Single-Valued Neutrosophic Planar Graphs

See discussions, stats, and author profiles for this publication at: https://www researchgate net/publication/311743641

Single-Valued Neutrosophic Planar Graphs

Article · December 2016

DOI: 10 20454/ijas 2016 1207

CITATIONS 8 READS 89

1 author:

Muhammad Akram University of the Punjab

231 PUBLICATIONS 1,483 CITATIONS

SEE PROFILE

Some of the authors of this publication are also working on these related projects:

Fuzzy grphs View project

Bipolar fuzzy set View project

All content following this page was uploaded by Muhammad Akram on 28 December 2016.

The user has requested enhancement of the downloaded file

InternationalJournalofAlgebraandStatistics

Volume5:2(2016),157-167

DOI:10.20454/ijas.2016.1207

PublishedbyModernSciencePublishers Availableat: http://www.m-sciences.com

Single-ValuedNeutrosophicPlanarGraphs

MuhammadAkrama

aDepartmentofMathematics,UniversityofthePunjab,NewCampus,Lahore,Pakistan (Received:15October2016;Accepted:15December2016)

Abstract.Weapplytheconceptofsingle-valuedneutrosophicsetstomultigraphs,planargraphsanddual graphs.Weintroducethenotionsofsingle-valuedneutrosophicmultigraphs,single-valuedneutrosophic planargraphs,andsingle-valuedneutrosophicdualgraphs.Weillustratetheseconceptswithexamples. Wealsoinvestigatesomeoftheirproperties.

1.Introduction

Zadeh[27]introducedtheconceptoffuzzyset.Attanassov[11]introducedtheintuitionisticfuzzysets whichisageneralizationoffuzzysets.Fuzzysettheoryand intuitionisticfuzzysettheoryareusefulmodels fordealingwithuncertaintyandincompleteinformation.Buttheymaynotbesufficientinmodeling ofindeterminateandinconsistentinformationencounteredinrealworld.Inordertocopewiththis issue,neutrosophicsettheorywasproposedbySmarandache [20]asageneralizationoffuzzysetsand intuitionisticfuzzysets.However,sinceneutrosophicsetsareidentifiedbythreefunctionscalledtruthmembership(T),indeterminacy-membership(I)andfalsity-membership(F)whosevaluesarerealstandard ornon-standardsubsetofunitinterval]0 , 1+[.

Therearesomedifficultiesinmodelingofsomeproblemsinengineeringandsciences.Toovercomethese difficulties,in2010,theconceptofsingle-valuedneutrosophicsetsanditsoperationsweredefinedbyWang etal.[22]asageneralizationofintuitionisticfuzzysets.Yang etal [23]introducedtheconceptofsinglevaluedneutrosophicrelationbasedonsingle-valuedneutrosophicset.Ye[25]introducedamulticriteria decisionmakingmethodusingaggregationoperators.

Rosenfeld[18]describedsomepropertiesoffuzzygraphs.Lateron,Bhattacharya[12]workedonfuzzy graphs.MordesonandNair[16]discussedsomeoperationsonfuzzygraphs.Abdul-jabbar etal. [1] introducedtheconceptofafuzzydualgraphanddiscussedsomeofitsinterestingproperties.Samanta andPal[17, 19]introducedandinvestigatedtheconceptoffuzzyplanargraphsandstudiedseveral properties.Onotherhand,AlshehriandAkram[10]introducedtheconceptofintuitionisticfuzzyplanar graphs.Akrametal.[5]discussedtheconceptofbipolarfuzzyplanargraphs.Dhavaseelanetal.[14] definedstrongneutrosophicgraphs.Broumietal.[13]proposedsingle-valuedneutrosophicgraphs. AkramandShahzadi[6]introducedthenotionsofneutrosophicgraphsandneutrosophicsoftgraphs. Theyalsopresentedapplicationofneutrosophicsoftgraphs.Representationofgraphsusingintuitionistic

2010 MathematicsSubjectClassification.03E72,05C72,05C78,05C99.

Keywords.Single-valuedneutrosophicgraphs,single-valuedneutrosophicmultigraphs,single-valuedneutrosophicplanargraphs, intuitionisticfuzzydualgraphs.

Emailaddress: m.akram@pucit.edu.pk (MuhammadAkram)

M.Akram / Int.J.ofAlgebraandStatistics5(2016),157-167 158

neutrosophicsoftsetswasdiscussedin[7].Akrametal.[8]introducedthenotionofsingle-valued neutrosophichypergraphs.Inthisarticle,weapplytheconceptofsingle-valuedneutrosophicsetsto multigraphs,planargraphsanddualgraphs.Weintroducethenotionsofsingle-valuedneutrosophic multigraphs,single-valuedneutrosophicplanargraphs,single-valuedneutrosophicdualgraphs.We presentsomeoftheirinterestingproperties.Thispaperis acontinuationofAlshehriandAkram’swork [10].

