutrosophicsoftgraphs.Inthispaper,weintroducethenotionsofbipolarsingle-valuedneutrosophic multigraphs,bipolarsingle-valuedneutrosophicplanargraphs,bipolarsingle-valuedneutrosophicdualgraphs, andinvestigatesomeoftheirinterestingproperties.Wealsodescribeapplicationsofbipolarneutrosophicgraphs inroadnetworkandelectricalconnections.
Smarandache[13]introducedneutrosophicsetsasageneralizationoffuzzysetsandintuitionisticfuzzysets.
Definition2.1. [13]Aneutrosophicset C onanon-emptyset X ischaracterizedbyatruthmembership function TC : X → [0, 1],indeterminacymembershipfunction IC : X → [0, 1]andafalsitymembershipfunction FC : X → [0, 1].Thereisnorestrictiononthesumof TC (x), IC (x)and FC (x)forall x ∈ X
Delietal.[9]definedbipolarneutrosophic(BN)setsageneralization ofbipolarfuzzysets. Definition2.2. [9]ABNsetonanonemptyset X isanobjectoftheform C = {(y,T P C (x),I P C (x),F P C (x),T N C (x),I N C (x),F N C (x)): y ∈ X} where, T P C ,I P C ,F P C : Y → [0, 1]and T N C ,I N C ,F N C : Y → [ 1, 0].Thepositivevalues T P C (x),I P C (x),F P C (x) denoterespectivelythetruth,indeterminacyandfalsemembershipdegreesofanelement y ∈ Y whereas T N C (x),I N C (x),F N C (x)denotetheimplicitcounterpropertyofthetruth,indeterminacy andfalsemembership degreesoftheelement y ∈ X correspondingtothebipolarneutrosophicset C.
WedefineBNmultisetsbasedontheconceptofYeandYe[17].
Definition2.3. Let X beanonemptysetwithgenericelementsin X denotedby x.ABNmultiset C drawn from X ischaracterizedbythethreepositivefunctions:counttruth-membershipof CT P C ,countindeterminacymembershipof CI P C ,andcountfalsity-membershipof CF P C suchthat CT P C (x): X → R+ , CI P C (x): X → R+ , CF P C (x): X → R+ for x ∈ X,where R+ isthesetofallrealnumbermultisetsintherealunitinterval[0, 1], andthreenegativefunctions:counttruth-membershipof CT N C ,countindeterminacy-membershipof CI N C ,and countfalsity-membershipof CF N C suchthat CT N C (x): X → R , CI N C (x): X → R , CF N C (x): X → R for x ∈ X,where R isthesetofallrealnumbermultisetsintherealunitinterval[ 1, 0],.Then,abipolarsingle valuedneutrosophicmultisetAisdenotedby A = { x, ((T 1)P C (x), (T 2)P C (x),..., (T q)P C (x)), ((I 1 )P C (x), (I 2)P C (x),..., (I q )P C (x)), ((F 1 )P C (x), (F 2)P C (x),..., (F q )P C (x)), (T 1)N C (x), (T 2)N C (x),..., (T q)N C (x)), ((I 1 )N C (x), (I 2)N C (x),..., (I q )N C (x)), ((F 1 )N C (x), (F 2 )N C (x),..., (F q)N C (x)) |x ∈ X}, wherethepositivetruth,indeterminacyandfalsity-membershipsequences ((T 1)P C (x), (T 2)P C (x),..., (T q)P C (x)), ((I 1 )P C (x), (I 2 )P C (x),..., (I q)P C (x)), ((F 1 )P C (x), (F 2)P C (x),..., (F q )P C (x))may beindecreasingorincreasingorder,andsumof(T i C )P (x),(I i)P C (x),(F i)P C (x) ∈ [0, 1]satisfiesthecondition
0 ≤ sup(T i)P C (x)+sup(I i)P C (x)+sup(F i)P C (x) ≤ 3for x ∈ X and i =1, 2,...,q,thenegativetruth,indeterminacyandfalsity-membershipsequences
((T 1)P C (x), (T 2)P C (x),..., (T q)P C (x)), ((I 1 )N C (x), (I 2)N C (x),..., (I q )N C (x)), ((F 1 )N C (x), (F 2 )N C (x),..., (F q )N C (x))may beindecreasingorincreasingorder,andsumof(T i C )N (x),(I i)N C (x),(F i)N C (x) ∈ [ 1, 0]satisfiesthecondition 3 ≤ inf(T i)N C (x)+inf(I i)N C (x)+inf(
(a) T P D (yz) ≤ min(T P C (y),T P C (z)),
(b) I P D (yz) ≤ min(I P C (y),I P C (z)), (c) F P D (yz) ≤ max(F P C (y),F P C (z)), (d) T N D (yz) ≥ max(T N C (y),T N C (z)), (e) I N D (yz) ≥ max(I N C (y),I N C (z)), (f) F N D (yz) ≥ min(F N C (y),F N C (z)) forall y,z ∈ X.Notethat D iscalledaBNrelationon C.
Example2.5. Considerabipolarneutrosophicgraph G =(C,D)on X = {x,y,z} asshowninFig.2.1. x(0. 3, 0. 5, 0. 4, −0. 5, −0. 5, −0. 2) y (0 . 5, 0. 3, 0. 3, − 0. 2, − 0. 7, − 0. 3) z(0 3, 0 3, 0 1, 0 4, 0 5, 0 5)
(0 3, 0 3, 0 4, 0 2, 0 5, 0 2) (0. 3, 0. 3, 0. 3, −0. 2, −0. 5, −0. 5) (0 2 , 0 3 , 0 3 , 0 3 , 0 . 5 , 0 5)
Figure2.1:Bipolarneutrosophicgraph
Definition2.6. Let C =(T P C ,I P C ,F P C ,T N C ,I N C ,F N C )beaBNseton V and let D = {(xy,T P D (xy)i,I P D (xy)i,F P D (xy)i,T N D (xy)i,I N D (xy)i,F N D (xy)i),i =1, 2,...,m|xy ∈ V × V } beaBN multisetof V × V suchthat (g) T P D (xy)i ≤ min{T P C (x),T P C (y)}, (h) T N D (xy)i ≥ max{T N C (x),T N C (y)}, (i) I P D (xy)i ≤ min{I P C (x),I P C (y)}, (j) I N D (xy)i ≥ max{I N C (x),I N C (y)}, (k) F P D (xy)i ≤ max{F P C (x),F P C (y)}, (l) F N D (xy)i ≥ min{F N C (x),F N C (y)} forall i =1, 2,...,m.Then G =(C,D)iscalled aBNmultigraph.