2.Preliminaries

Let X beanonemptyset.A fuzzyset [27] A drawnfrom X isdefinedas A = {< x : µA(x) >: x ∈ X},where µ : X → [0, 1]isthemembershipfunctionofthefuzzyset A.A fuzzybinaryrelation [28]on X isafuzzysubset µ on X × X.Byafuzzyrelation,wemeanafuzzybinaryrelationgivenby µ : X × X → [0, 1].A fuzzygraph [15] G = (V,σ,µ)isanon-emptyset V togetherwithapairoffunctions σ : V → [0, 1]and µ : V × V → [0, 1] suchthatforall x, y ∈ V, µ(x, y) ≤ min(σ(x),σ(y)),where σ(x)and µ(x, y)representthemembershipvalues ofthevertex x andoftheedge(x, y)in G,respectively.Aloopatavertex x inafuzzygraphisrepresented by µ(x, x) 0.Anedgeisnon-trivialif µ(x, y) 0.Let G beafuzzygraphandforacertaingeometric representation,thegraphhasonlyonecrossingbetweentwo fuzzyedges((w, x),µ(w, x))and((y, z),µ(y, z)). If µ(w, x) = 1and µ(y, z) = 0,thenwesaythatthefuzzygraphhasnocrossing.Similarly,if µ(w, x)hasvalue nearto1and µ(y, z)hasvaluenearto0,thecrossingwillnotbeimportantfortheplanarity.If µ(y, z)has valuenearto1and µ(w, x)hasvaluenearto1,thenthecrossingbecomesveryimportantfortheplanarity. Let X beanonemptyset.A fuzzymultiset [24] A drawnfrom X ischaracterizedbyafunction,‘count membership’of A denotedby CMA suchthat CMA : X → Q,where Q isthesetofallcrispmultisetsdrawn fromtheunitinterval[0, 1].Thenforany x ∈ X,thevalue CMA(x)isacrispmultisetdrawnfrom[0, 1]. Foreach x ∈ X,themembershipsequenceisdefinedasthedecreasinglyorderedsequenceofelementsin CMA(x).Itisdenotedby(µ1 A(x),µ2 A(x),µ3 A(x), ··· ,µp A(x))where µ1 A(x) ≥ µ2 A(x) ≥ µ3 A(x) ≥ ... ≥ µp A(x).

Let V beanon-emptysetand σ : V → [0, 1]beamappingandlet µ = {((x, y),µ(x, y)j), j = 1, 2,..., pxy|(x, y) ∈ V × V} beafuzzymulti-setof V × V suchthat µ(x, y)j ≤ min{σ(x),σ(y)} forall j = 1, 2,..., pxy,where pxy = max{ j|µ(x, y)j 0}.Then G = (V,σ,µ)isdenotedas fuzzymultigraph [17]where σ(x)and µ(x, y)j representthemembershipvalueofthevertex x andthemembershipvalueoftheedge(x, y)in G,respectively.

Definition2.1. [20]LetXbeaspaceofpoints(objects).AneutrosophicsetAinX ischaracterizedbyatruthmembershipfunctionTA(x),anindeterminacy-membershipfunctionIA(x) andafalsity-membershipfunctionFA(x) ThefunctionsTA(x),IA(x),andFA(x) arerealstandardornon-standardsubsetsof ]0 , 1+[ .Thatis,TA(x): X → ]0 , 1+[, IA(x): X → ]0 , 1+[ andFA(x): X → ]0 , 1+[ and 0 ≤ TA(x) + IA(x) + FA(x) ≤ 3+ .

Fromphilosophicalpointofview,theneutrosophicsettakesthevaluefromrealstandardornon-standard subsetsof]0 , 1+[.Inreallifeapplicationsinscientificandengineeringproblems,itisdifficulttouse neutrosophicsetwithvaluefromrealstandardornon-standardsubsetof]0 , 1+[.Toapplyneutrosophic setsinreal-lifeproblemsmoreconveniently,Wangetal.[22]definedsingle-valuedneutrosophicsetswhich takesthevaluefromthesubsetof[0, 1].

Definition2.2. [26]LetXbeanonemptysetwithgenericelementsinXdenotedbyx.AsinglevaluedneutrosophicmultisetAdrawnfromXischaracterizedbythethreefunctions:counttruth-membershipofCTA,count indeterminacy-membershipofCIA,andcountfalsity-membershipofCFA suchthatCTA(

forx ∈ Xandi = 1, 2,..., q.Forconvenience,asinglevaluedneutrosophicmultisetA canbedenotedbythesimplified form: A = { x, TA(x)i, IA(x)i, FA(x)i |x ∈ X, i = 1, 2,..., q}

Definition2.3. [26]LetA = { x, TA(x)i, IA(x)i, FA(x)i |x ∈ X, i = 1, 2, , q} andB = { x, TB(x)i, IB(x)i, FB(x)i |x ∈ X, i = 1, 2, ··· , q} betwosingle-valuedneutrosophicmultisetsinX.Then,therearethefollowingoperations:

(1) A ⊆ BifandonlyifTA(x)i ≤ TB(x)i,IA(x)i ≥ IB(x)i,FA(x)i ≥ FB(x)i fori = 1, 2, 3, , qandx ∈ X;

(2) A = BifandonlyifA ⊆ BandB ⊆ A, (3) Ac = { x, FA(x)i, 1 IA(x)i, TA(x)i |x ∈ X, i = 1, 2, , q}, (4) A B={ x, TA(x)i ∨ TB(x)i, IA(x)i ∧ IB(x)i, FA(x)i ∧ FB(x)i|x ∈ X, i = 1, 2, , q}, (5) A B={ x, TA(x)i ∧ TB(x)i, IA(x)i ∨ IB(x)i, FA(x)i ∨ FB(x)i|x ∈ X, i = 1, 2, , q}.

AkramandShahzadi[6]introducednotionofsingle-valuedneutrosophicsoftgraphs.AkramandShahzadi [9]highlightedsomeflawsinthedefinitionsofBroumietal.[13]andShah-Hussain[21].

Definition2.4. [6, 9]Asingle-valuedneutrosophicgraphisapairG = (A, B),whereA : V → [0, 1] issingle-valued neutrosophicsetinVandB : V × V → [0, 1] issingle-valuedneutrosophicrelationonVsuchthat

TB(xy) ≤ min{TA(x), TA(y)},

IB(xy) ≤ min{IA(x), IA(y)},

FB(xy) ≤ max{FA(x), FA(y)} forallx, y ∈ V.

3.Single-valuedNeutrosophicplanargraphs

Wefirstintroducethenotionofasingle-valuedneutrosophicmultigraphusingtheconceptofasinglevaluedneutrosophicmultiset.

Definition3.1. Let A = (TA, IA, FA)beasingle-valuedneutrosophicseton V andlet B = {(xy, TB(xy)i, IB(xy)i, FB(xy)i), i = 1, 2,..., m|xy ∈ V × V}

beasingle-valuedneutrosophicmultisetof V × V suchthat

TB(xy)i ≤ min{TA(x), TA(y)},

IB(xy)i ≤ min{IA(x), IA(y)}, FB(xy)i ≤ max{FA(x), FA(y)} forall i = 1, 2,..., m.Then G = (A, B)iscalled asingle-valuedneutrosophicmultigraph

Notethattheremaybemorethanoneedgebetweenthevertices x and y TB(xy)i, IB(xy)i, FB(xy)i represent truth-membershipvalue,indeterminacy-membershipvalue andfalsity-membershipvalueoftheedge xy in G,respectively. m denotesthenumberofedgesbetweenthevertices.Insingle-valuedneutrosophic multigraph G, B issaidtobesingle-valuedneutrosophicmultiedgeset.

Example3.2. ConsideramultigraphG∗ = (V, E) suchthatV = {a, b, c, d},E = {ab, ab, ab, bc, bd}.LetA = (TA, IA, FA) beasingle-valuedneutrosophicsetonVandB = (TB, IB, FB) beasingle-valuedneutrosophicmultiedge setonV × VdefinedinTable 1 andTable 2.

Table1:Single-valuedneutrosophicset A A abcd

TA 0 50 40 50 4 IA 0 30 20 40 3 FA 0.30.40.30.4

Table2:single-valuedneutrosophicmultiedgeset B B abababbcbd TB 0 20 10 20 30 1 IB 0 20 10 20 10 2 FB 0.200.20.30.2

(0 2, 0 2, 0 2) (0 1, 0 1, 0) (0.2, 0.2, 0.2)

(0 3, 0 1, 0 3) (0 .1 ,0 .2 ,0 .2)

Byroutinecalculations,itiseasytoseefromFig. 1 thatitisasingle-valuedneutrosophicmultigraph. a(0 5, 0 3, 0 3) b(0 . 4, 0. 2, 0. 4) c(0 5, 0 4, 0 3) d(0 4, 0 3, 0 4)

Figure1:Single-valuedneutrosophicmultigraph.