Theremaybemorethanoneedgebetweenthevertices x and y.Thepositivevalues T P D (xy)i, I P D (xy)i, F P D (xy)i representtruth,indeterminacyandfalsityoftheedge xy in G,whereasthenegativevalues T N D (xy)i, I N D (xy)i, F N D (xy)i representtheimplicitcounterpropertyofthetruth,indeterminacyandfalsemembershipdegreesof theedge xy in G m denotesthenumberofedgesbetweenthevertices.InBNmultigraph G, D issaidtobe BNmultiedgeset. Example2.7. Let G∗ =(V,E),where V = {a,b,c,d}, E = {ab,ab,ab,bc,bd}.Let C =(T P C ,I P C ,F P C ,T N C ,I N C ,F N C ) beaBNseton V and D =(T P D ,I P D ,F P D ,T N D ,I N D ,F N D )beaBNmultiedgeseton E ⊆ V × V definedinTable1 andTable2.
Table1:Single-valuedneutrosophicset C C abcd
T P C 0 50 40 50 4 I P C 0.30.20.40.3 F P C 0.30.40.30.4 T N C 0.5 0.4 0.5 0.4 I N C 0 3 0 2 0 4 0 3 F N C 0 3 0 4 0 3 0 4
Table2:BNmultiedgeset D D abababbcbd
T P D 0.20.10.20.30.1 I P D 0.20.10.20.10.2 F P D 0 200 20 30 2 T N D 0 2 0 1 0 2 0 3 0 1 I N D 0 2 0 1 0 2 0 1 0 2 F N D 0 2 0 0 2 0 3 0 2
Bydirectcalculations,weseefromFig.2.2thatitisaBNmultigraph.
a(0 5, 0 3, 0 3, 0 5, 0 3, 0 3)
b(0. 4, 0. 2, 0. 4, −0. 4, −0. 2, −0. 4) (0 .1 ,0 .2 ,0 .2 , −0 .1 , −0 .2 , −0 .2)
c(0 5, 0 4, 0 3, 0 5, 0 4, 0 3)
(0 1, 0 1, 0, 0 1, 0 1, 0) (0 3, 0 1, 0 3, 0 3, 0 1, 0 3)
(0 2, 0 2, 0 2, 0 2, 0 2, 0 2) (0 2, 0 2, 0 2, 0 2, 0 2, 0 2) d(0 4, 0 3, 0 4, 0 4, 0 3, 0 4)
Figure2.2:Neutrosophicmultigraph Definition2.8. Let D = {(xy,T P D (xy)i,I P D (xy)i,F P D (xy)i,T N D (xy)i,I N D (xy)i,F N D (xy)i),i =1, 2,...,m|xy ∈ V × V } beaBNmultiedgesetinBNmultigraph G.Thedegreeofavertex x ∈ V ,denotedbydeg(x),isdefined by deg(x)=( m i=1 T P D (xy)i, m i=1 I P D (xy)i, m i=1 F P D (xy)i, m i=1 T N D (xy)i, m i=1 I N D (xy)i, m i=1 F N D (xy)i)
Example2.9. InExample2.7,thedegreeofvertices a,b,c,d are deg(a)=(0 5, 0 5, 0 4, 0 5, 0 5, 0 4), deg(b)=(0 9, 0 8, 0 9, 0 9, 0 8, 0 9), deg(c)=(0 3, 0 1, 0 3, 0 3, 0 1, 0 3)and deg(d)=(0 1, 0 2, 0 2, 0 1, 0 2, 0 2). Definition2.10. Let D = {(xy,T P D (xy)i,I P D (xy)i,F P D (xy)i,T N D (xy)i,I N D (xy)i,F N D (xy)i),i =1, 2,...,m|xy ∈ V × V } beaBNmultiedgesetinBNmultigraph G.Amultiedge xy of G is strong ifthefollowingconditions aresatisfied:
(m) 1 2 min{T P C (x),T P C (y)}≤ T P D (xy)i, (n) 1 2 max{T N C (x),T N C (y)}≥ T N D (xy)i, (o) 1 2 min{I P C (x),I P C (y)}≤ I P D (xy)i, (p) 1 2 max{I N C (x),I N C (y)}≥ I N D (xy)i, (q) 1 2 max{F P C (x),F P C (y)}≥ F P D (xy)i, (r) 1 2 min{F N C (x),F N C (y)}≤ F N D (xy)i forall i =1, 2,...,m. Definition2.11. Let D = {(xy,T P D (xy)i,I P D (xy)i,F P D (xy)i,T N D (xy)i,I N D (xy)i,F N D (xy)i),i =1, 2,...,m|xy ∈ V × V } beaBNmultiedgesetinBNmultigraph G.ABNmultigraph G is complete ifthefollowingconditions aresatisfied: (s)min{T P C (x),T P C (y)} = T P D (xy)i, (t)max{T N C (x),T N C (y)} = T N D (xy)i, (u)min{I P C (x),I P C (y)} = I P D (xy)i, (v)max{I N C (x),I N C (y)} = I N D (xy)i, (w)max{F P C (x),F P C (y)} = F P D (xy)i (x)min{F N C (x),F N C (y)} = F N D (xy)i forall i =1, 2,...,m andforall x,y ∈ V .
Example2.12. ConsideraBNmultigraph G asshowninFig.2.3.Byroutinecalculations,itiseasytosee thatFig.2.3isaBNcompletemultigraph.
b(0. 5, 0. 4, 0. 3, −0. 5, −0. 4, −0. 3) (0 4, 0 3, 0 3, 0 4, 0 3, 0 3) (0 4, 0 4, 0 3, 0 4, 0 4, 0 3)
a(0 4, 0 3, 0 2, 0 4, 0 3, 0 2) c(0. 4, 0. 4, 0. 3, −0. 4, −0. 4, −03) (0 4, 0 3, 0 3, 0 4, 0 3, 0 3) (0 4, 0 2, 0 3, 0 4, 0 2, 0 3)
Figure2.3:Bipolarneutrosophiccompletemultigraph.
SupposethatgeometricinsightforBNgraphshasonlyonecrossing betweensinglevaluedneutrosophicedges (ab,T P D (ab)i,I P D (ab)i,F P D (ab)i,T N D (ab)i,I N D (ab)i,F N D (ab)i)and (cd,T P D (cd)i,I P D (cd)i,F P D (cd)i,T N D (cd)i,I N D (cd)i,F N D (cd)i).Wenotethat:
•
If(ab,T P D (ab)i,I P D (ab)i,F P D (ab)i,T N D (ab)i,I N D (ab)i,F N D (ab)i)=(1, 1, 1, 1, 1, 1)and(cd,T P D (cd)i,I P D (cd)i, F P D (cd)i,T N D (cd)i,I N D (cd)i, F N D (cd)i)=(0, 0, 0, 0, 0, 0)or(ab,T P D (ab)i,I P D (ab)i, F P D (ab)i,T N D (ab)i,I N D (ab)i, F N D (ab)i)=(0, 0, 0, 0, 0, 0),(cd,T P D (cd)i,I P D (cd)i,F P D (cd)i,T N D (cd)i,I N D (cd)i,F N D (cd)i)=(1, 1, 1, 1, 1, 1), thenBNgraphhasnocrossing,
• If(ab,T P D (ab)i,I P D (ab)i,F P D (ab)i,T N D (ab)i,I N D (ab)i,F N D (ab)i)=(1, 1, 1, 1, 1, 1)and (cd,T P D (cd)i,I P D (cd)i,F P D (cd)i,T N D (cd)i,I N D (cd)i,F N D (cd)i)=(1, 1, 1, 1, 1, 1),thenthereexistsacrossingfortherepresentationofthegraph.
Definition2.13. The strengthoftheBNedge ab canbemeasuredbythevalue Sab =((ST P )ab, (SI P )ab, (SF P )ab, (ST N )ab, (SI N )ab, (SF N )ab) =( T P D (ab)i min(T P C (a),T P C (b)) , I P D (ab)i min(I P C (a),I P C (b)) , F P D (ab)i max(F P C (a),F P C (b)) , T N D (ab)i max(T N C (a),T N C (b)) , I N D (ab)i max(I N C (a),I N C (b)) , F N D (ab)i min(F N C (a),F N C (b)) ) Definition2.14. Let G beaBNmultigraph.Anedge ab issaidtobea BNstrong if(ST P )ab ≥ 0 5,(SI P )ab ≥ 0 5,(SF P )ab ≥ 0 5,(ST N )ab ≤−0 5,(SI N )ab ≤−0 5,(SF N )ab ≤−0 5otherwise,wecallweakedge. Definition2.15. Let G =(C,D)beaBNmultigraphsuchthat D containstwoedges (ab,T P D (ab)i,I P D (ab)i,F P D (ab)i,T N D (ab)i,I N D (ab)i,F N D (ab)i)and(cd,T P D (cd)j ,I P D (cd)j ,F P D (cd)j ,T N D (cd)j ,I N D (cd)j ,F N D (cd)j ) intersectedatapoint P ,where i and j arefixedintegers.Wedefinetheintersectingvalueatthepoint Q by SQ =((ST P )Q, (SIP )Q, (SF P )Q, (ST N )Q, (SIN )Q, (SF N )Q) =( (ST P )ab +(ST P )cd 2 , (SIP )ab +(SIP )cd 2 , (SF P )ab +(SF P )cd 2 , (ST N )ab+(ST N )cd 2 , (SIN )ab +(SIN )cd 2 , (SF N )ab +(SF N )cd 2 ).
IfthenumberofpointofintersectionsinaBNmultigraphincreases, planaritydecreases.ThusforBNmultigraph, SQ isinverselyproportionaltotheplanarity.WenowintroducetheconceptofaBNplanargraph. Definition2.16. Let G beaBNmultigraphand Q1,Q2,...,Qz bethepointsofintersectionbetweenthe edgesforacertaingeometricalrepresentation, G issaidtobeaBNplanargraphwithBNplanarityvalue f =(fT P ,fI P ,fF P ,fT N ,fI N ,fF N ),where f =(fT P ,fI P ,fF P ,fT N ,fI N ,fF N ) =( 1 1+ {(ST P )Q1 +(ST P )Q2 + +(ST P )Qz } , 1 1+ {(SI P )Q1 +(SI P )Q2 + +(SI P )Qz } , 1 1+ {(SF P )Q1 +(SF P )Q2 + ... +(SF P )Qz } , 1 1 −{(ST N )Q1 +(ST N )Q2 + ... +(ST N )Qz } , 1 1 −{(SI N )Q1 +(SI N )Q2 + ... +(SI N )Qz } , 1 1 −{(SF N )Q1 +(SF N )Q2 + ... +(SF N )Qz } )
Clearly, f =(fT P ,fI P ,fF P ,fT N ,fI N ,fF N )isboundedand0 <fT P ≤ 1,0 <fI P ≤ 1,0 <fF P ≤ 1, 1 <fT N ≤ 0, 1 <fI N ≤ 0, 1 <fF N ≤ 0 IfthereisnopointofintersectionforacertaingeometricalrepresentationofaBNplanargraph,thenitsBN planarityvalueis(1, 1, 1, 1, 1, 1).WeconcludethateveryBNgraphisaBNplanargraphwithcertain BN planarityvalue. Example2.17. Consideramultigraph G∗ =(V,E)suchthat V = {a,b,c,d,e}, E = {ab,ac,ad,ad,bc,bd,cd,ce,ae,de,be}. Let C =(T P C ,I P C ,F P C ,T N C ,I N C ,F N C )beaBNsetof V andlet D =(T P D ,I P D ,F P D ,T N D ,I N D ,F N D )beaBNmultiedge setof V × V definedinTable3andTable4.
Table3:BNset C A abcde
T P C 0.50.40.30.60.6
I P C 0.50.40.30.60.6
F P C 0.20.10.10.20.1
T N C 0 5 0 4 0 3 0 6 0 6 I N C 0 5 0 4 0 3 0 6 0 6 F N C 0.2 0.1 0.1 0.2 0.1
Table4:BNmultiedgeset D
B abacadadbcbdcdaecedebe
T P D 0.20.20.20.30.20.20.20.20.20.20.2
I P D 0.20.20.20.30.20.20.20.20.20.20.2
F P D 0.10.10.10.10.10.10.10.10.10.10.1
T N D 0 2 0 2 0 2 0 3 0 2 0 2 0 2 0 2 0 2 0 2 0 2 I N D 0 2 0 2 0 2 0 3 0 2 0 2 0 2 0 2 0 2 0 2 0 2 F N D 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
TheBNmultigraphasshowninFig.2.4hastwopointofintersections P1 and P2. P1 isapointbetweentheedges(ad, 0 2, 0 2, 0 1, 0 2, 0 2, 0 1)and(bc, 0 2, 0 2, 0 1, 0 2, 0 2, 0 1)and P2 isbetween (ad, 0 3, 0 3, 0 1, 0 3, 0 3, 0 1)and(bc, 0 2, 0 2, 0 1, 0 2, 0 2, 0 1). Fortheedge(ad, 0 2, 0 2, 0 1, 0 2, 0 2, 0 1), Sad =(0 4, 0 4, 0 5, 0 4, 0 4, 0 5), Fortheedge(ad, 0 3, 0 3, 0 1, 0 3, 0 3, 0 1), Sad =(0 6, 0 6, 0 5, 0 6, 0 6, 0 5)and fortheedge(bc, 0 2, 0 2, 0 1, 0 2, 0 2, 0 1), Sbc =(0 6667, 0 6667, 1, 0 6667, 0 6667, 1).