Definition3.3. LetB = {(xy, TB(xy)i, IB(xy)i, FB(xy)i), i = 1, 2,..., m|xy ∈ V × V} beasingle-valuedneutrosophic multiedgesetinsingle-valuedneutrosophicmultigraphG. Thedegreeofavertexx ∈ Visdenotedby deg(x) andis definedby deg(x) = ( m i=1 TB(xy)i, m i=1 IB(xy)i, m i=1 FB(xy)i) forally ∈ V.

Example3.4. InExample 3.2,thedegreeofverticesa, b, c, darede (a) = (0 5, 0 5, 0 4),de (b) = (0 9, 0 8, 0 9), de (c) = (0 3, 0 1, 0 3) andde (d) = (0 1, 0 2, 0 2)

Definition3.5. LetB = {(xy, TB(xy)i, IB(xy)i, FB(xy)i), i = 1, 2,..., m|xy ∈ V × V} beasingle-valuedneutrosophic multiedgesetinsingle-valuedneutrosophicmultigraphG. AmultiedgexyofGisstrongif 1 2 min{TA(x), TA(y)}≤ TB(xy)i, 1 2 min{IA(x), IA(y)}≤ IB(xy)i, 1 2 max{FA(x), FA(y)}≥ FB(xy)i, foralli = 1, 2,..., m. Definition3.6. LetB = {(xy, TB(xy)i, IB(xy)i, FB(xy)i), i = 1, 2,..., m|xy ∈ V × V} beasingle-valuedneutrosophic multiedgesetinsingle-valuedneutrosophicmultigraphG. Ansingle-valuedneutrosophicmultigraphGiscomplete if min{TA(x), TA(y)} = TB(xy)i, min{IA(x), IA(y)} = IB(xy)i, max{FA(x), FA(y)} = FB(xy)i foralli = 1, 2,..., mandforallx, y ∈ V.

Example3.7. Considerasingle-valuedneutrosophicmultigraphGasshowninFig. ??.Byroutinecalculations,it iseasytoseethatFig. ?? isasingle-valuednutrosophiccompletemultigraph.

a(0 4, 0 3, 0 2) b(0 . 5, 0. 4, 0. 3) c(0 4, 0 4, 0 3)

(0 4, 0 3, 0 3) (0 4, 0 4, 0 3) (0 4, 0 3, 0 3) (0 4, 0 2, 0 3)

Supposethatgeometricinsightforsingle-valuedneutrosophicgraphshasonlyonecrossingbetweensingle valuedneutrosophicedges(ab, TB(ab)i, IB(ab)i, FB(ab)i)and(cd, TB(cd)i, IB(cd)i, FB(cd)i).Wenotethat:

• If(ab, TB(ab)i, IB(ab)i, FB(ab)i) = (1, 1, 1)and(cd, TB(cd)i, IB(cd)i, FB(cd)i) = (0, 0, 0)or(ab, TB(ab)i, IB(ab)i, FB(ab)i) = (0, 0, 0),(cd, TB(cd)i, IB(cd)i, FB(cd)i) = (1, 1, 1),thensingle-valuedneutrosophicgraphhasno crossing,

• If(ab, TB(ab)i, IB(ab)i, FB(ab)i) = (1, 1, 1)and(cd, TB(cd)i, IB(cd)i, FB(cd)i) = (1, 1, 1),thenthereexistsa crossingfortherepresentationofthegraph.

Definition3.8. Thestrengthofthesingle-valuedneutrosophicedgeabcanbemeasuredbythevalue

Sab = ((ST)ab, (SI)ab, (SF)ab) = ( TB(ab)i min(TA(a), TA(b)) , IB(ab)i min(IA(a), IA(b)) , FB(ab)i max(FA(a), FA(b)) ).

Definition3.9. LetGbeasingle-valuedneutrosophicmultigraph.Anedgeab issaidtobeansingle-valued neutrosophicstrongif (ST)ab ≥ 0 5, (SI)ab ≥ 0 5, (SF)ab ≥ 0 5,otherwise,wecallweakedge.