Forthefirstpointofintersection P1,intersectingvalue SP1 is(0.5334, 0.5334,, 0.75, 0.5334, 0.5334, 0.75)and thatforthesecondpointofintersection P2, SP2 =(0.63335, 0.63335,, 0.75, 0.63335, 0.63335, 0.75).Therefore,theBNplanarityvaluefortheBNmultigraphshowninFig.2.4is(0 461, 0 461, 0 4, 0 461, 0 461, 0 4).
a(0 5, 0 5, 0 2, 0 5, 0 5, 0 2)
b(0 4, 0 4, 0 1, 0 4, 0 4, 0 1)
(0 2 , 0 2 , 0 1 , 0 2 , 0 2 , 0 1)
(0 .3 ,0 .3 ,0 .1 , −0 .3 , −0 .3 , −0 . 1)
(0 2, 0 2, 0 1, 0 2, 0 2, 0 1) (0 2 , 0 2 , 0 1 , 0 2 , 0 2 , 0 1)
c(0 3, 0 3, 0 1, 0 3, 0 3, 0 1)
(0. 2, 0. 2, 0. 1, −0. 2, −0. 2, −0. 1) (0 .2 ,0 .2 ,0 .1 , −0 .2 , −0 .2 , −0 .1)
P1
(0 .2 ,0 .2 ,0 .1 , −0 .2 , −0 .2 , −0 . 1) (0 . 2, 0. 2, 0. 1, −0. 2, −0. 2, −0. 1)
P2 (0. 2, 0. 2, 0. 1, −0. 2, −0. 2, −0. 1) (0 .2 ,0 .2 ,0 .1 , −0 .2 , −0 .2 , −0 .1)
(0 2, 0 2, 0 1, 0 2, 0 2, 0 1)
d(0 6, 0 6, 0 2, 0 6, 0 6, 0 2)
e(0 6, 0 6, 0 1, 0 6, 0 6, 0 1)
Figure2.4:Neutrosophicplanargraph
Theorem2.18. Let G beaBNcompletemultigraph.Theplanarityvalue, f =(fT P ,fI P ,fF P ,fT N ,fI N ,fF N ) of G isgivenby fT P = 1 1+nQ , fI P = 1 1+nQ and fF P = 1 1+nQ suchthat fT P + fI P + fF P ≤ 3, fT N = 1 1 nQ , fI N = 1 1 nQ and fF N = 1 1 nQ suchthat 3 ≤ fT N + fI N + fF N ≤ 0 where nQ isthenumberofpointof intersectionsbetweentheedgesin G Definition2.19. ABNplanargraph G iscalled strongBNplanargraph iftheBNplanarityvalue f = (fT P ,fI P ,fF P ,fT N ,fI N ,fF N )ofthegraphis fT P ≥ 0.5, fI P ≥ 0.5, fF P ≤ 0.5, fT N ≤−0.5, fI N ≤−0.5, fF P ≥−0.5.
Theorem2.20. Let G beastrongBNplanargraph.Thenumberofpointofintersectionsbetweenstrongedges in G isatmostone.
Proof. Let G beastrongBNplanargraph.Assumethat G hasatleasttwopointofintersections P1 and P2 betweentwostrongedgesin G.Foranystrongedge (ab,T P D (ab)i,I P D (ab)i,F P D (ab)i,T N D (ab)i,I N D (ab)i,F N D (ab)i), T P D (ab)i ≥ 1 2 min{T P C (a),T P C (b)},I P D (ab)i ≥ 1 2 min{I P C (a),I P C (b)},F P D (ab)i ≤ 1 2 max{F P C (a),F P C (b)}, T N D (ab)i ≤ 1 2 max{T N C (a),T N C (b)},I N D (ab)i ≤ 1 2 max{I N C (a),I N C (b)},F N D (ab)i ≥ 1 2 min{F N C (a),F N C (b)}
Thisshowsthat(ST P )ab ≥ 0.5,(SI P )ab ≥ 0.5,(SF P )ab ≤ 0.5,(ST N )ab ≤−0.5,(SI N )ab ≤−0.5,(SF N )ab ≥ 0.5.
Thusfortwointersectingstrongedges(ab,T P D (ab)i,I P D (ab)i,F P D (ab)i,T N D (ab)i,I N D (ab)i,F N D (ab)i)and (cd,T P D (cd)j ,I P D (cd)j ,F P D (cd)j ,T N D (cd)j ,I N D (cd)j ,F N D (cd)j ),
(ST P )ab +(ST P )cd 2 ≥ 0.5, (SI P )ab +(SI P )cd 2 ≥ 0.5, (SF P )ab +(SF P )cd 2 ≤ 0.5, (ST N )ab +(ST N )cd 2 ≤−0.5, (SI N )ab +(SI N )cd 2 ≤−0.5, (SF N )ab +(SF N )cd 2 ≥−0.5.
ItcontradictsthefactthattheBNgraphisastrongBNplanargraph.Thusnumberofpointofintersections betweenstrongedgescannotbetwo.Obviously,ifthenumberofpointofintersectionsofstrongBNedges increases,theBNplanarityvaluedecreases.Similarly,ifthenumber ofpointofintersectionofstrongedgesis one,thentheBNplanarityvalue fT P > 0.5, fI P > 0.5, fI P > 0.5, fT N < 0.5, fI N < 0.5, fI N < 0.5.Any BNplanargraphwithoutanycrossingbetweenedgesisastrongBNplanargraph.Thus,weconcludethatthe maximumnumberofpointofintersectionsbetweenthestrongedgesin G isone.
FaceofaBNplanargraphisanimportantparameter.FaceofaBNgraphisaregionboundedbyBNedges. EveryBNfaceischaracterizedbyBNedgesinitsboundary.IfalltheedgesintheboundaryofaBNfacehave T P , I P , F P , T N , I N and F N values(1, 1, 1, 1, 1, 1)and(0, 0, 0, 0, 0, 0),respectively,itbecomescrispface.If oneofsuchedgesisremovedorhas T P , I P , F P , T N , I N and F N values(0, 0, 0, 0, 0, 0)and(1, 1, 1, 1, 1, 1), respectively,theBNfacedoesnotexist.SotheexistenceofaBNfacedependsontheminimumvalueof strengthofBNedgesinitsboundary.ABNfaceandits T P , I P , F P , T N , I N ,and F N valuesofaBNgraph aredefinedbelow.
Let G beaBNplanargraphand
G.Thetruth, indeterminacyandfalsityvaluesoftheBNfaceare:
6.min F N D (xy)i min{F N C (x),F N C (y)} ,i =1, 2,...,m| xy ∈ E′
Definition2.22. ABNfaceiscalled strongBNface ifitspositivetrueandindeterminacyvalueisgreater than0 5butfalsevalueislesserthan0.5,andnegativetrueandindeterminacyvalueislessthan 0 5butfalse valueisgreaterthan-0.5.Otherwise,faceisweak.EveryBNplanar graphhasaninfiniteregionwhichiscalled outerBNface.Otherfacesarecalled innerBNfaces.
Example2.23. ConsideraBNplanargraphasshowninFig.2.5.TheBNplanargraphhasthefollowing faces:
• BNface F1 isboundedbytheedges
(v1v2, 0 5, 0 5, 0 1, 0 5, 0 5, 0 1), (v2v3, 0 6, 0 6, 0 1, 0 6, 0 6, 0 1), (v1v3, 0 5, 0 5, 0 1, 0 5, 0 5, 0 1)
• outerBNface F2 surroundedbyedges
(v1v3, 0.5, 0.5, 0.1, 0.5, 0.5, 0.1), (v1v4, 0.5, 0.5, 0.1, 0.5, 0.5, 0.1), (v2v4, 0.6, 0.6, 0.1, 0.6, 0.6, 0.1),
(v2v3, 0.6, 0.6, 0.1, 0.6, 0.6, 0.1),
• BNface F3 isboundedbytheedges
(v1v2, 0 5, 0 5, 0 1, 0 5, 0 5, 0 1), (v2v4, 0 6, 0 6, 0 1, 0 6, 0 6, 0 1), (v1v4, 0 5, 0 5, 0 1, 0 5, 0 5, 0 1)
Clearly,thepositivetruth,indeterminacyandfalsityvaluesofaBNface F1 are0.833,0.833and0.333,respectively,andthenegativetruth,indeterminacyandfalsityvaluesofa BNface F1 are-0.833,-0.833and-0.333, respectively.Thepositivetruth,indeterminacyandfalsityvaluesofaBNface F3 are0.833,0.833and0.333, respectively,andthenegativetruth,indeterminacyandfalsityvaluesofaBNface F3 are-0.833,-0.833and -0.333,respectively.Thus F1 and F3 arestrongBNfaces.
v1 (0 . 6, 0. 6, 0. 3, −0. 6, −0. 6, −0. 3)
(0 .5 ,0 .5 ,0 .1 , −0 .5 , −0 .5 , −0 . 1) (0 5 , 0 5 , 0 . 1 , 0 5 , 0 5 , 0 . 1)
F3 F2
(0 5, 0 5, 0 1, 0 5, 0 5, 0 1)
(0 . 6 , 0 . 6 , 0 1 , 0 6 , 0 6 , 0 1) (0. 6, 0. 6, 0. 1, −0. 6, −0. 6, −0. 1) F1
v4 (0 . 7, 0. 7, 0. 1, −0. 7, −0. 7, −0. 1)
v2(0 .7 ,0 .7 ,0 .3 , −0 .7 , −0 .7 , −0 . 3) v3(0 .8 ,0 .8 ,0 .1 , −0 .8 , −0 .8 , −0 .1)
Figure2.5:FacesinBNplanargraph 10
WenowintroducedualofBNplanargraph.InBNdualgraph,verticesarecorrespondingtothestrongBNfaces oftheBNplanargraphandeachBNedgebetweentwoverticesiscorrespondingtoeachedgeintheboundary betweentwofacesofBNplanargraph.Theformaldefinitionisgiven below.
Definition2.24. Let G beaBNplanargraphandlet D = {(xy,T P D (xy)i,I P D (xy)i,F P D (xy)i,T N D (xy)i,I N D (xy)i,F N D (xy)i),i =1, 2,...,m|xy ∈ V ×V }.Let F1,F2,...,Fk bethestrongBNfacesof G.TheBNdualgraphof G isaBNplanargraph G′ =(V ′,C ′,D′),where V ′ = {xi,i = 1, 2,...,k},andthevertex xi of G′ isconsideredfortheface Fi of G.Thetruth-membership,indeterminacy andfalse-truth-membershipvaluesofverticesaregivenbythemapping C ′ =(T P C′ ,I P C′ ,F P C′ ,T N C′ ,I N C′ ,F N C′ ): V ′ → [0, 1] × [0, 1] × [0, 1] × [ 1, 0] × [ 1, 0] × [ 1, 0]suchthat T P C′ (xi)=max{T P D′ (uv)i,i =1, 2,...,p|uv isanedgeoftheboundaryofthestrongBNface Fi}, T N C′ (xi)=min{T N D′ (uv)i,i =1, 2,...,p|uv isanedgeoftheboundaryofthestrongBNface Fi}, I P C′ (xi)=max{I P D′ (uv)i,i =1, 2,...,p|uv isanedgeoftheboundaryofthestrongBNface Fi}, I N C′ (xi)=min{I N D′ (uv)i,i =1, 2,...,p|uv isanedgeoftheboundaryofthestrongBNface Fi}, F P C′ (xi)=min{F P D′ (uv)i,i =1, 2,...,p|uv isanedgeoftheboundaryofthestrongBNface Fi}, F N C′ (xi)=max{F N D′ (uv)i,i =1, 2,...,p|uv isanedgeoftheboundaryofthestrongBNface Fi}.
Theremayexistmorethanonecommonedgesbetweentwofaces Fi and Fj of G.Thustheremaybemorethan oneedgesbetweentwovertices xi and xj inBNdualgraph G′.Let(T P )l D (xixj ),(I P )l D (xixj )and(F P )l D (xixj ) denotethepositivetruth,indeterminacyandfalsitymembershipvaluesofthe l-thedgebetween xi and xj ,andlet (T N )l D (xixj ),(I N )l D (xixj )and(F N )l D (xixj )denotethenegativetruth,indeterminacyandfalsitymembership valuesofthe l-thedgebetween xi and xj .Thepositiveandnegativetruth,indeterminacyandfalsityvalues oftheBNedgesoftheBNdualgrapharegivenby T P D′ (xixj )l =(T P )l D (uv)j , I P D′ (xixj )l =(I P )l D (uv)j , F P D′ (xixj )l =(F P )l D (uv)j , T N D′ (xixj )l =(T N )l D (uv)j , I N D′ (xixj )l =(I N )l D (uv)j , F N D′ (xixj )l =(F N )l D (uv)j , where(uv)l isanedgeintheboundarybetweentwostrongBNfaces Fi and Fj and l =1, 2,...,s,where s is thenumberofcommonedgesintheboundarybetween Fi and Fj orthenumberofedgesbetween xi and xj .If therebeanystrongpendantedgeintheBNplanargraph,thentherewillbeaselfloopin G′ correspondingto thispendantedge.Theedgetruth-membership,indeterminacy-membershipandfalsity-membershipvalueof theselfloopisequaltothetruth-membership,indeterminacy-membershipandfalsity-membershipvalueofthe pendantedge.Single-valuedneutrosophicdualgraphofBNplanar graphdoesnotcontainpointofintersection ofedgesforacertainrepresentation,soitisBNplanargraphwithplanarityvalue(1, 1, 1, 1, 1, 1).Thus theBNfaceofBNdualgraphcanbesimilarlydescribedasinBNplanargraphs.