Definition3.10. LetG = (A, B) beasingle-valuedneutrosophicmultigraphsuchthatBcontainstwoedges (ab, TB(ab)i, IB(ab)i, FB(ab)i) and (cd, TB(cd)j, IB(cd)j, FB(cd)j) ,intersectedatapointP,whereiandjarefixed integers.WedefinetheintersectingvalueatthepointPby

SP = ((ST )P, (SI)P, (SF )P) = ( (ST)ab + (ST)cd 2 , (SI)ab + (SI)cd 2 , (SF)ab + (SF)cd 2 )

Ifthenumberofpointofintersectionsinasingle-valuedneutrosophicmultigraphincreases,planarity decreases.Thusforsingle-valuedneutrosophicmultigraph, SP isinverselyproportionaltotheplanarity. Wenowintroducetheconceptofasingle-valuedneutrosophicplanargraph.

Definition3.11. LetGbeasingle-valuedneutrosophicmultigraphandP1, P2,..., Pz bethepointsofintersection betweentheedgesforacertaingeometricalrepresentation,Gissaidtobeasingle-valuedneutrosophicplanargraph withsingle-valuedneutrosophicplanarityvaluef = ( fT , fI, fF),where f = ( fT , fI , fF) = ( 1 1 + {(ST)P1 + (ST)P2 + ... + (ST)Pz } , 1 1

Ifthereisnopointofintersectionforacertaingeometricalrepresentationofasingle-valuedneutrosophic planargraph,thenitssingle-valuedneutrosophicplanarityvalueis(1, 1, 1).Inthiscase,theunderlying crispgraphofthissingle-valuedneutrosophicgraphisthe crispplanargraph.If fT and fI decreaseand fF increases,thenthenumberofpointsofintersectionbetweentheedgesincreasesanddecreases,respectively, andthenatureofplanaritydecreasesandincreases,respectively.Weconcludethateverysingle-valued neutrosophicgraphisasingle-valuedneutrosophicplanar graphwithcertainsingle-valuedneutrosophic planarityvalue.

Example3.12. ConsideramultigraphG∗ = (V, E) suchthatV = {a, b, c, d, e}, E = {ab, ac, ad, ad, bc, bd, cd, ce, ae, de, be}.

LetA = (TA, IA, FA) beasingle-valuedneutrosophicsetofVandletB = (TB, IB, FB) beasingle-valuedneutrosophic multiedgesetofV × VdefinedinTable 3 andTable 4.

Table3:single-valuedneutrosophicset A A abcde

TA 0.50.40.30.60.6 IA 0.50.40.30.60.6 FA 0.20.10.10.20.1

Table4:single-valuedneutrosophicmultiedgeset B B abacadadbcbdcdaecedebe TB 0.20.20.20.30.20.20.20.20.20.20.2 IB 0.20.20.20.30.20.20.20.20.20.20.2 FB 0.10.10.10.10.10.10.10.10.10.10.1

Thesingle-valuedneutrosophicmultigraphasshowninFig. 2 hastwopointofintersectionsP1 andP2.P1 isapoint betweentheedges (ad, 0 2, 0 2, 0 1) and (bc, 0 2, 0 2, 0 1) andP2 isbetween (ad, 0 3, 0 3, 0 1) and (bc, 0 2, 0 2, 0 1).For theedge (ad, 0 2, 0 2, 0 1),Sad = (0 4, 0 4, 0 5),Fortheedge (ad, 0 3, 0 3, 0 1),Sad = (0 6, 0 6, 0 5) andfortheedge (bc, 0 2, 0 2, 0 1),Sbc = (0 6667, 0 6667, 1)

ForthefirstpointofintersectionP1,intersectingvalue SP1 is (0 5334, 0 5334,, 0 75) andthatforthesecondpoint ofintersectionP2, SP2 = (0 63335, 0 63335,, 0 75).Therefore,thesingle-valuedneutrosophicplanarityvalueforthe single-valuedneutrosophicmultigraphshowninFig. 2 is (0 461, 0 461, 0 4)

(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) (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) P1 P2

a(0 5, 0 5, 0 2) b(0 4, 0 4, 0 1) d(0 6, 0 6, 0 2) c(0 3, 0 3, 0 1) e(0 6, 0 6, 0 1)

Figure2:Single-valuedneutrosophicplanargraph

Theorem3.13. LetGbeasingle-valuedneutrosophiccompletemultigraph. Theplanarityvalue,f = ( fT , fI , fF) of GisgivenbyfT = 1 1+np ,fI = 1 1+np andfF = 1 1+np suchthatfT + fI + fF ≤ 3,wherenp isthenumberofpointof intersectionsbetweentheedgesinG

Definition3.14. Ansingle-valuedneutrosophicplanargraphGiscalledstrongsingle-valuedneutrosophicplanar graphifthesingle-valuedneutrosophicplanarityvaluef = ( fT , fI , fF) ofthegraphisfT ≥ 0 5,fI ≥ 0 5,fF ≤ 0 5

Theorem3.15. LetGbeastrongsingle-valuedneutrosophicplanargraph.Thenumberofpointofintersections betweenstrongedgesinGisatmostone.

Proof. Let G beastrongsingle-valuedneutrosophicplanargraph.Assumethat G hasatleasttwopointof intersections P1 and P2 betweentwostrongedgesin G.Foranystrongedge(ab, TB(ab)i, IB(ab)i, FB(ab)i), TB(ab)i ≥ 1 2 min{TA(a), TA(b)}, IB(ab)i ≥ 1 2 min{IA(a), IA(b)}, FB(ab)i ≤ 1 2 max{FA(a), FA(b)}.

Thisshowsthat(ST )ab ≥ 0 5,(SI)ab ≥ 0 5,(SF)ab ≤ 0 5.Thusfortwointersectingstrongedges(ab, TB (ab)i, IB(ab)i, FB(ab)i) and(cd, TB(cd)j, IB(cd)j, FB(cd)j), (ST)ab + (ST)cd 2 ≥ 0 5, (SI)ab + (SI)cd 2 ≥ 0 5, (SF)ab + (SF)cd 2 ≤ 0 5, thatis,(ST)P1 ≥ 0 5,(SI)P1 ≥ 0 5,(SF)P1 ≤ 0 5.Similarly,(ST)P2 ≥ 0 5,(SI)P2 ≥ 0 5,(SF)P2 ≤ 0 5.Thisimplies that1 + (ST)P1 + (ST)P2 ≥ 2,1 + (SI)P1 + (SI)P2 ≥ 2,1 + (SF)P1 + (SF)P2 ≤ 2.Therefore, fT = 1 1+(ST )P1 +(ST )P2 ≤ 0 5, fI = 1 1+(SI )P1 +(SI )P2 ≤ 0 5, fF = 1 1+(SF )P1 +(SF)P2 ≥ 0 5.Itcontradictsthefactthatthesingle-valuedneutrosophic graphisastrongsingle-valuedneutrosophicplanargraph. Thusnumberofpointofintersectionsbetween strongedgescannotbetwo.Obviously,ifthenumberofpoint ofintersectionsofstrongsingle-valued neutrosophicedgesincreases,thesingle-valuedneutrosophicplanarityvaluedecreases.Similarly,ifthe numberofpointofintersectionofstrongedgesisone,thenthesingle-valuedneutrosophicplanarityvalue fT > 0 5, fI > 0 5, fI > 0 5.Anysingle-valuedneutrosophicplanargraphwithoutany crossingbetween edgesisastrongsingle-valuedneutrosophicplanargraph. Thus,weconcludethatthemaximumnumber ofpointofintersectionsbetweenthestrongedgesin G isone.

Faceofasingle-valuedneutrosophicplanargraphisanimportantparameter.Faceofasingle-valued neutrosophicgraphisaregionboundedbysingle-valuedneutrosophicedges.Everysingle-valuedneutrosophicfaceischaracterizedbysingle-valuedneutrosophicedgesinitsboundary.Ifalltheedgesin theboundaryofasingle-valuedneutrosophicfacehave T, I, F values(1, 1, 1)and(0, 0, 0),respectively,it becomescrispface.Ifoneofsuchedgesisremovedorhas T, I and F values(0, 0, 0)and(1, 1, 1),respectively,thesingle-valuedneutrosophicfacedoesnotexist.Sotheexistenceofasingle-valuedneutrosophic facedependsontheminimumvalueofstrengthofsingle-valuedneutrosophicedgesinitsboundary.A single-valuedneutrosophicfaceandits T, I and F valuesofasingle-valuedneutrosophicgrapharedefined below.

Definition3.16. LetGbeasingle-valuedneutrosophicplanargraphandB = {(xy, TB(xy)i, IB(xy)i, FB(xy)i), i = 1, 2,..., m| xy ∈ V × V}.Asingle-valuedneutrosophicfaceofGisaregion,bounded bythesetofsingle-valued neutrosophicedgesE′ ⊂ E,ofageometricrepresentationofG.Themembershipandnonmembershipvaluesofthe single-valuedneutrosophicfaceare:

min TB(xy)i min{TA(x), TA(y)} , i = 1, 2,..., m| xy ∈ E′ , min IB(xy)i min{IA(x), IA(y)} , i = 1, 2,..., m| xy ∈ E′ , max FB(xy)i max{FA(x), FA(y)} , i = 1, 2,..., m| xy ∈ E′ .