Example2.25. ConsideraBNplanargraph G =(V,A,B)asshowninFig.2.6suchthat V = {a,b,c,d}, C =(a, 0.6, 0.6, 0.2, 0.6, 0.6, 0.2), (b, 0.7, 0.7, 0.2, 0.7, 0.7, 0.2), (c, 0.8, 0.8, 0.2, 0.8, 0.8, 0.2), (d, 0.9, 0.9, 0.1, 0.9, 0.9, 0.1), and D = {(ab, 0.5, 0.5, 0.01, 0.5, 0.5, 0.01), (ac, 0.4, 0.4, 0.01, 0.4, 0.4, 0.01), (ad, 0.55, 0.55, 0.01, 0.55, 0.55, 0.01), (bc, 0.45, 0.45, 0.01, 0.45, 0.45, 0.01), (bc, 0.6, 0.6, 0.01, 0.6, 0.6, 0.01), (cd, 0.7, 0.7, 0.01, 0.7, 0.7, 0.01)}.
a b c d
x1 x2 x3 x4
Figure2.6:Neutrosophicdualgraph
TheBNplanargraphhasthefollowingfaces:
• BNface F1 isboundedby (ab, 0 5, 0 5, 0 01, 0 5, 0 5, 0 01), (ac, 0 4, 0 4, 0 01, 0 4, 0 4, 0 01), (bc, 0 45, 0 45, 0 01, 0 45, 0 45, 0 01).
• BNface F2 isboundedby (ad, 0 55, 0 55, 0 01, 0 55, 0 55, 0 01), (cd, 0 7, 0 7, 0 01, 0 7, 0 7, 0 01), (ac, 0 4, 0 4, 0 01, 0 4, 0 4, 0 01).
• BNface F3 isboundedby (bc, 0 45, 0 45, 0 01, 0 45, 0 45, 0 01), (bc, 0 6, 0 6, 0 01, 0 6, 0 6, 0 01)
• outerBNface F4 issurroundedby (ab, 0.5, 0.5, 0.01, 0.5, 0.5, 0.01), (bc, 0.6, 0.6, 0.01, 0.6, 0.6, 0.01), (cd, 0.7, 0.7, 0.01, 0.7, 0.7, 0.01), (ad, 0.55, 0.55, 0.01, 0.55, 0.55, 0.01). RoutinecalculationsshowthatallfacesarestrongBNfaces.ForeachstrongBNface,weconsideravertexfor theBNdualgraph.Sothevertexset V ′ = {x1,x2,x3,x4},wherethevertex xi istakencorrespondingtothe strongBNface Fi, i =1, 2, 3
F N C′ (x1)=max{−0 01, 0 01, 0 01} = 0 01,F N C′ (x2)=max{−0 01, 0 01, 0 01} = 0 01,
T P C′ (x3)=max{0 45, 0 6} =0 6,T P C′ (x4)=max{0 5, 0 6, 0 7, 0 55} =0 7,
T N C′ (x3)=min{−0 45, 0 6} = 0 6,T N C′ (x4)=min{−0 5, 0 6, 0 7, 0 55} = 0 7,
I P C′ (x3)=max{0 45, 0 6} =0 6,I P C′ (x4)=max{0 5, 0 6, 0 7, 0 55} =0 7,
F P C′ (x3)=min{0.01, 0.01} =0.01,F P C′ (x4)=min{0.01, 0.01, 0.01, 0.01} =0.01.
F N C′ (x3)=max{−0 01, 0 01} = 0 01,F N C′ (x4)=max{−0 01, 0 01, 0 01, 0 01} = 0 01
Therearetwocommonedges ad and cd betweenthefaces F2 and F4 in G.Hencebetweenthevertices x2 and x4,thereexisttwoedgesintheBNdualgraphof G.Truth-membership,indeterminacy-membershipand falsity-membershipvaluesoftheseedgesaregivenby
T P D′ (x2x4)= T P D (cd)=0.7,T P D′ (x2x4)= T P D (ad)=0.55,I P D′ (x2x4)= I P D (cd)=0.7,I P D′ (x2x4)= I P D (ad)=0.55,
F P D′ (x2x4)= F P D (cd)=0 01,F P D′ (x2x4)= F P D (ad)=0 01
T N D′ (x2x4)= T N D (cd)= 0 7,T N D′ (x2x4)= T N D (ad)= 0 55,I N D′ (x2x4)= I N D (cd)= 0 7,I N D′ (x2x4)= I N D (ad)= 0 55, F N D′ (x2x4)= F N D (cd)= 0.01,F N D′ (x2x4)= F N D (ad)= 0.01.
Thetruth-membership,indeterminacy-membershipandfalsity-membershipvaluesofotheredgesoftheBN dualgrapharecalculatedas
T P D′ (x1x3)= T P D (bc)=0 45,T P D′ (x1x2)= T P D (ac)=0 4,T P D′ (x1x4)= T P D (ab)=0 5,T P D′ (x3x4)= T P D′ (bc)=0 6,
T N D′ (x1x3)= T N D (bc)= 0.45,T N D′ (x1x2)= T N D (ac)= 0.4,T N D′ (x1x4)= T N D (ab)= 0.5,T N D′ (x3x4)= T N D′ (bc)= 0.6, I P D′ (x1x3)= I P D (bc)=0.45,I P D′ (x1x2)= I P D (ac)=0.4,I P D′ (x1x4)= I P D (ab)=0.5,I P D′ (x3x4)= I P D′ (bc)=0.6, I N D′ (x1x3)= I N D (bc)= 0 45,I N D′ (x1x2)= I N D (ac)= 0 4,I N D′ (x1x4)= I N D (ab)= 0 5,I N D′ (x3x4)= I N D′ (bc)= 0 6,
F P D′ (x1x3)= T P D (bc)=0 01,F P D′ (x1x2)= F P D (ac)=0 01,F P D′ (x1x4)= F P D (ab)=0 01,F P D′ (x3x4)= F P D (bc)=0 01 F N D′ (x1x3)= T N D (bc)=0 01,F N D′ (x1x2)= F N D (ac)=0 01,F N D′ (x1x4)= F N D (ab)=0 01,F N D′ (x3x4)= F N D (bc)=0 01
ThustheedgesetofBNdualgraphis
D′ = {(x1x3, 0 45, 0 45, 0 01, 0 45, 0 45, 0 01), (x1x2, 0 4, 0 4, 0 01, 0 4, 0 4, 0 01), (x1x4, 0 5, 0 5, 0 01, 0 5, 0 5, 0 01), (x3x4, 0 6, 0 6, 0 01, 0 6, 0 6, 0 01), (x2x4, 0 7, 0 7, 0 01, 0 7, 0 7, 0 01), (x2x4, 0 55, 0 55, 0 01, 0 55, 0 55, 0 01)} InFig.2.6,theBNdualgraph G′ =(V ′,C ′,D′)of G isdrawnby dottedline.