Definition3.17. Ansingle-valuedneutrosophicfaceiscalledstrongsingle-valuedneutrosophicfaceifitstruemembershipvalueisgreaterthan 0 5,indeterminacyvalueisgreaterthan 0 5,false-membershipvalueislesserthan 0.5,andweakfaceotherwise.Everysingle-valuedneutrosophicplanargraphhasaninfiniteregionwhichiscalled outersingle-valuedneutrosophicface.Otherfacesarecalledinnersingle-valuedneutrosophicfaces.

Example3.18. Considerasingle-valuedneutrosophicplanargraphasshowninFig. 3.Thesingle-valuedneutrosophicplanargraphhasthefollowingfaces:

• single-valuedneutrosophicfaceF1 isboundedbytheedges (v1v2, 0.5, 0.5, 0.1), (v2v3, 0.6, 0.6, 0.1), (v1v3, 0.5, 0.5, 0.1).

• outersingle-valuedneutrosophicfaceF2 surroundedbyedges

(v1v3, 0.5, 0.5, 0.1), (v1v4, 0.5, 0.5, 0.1), (v2v4, 0.6, 0.6, 0.1), (v2v3, 0.6, 0.6, 0.1),

• single-valuedneutrosophicfaceF3 isboundedbytheedges

(v1v2, 0.5, 0.5, 0.1), (v2v4, 0.6, 0.6, 0.1), (v1v4, 0.5, 0.5, 0.1).

Clearly,thetruth-membershipvalue,indeterminacy-membershipvalueandfalsity-membershipvalueofasinglevaluedneutrosophicfaceF1 are0.833,0.833and0.333,respectively.Thetruth-membershipvalue,indeterminacymembershipvalueandfalsity-membershipvalueofasingle-valuedneutrosophicfaceF3 arealso0.833,0.833and 0.333,respectively.ThusF1 andF3 arestrongsingle-valuedneutrosophicfaces.

F3

v1(0 6, 0 6, 0 3)

(0 5, 0 5, 0 1) (0 5, 0 5, 0 1)

F1 F2

v2(0 7, 0 7, 0 3)

(0 6 , 0 . 6 , 0 1)

(0 5, 0 5, 0 1) (0 . 6, 0. 6, 0. 1)

v3(0 8, 0 8, 0 1) v4(0 7, 0 7, 0 1)

Figure3:Facesinsingle-valuedneutrosophicplanargraph

Wenowintroducedualofsingle-valuedneutrosophicplanar graph.Insingle-valuedneutrosophicdual graph,verticesarecorrespondingtothestrongsingle-valuedneutrosophicfacesofthesingle-valued neutrosophicplanargraphandeachsingle-valuedneutrosophicedgebetweentwoverticesiscorresponding toeachedgeintheboundarybetweentwofacesofsingle-valuedneutrosophicplanargraph.Theformal definitionisgivenbelow.

Definition3.19. LetGbeasingle-valuedneutrosophicplanargraphandlet

LetF1, F2,..., Fk bethestrongsingle-valuedneutrosophicfacesofG.Thesingle-valuedneutrosophicdualgraphofG isasingle-valuedneutrosophicplanargraphG′ = (V′ , A′ , B′),whereV′ = {xi, i = 1, 2,..., k},andthevertexxi ofG′ isconsideredforthefaceFi ofG.Thetrue-truth-membership,indeterminacyandfalse-truth-membershipvaluesof verticesaregivenbythemappingA′ = (TA′ , IA′ , FA′ ): V′ → [0, 1] × [0, 1] × [0, 1] suchthat TA′ (xi) = max{TB′ (uv)i, i = 1, 2,..., p|uvisanedgeoftheboundaryofthestrongsingle-valuedneutrosophicfaceFi}, IA′ (xi) = max{IB′ (uv)i, i = 1, 2,..., p|uvisanedgeoftheboundaryofthestrongsingle-valuedneutrosophicfaceFi}, FA′ (xi) = min{FB′ (uv)i, i = 1, 2,..., p|uvisanedgeoftheboundaryofthestrongsingle-valuedneutrosophicfaceFi}

Theremayexistmorethanonecommonedgesbetweentwofaces Fi and Fj of G.Thustheremaybe morethanoneedgesbetweentwovertices xi and xj insingle-valuedneutrosophicdualgraph G′.Let Tl B(xixj)denotethetruth-membershipvalueofthe l-thedgebetween xi and xj,and Fl B(xixj)denotethe falsity-membershipvalueofthe l-thedgebetween xi and xj