Weakedgesinplanargraphsarenotconsideredforanycalculationin BNdualgraphs.Westatethefollowing Theoremwithoutitsproof.
Theorem2.26. Let G =(V,C,D) beaBNplanargraphwithoutweakedgesandtheBNdualgraphof G be G′ =(V ′,C ′,D′).Thetruth-membershipindeterminacy-membershipandfalsity-membershipvaluesofBN edgesof G′ areequaltotruth-membership,indeterminacy-membership andfalsity-membershipvaluesoftheBN edgesof G
GraphisconsideredanimportantpartofMathematicsforsolvingcountlessrealWorldproblemsininformation technology,psychology,engineering,combinatoricsandmedicalsciences.EverythinginthisWorldisconnected, forinstance,citiesandcountriesareconnectedbyroads,railwaysarelinkedbyrailwaylines,flightnetworks areconnectedbyair,electricaldevicesareconnectedbywires,pagesoninternetbyhyperlinks,componentsof electriccircuitsbyvariouspaths,andmanymore.Scientists,analystsandengineersaretryingtooptimizethese networkstofindawaytosavemillionsoflivesbyreducingtrafficaccidents,planecrashesandcircuitshots. Planargraphsareusedtofindsuchgraphicalrepresentationsof networkswithoutanycrossingorminimum numberofcrossings.Butthereisalwaysanuncertaintyanddegreeofindeterminacyindatawhichcanbe dealtusingbipolarneutrosophicgraphs.Wenowpresentapplicationsofbipolarneutrosophicgraphsinroad networks.
1.Roadnetworkmodeltomonitortraffic: Roadsareameanoffrequentandunacceptablenumber offatalitieseveryyear.Roadaccidentsareincreasingduetodensetraffic,negligenceofdriversandspeedof vehicles.Trafficaccidentscanbeminimizedbymodelingroadnetworks tomonitorthetraffic,applyquick emergencyservicesandtotakeactionagainstthespeedilygoingvehiclesquickly.Thepracticalapproachof bipolarneutrosophicplanargraphscanbeappliedtoconstructroadnetworks,asthesearethecombinationof verticesandedgesalongwiththedegreeoftruth,indeterminacyandfalsity.Themethodfortheconstruction ofroadnetworkisgiveninAlgorithm1.
Algorithm1
Table5:Bipolarneutrosophicsetofcities
A L1 L2 L3 L4 L5 L6
T p A 0 70 50 80 60 50 4
I p A 0.40.40.20.10.40.5
F p A 0.20.30.20.10.40.5
T n A 0.2 0.3 0.2 0.1 0.4 0.5
I n A 0.4 0.4 0.2 0.1 0.4 0.5
F n A 0 7 0 5 0 8 0 6 0 5 0 4
Table6:Bipolarneutrosophicsetofroads
A L1L3 L1L6 L2L3 L2L4 L3L5 L5L6 L2L5 L3L6 L4L6
T p B 0.40.40.50.50.50.40.50.40.4
I p B 0 20 40 20 10 20 40 40 20 1
F p B 0 20 50 30 10 40 40 30 50 5
T n B 0 2 0 2 0 3 0 1 0 2 0 4 0 3 0 2 0 1
I n B 0 4 0 4 0 2 0 1 0 2 0 4 0 4 0 2 0 1
L3 (0 8, 0 2, 0 2, 0 2, 0 2, 0 8)
(0 4 , 0 2 , 0 5 , 0 2 , 0 2 , 0 8) (0 4 , 0 2 , 0 5 , 0 2 , 0 2 , 0 8)
(0. 4, 0. 4, 0. 5, −0. 2, −0. 4, −0. 4) (0. 4, 0. 2, 0. 2, −0. 2, −04, −07) (0.5, 0.2, 0.4, −02, −0.2, −08) (0. 4, 0. 4, 0. 5, −0. 4, −0. 4, −0. 4) (0 . 5, 0. 2, 0. 3, −0. 2, −0. 2, −0. 8) (0.5, 01, 01, −0.1, −01, −0.6)
(0 . 5, 0. 4, 0. 3, −0. 3, −0. 4, −0. 5)
L6 (0 . 4, 0. 5, 0. 5, −0. 5, −0. 5, −0. 4)
L4 (0 6, 0 1, 0 1, 0 1, 0 1, 0 6)
L 5 (0 5 , 0 4 , 0 4 , 0 4 , 0 4 , 0 5)
F n B 0 7 0 4 0 8 0 6 0 8 0 4 0 5 0 8 0 6 L 1 (0 7 , 0 4 , 0 2 , 0 2 , 0 4 , 0 . 7) L 2 (0 5 , 0 . 4 , 0 3 , 0 3 , 0 4 , 0 . 5)
Figure3.1:Bipolarneutrosophicroadmodel
representtheindeterminacyandfalsityinthispercentage.Thepositivedegreeofmembershipsofeachedge xy indicatethepercentageoftruth,indeterminacyandfalsityofroadaccidentsthroughthisroad.Thenegative degreeofmembershipsof xy showthepercentageoftruth,indeterminacyandfalsitythatthe roadissafer. ThebipolarneutrosophicmodelofroadconnectionsbetweenthecitiesisshowninFig.3.1.Thisbipolar neutrosophicmodelcanbeusedtocheckandmonitorthepercentageofannualaccidents.Also,bymonitoring andtakingspecialsecurityactions,thetotalnumberofaccidentscanbeminimized.
2.Electricalconnections: Graphtheoryisextensivelyusedindesigningcircuitconnectionsand installation ofwiresinordertopreventcrossingwhichcancausedangerouselectricalhazards.Thetwistedandcrossing wiresareaserioussafetyrisktohumanlife.Thereisaneedtoinstall electricalwirestoreducecrossing.Bipolar neutrosophicplanargraphscanbeusedtomodelelectricalconnectionsandtostudythedegreeofdamagethat cancauseduetotheconnection.
E2 (0 8, 0 2, 0 2, 0 2, 0 2, 0 8)
Considertheproblemofsettingelectricalwiresbetween5electricalutilitiesandpowerplugs E1,E2,E3,E4,E5 inafactoryasshowninFig.3.2. E 1 (0 7 , 0 4 , 0 2 , 0 2 , 0 4 , 0 7)
(0 .4 ,0 .4 ,0 .5 , −0 .2 , −0 .4 , −0 .4)
(0. 4, 0. 2, 0. 2, −0. 2, −0. 4, −0. 7) (0.5, 0.2, 0.4, −0.2, −0.2, −0.8) (0. 4, 0. 4, 0. 5, −0. 4, −0. 4, −0. 4) (0 6 , 0 1 , 0 .2 , − 0 .1 , 0 1 , 0 8) (0 .6 ,0 .1 ,0 .2 , −0 .1 , −0 .1 , −0 . 7)
E 3 (0 5 , 0 4 , 0 4 , 0 4 , 0 4 , 0 5)
(0 5, 0 4, 0 4, 0 2, 0 4, 0 7)
E4(04,0 5,0.5, −0.5, −0.5, −0.4)
E5 (0 6, 0 1, 0 1, 0 1, 0 1, 0 6)
Figure3.2:Electricalconnections
Thepositivedegreeofmembership T p(Ei)ofeachvertex Ei representsthepercentageoffaultsandelectrical sparksofutilityorpowerplug Ei , I p(Ei)and F p(Ei)representtheindeterminacyandfalsityinthispercentage. Thenegativedegreeofmembership T n(Ei)representsthepercentagethat Ei isupdateandsafer, I n(x)and F n(x)representtheindeterminacyandfalsityinthispercentage.Thepositivedegreeofmembershipsof eachedge EiEj indicatethepercentageoftruth,indeterminacyandfalsityofelectricalhazardsthroughthis connection.Thenegativedegreeofmembershipsof EiEj showthepercentageoftruth,indeterminacyandfalsity thattheconnectionissafer.Thecrossingofwirescanbereduced ifwechangethegeometricalrepresentation ofFig.3.2.TheotherrepresentationisshowninFig.3.3whichhasonly onecrossing,atpoint P1,between theedges E1E4 and E2E5.Theelectricaldamageatcrossingpoint P1 canbereducedbyusingbetterelectrical wiresbetween E1 and E4, E2 and E5
(0 5, 0 4, 0 4, 0 2, 0 4, 0 7) P 1
E 1 (0 . 7 , 0 4 , 0 2 , 0 2 , 0 4 , 0 7) E2 (0 8, 0 2, 0 2, 0 2, 0 2, 0 8) E5 (0 6, 0 1, 0 1, 0 1, 0 1, 0 6)
(04, 02, 0. 2, −0. 2, −0. 4, −0. 7) (05, 02, 0.4, −0.2, −02, −0.8) (0. 4, 0. 4, 0. 5, −0. 4, −0. 4, −0. 4) (0 .6 , 0 .1 , 0 2 , − 0 1 , − 0 1 , − 0 . 8) (0 .6 ,0 .1 ,0 .2 , −0 .1 , −0 .1 , −0 . 7)
(0 .4 ,0 .4 ,0 .5 , −0 .2 , −0 .4 , −0 .4)
E 3 (0 5 , 0 4 , 0 4 , 0 4 , 0 4 , 0 5) E4(0.4,0.5,0 5, −0 5, −0 5, −04)
Figure3.3:Bipolarneutrosophicplanargraph
ThemethodfortheconstructionofbipolarneutrosophicplanargraphingiveninAlgorithm2. Algorithm2
n numberofutilities E1,E2,...,En and p numberofconnections e1,e2,...,ep
[1]N.Abdul-Jabbar,J.H.NaoomandE.H.Ouda, Fuzzydualgraph,JournalofAl-NahrainUniversity, 12(4)(2009),168-171.
[2]M.AkramandS.Shahzadi, Neutrosophicsoftgraphswithapplication,JournalofIntelligentandFuzzy Systems, 32 (2017),841-858.
[3]M.AkramandS.Shahzadi, Representationofgraphsusingintuitionisticneutrosophicsoftsets,Journalof MathematicalAnalysis, 7(6)(2016)31-53.
[4]M.AkramandG.Shahzadi, Operationsonsingle-valuedneutrosophicgraphs,JournalofUncertainSystem, 11(2017)1-26.
[5]M.Akram,S.SamantaandM.Pal, Applicationofbipolarfuzzysetsinplanargraphs,InternationalJournal ofAppliedandComputationalMathematics,2016,doi:10.1007/s40819-016-0132-4.
[6]N.AlshehriandM.Akram, Intuitionisticfuzzyplanargraphs,DiscreteDynamicsinNatureandSociety, 2014(2014),ArticleID397823,9pages.
[7]M.Akram, Single-valuedneutrosophicplanargraphs,InternationalJournalofAlgebraandStatistics, 5(2)2016),157-167.
[8]P.Bhattacharya, Someremarksonfuzzygraphs,PatternRecognitionLetter 6(1987),297-302.
[9]I.Deli,M.AliandF.Smarandache, Bipolarneutrosophicsetsandtheirapplicationbasedonmulti-criteria decisionmakingproblems,arXivpreprintarXiv:1504 02773, 2015.
[10]R.Dhavaseelan,R.Vikramaprasad,andV.Krishnaraj, Certaintypesofneutrosophicgraphs,IntJr.of MathematicalSciencesandApplications, 5(2)(2015),333 339.
[11]A.Kauffman, IntroductionalaTheoriedesSous-emsemblesFlous,MassonetCie,Vol.1,1973.
[12]J.N.MordesonandP.S.Nair, Fuzzygraphsandfuzzyhypergraphs,PhysicaVerlag,Heidelberg1998;Second Edition2001.
[13]F.Smarandache,AUnifyingfieldinlogicsneutrosophy:Neutrosophicprobability,setandlogic,Rehoboth: AmericanResearchPress,1998.
[14]A.Rosenfeld, Fuzzygraphs,FuzzySetsandtheirApplications(L.A.Zadeh,K.S.Fu,M.Shimura, Eds.), AcademicPress,NewYork(1975)77-95.
[15]A.Pal,S.SamantaandM.Pal, Conceptoffuzzyplanargraphs,ProceedingsofScienceandInformation Conference2013,October7-9,2013,London,UK,557-563,2013.
[16]S.Samanta,M.PalandA.Pal, Newconceptsoffuzzyplanargraph,InternationalJournalofAdvanced ResearchinArtificialIntelligence, 3(1)(2014)52-59.
[17]S.YeandJ.Ye, Dicesimilaritymeasurebetweensinglevaluedneutrosophicmultisetsanditsapplication inmedicaldiagnosis,NeutrosophicSetsandSystems, 6 (2014),48-52.
[18]L.A.Zadeh, Fuzzysets,InformationandControl, 8(1965)338-353.
[19]W.-R.Zhang, Bipolarfuzzysets,Proc.ofFUZZ-IEEE,1998,835-840.