Thetruth-membership,indeterminacy-membershipandfalsity-membershipvaluesofthesingle-valued neutrosophicedgesofthesingle-valuedneutrosophicdual grapharegivenby TB′ (xixj)l = Tl B(uv)j, IB′ (xixj)l = Il B(uv)j, FB′ (xixj)l = Fl B(uv)j where(uv)l isanedgeintheboundarybetweentwostrongsingle-valued neutrosophicfaces Fi and Fj and l = 1, 2,..., s,where s isthenumberofcommonedgesintheboundary between Fi and Fj orthenumberofedgesbetween xi and xj

Iftherebeanystrongpendantedgeinthesingle-valuedneutrosophicplanargraph,thentherewillbeaself loopin G′ correspondingtothispendantedge.Theedgetruth-membership,indeterminacy-membership andfalsity-membershipvalueoftheselfloopisequaltothe truth-membership,indeterminacy-membership andfalsity-membershipvalueofthependantedge.

Single-valuedneutrosophicdualgraphofsingle-valuedneutrosophicplanargraphdoesnotcontainpoint ofintersectionofedgesforacertainrepresentation,soit issingle-valuedneutrosophicplanargraphwith planarityvalue(1, 1, 1).Thusthesingle-valuedneutrosophicfaceofsingle-valuedneutrosophicdualgraph canbesimilarlydescribedasinsingle-valuedneutrosophicplanargraphs.

Example3.20. Considerasingle-valuedneutrosophicplanargraphG = (V, A, B) asshowninFig. 4 suchthat V = {a, b, c, d},A = (a, 0.6, 0.6, 0.2), (b, 0.7, 0.7, 0.2), (c, 0.8, 0.8, 0.2), (d, 0.9, 0.9, 0.1), and B = {(ab, 0 5, 0 5, 0 01), (ac, 0 4, 0 4, 0 01), (ad, 0 55, 0 55, 0 01), (bc, 0 45, 0 45, 0 01), (bc, 0 6, 0 6, 0 01), (cd, 0 7, 0 7, 0 01)} a b c d

x1 x2 x3 x4

Figure4:Single-valuedneutrosophicdualgraph

Thesingle-valuedneutrosophicplanargraphhasthefollowingfaces:

• single-valuedneutrosophicfaceF1 isboundedby (ab, 0 5, 0 5, 0 01), (ac, 0 4, 0 4, 0 01), (bc, 0 45, 0 45, 0 01),

• single-valuedneutrosophicfaceF2 isboundedby (ad, 0 55, 0 55, 0 01), (cd, 0 7, 0 7, 0 01), (ac, 0 4, 0 4, 0 01),

• single-valuedneutrosophicfaceF3 isboundedby (bc, 0.45, 0.45, 0.01), (bc, 0.6, 0.6, 0.01)and

• outersingle-valuedneutrosophicfaceF4 issurroundedby (ab, 0.5, 0.5, 0.01), (bc, 0.6, 0.6, 0.01), (cd, 0.7, 0.7, 0.01), (ad, 0.55, 0.55, 0.01).

Routinecalculationsshowthatallfacesarestrongsingle-valuedneutrosophicfaces.Foreachstrongsingle-valued neutrosophicface,weconsideravertexforthesingle-valuedneutrosophicdualgraph.SothevertexsetV′ = {x1, x2, x3, x4},wherethevertexxi istakencorrespondingtothestrongsingle-valuedneutrosophicfaceFi,i = 1, 2, 3, 4 Thus

TA′ (x1) = max{0.5, 0.4, 0.45} = 0.5, TA′ (x2) = max{0.55, 0.7, 0.4} = 0.7,

IA′ (x1) = max{0 5, 0 4, 0 45} = 0 5, IA′ (x2) = max{0 55, 0 7, 0 4} = 0 7,

FA′ (x1) = min{0.01, 0.01, 0.01} = 0.01, FA′ (x2) = min{0.01, 0.01, 0.01} = 0.01,

TA′ (x3) = max{0.45, 0.6} = 0.6, TA′ (x4) = max{0.5, 0.6, 0.7, 0.55} = 0.7,

IA′ (x3) = max{0.45, 0.6} = 0.6, IA′ (x4) = max{0.5, 0.6, 0.7, 0.55} = 0.7,

F′ A(x3) = min{0 01, 0 01} = 0 01, FA′ (x4) = min{0 01, 0 01, 0 01, 0 01} = 0 01

TherearetwocommonedgesadandcdbetweenthefacesF2 andF4 inG.Hencebetweentheverticesx2 andx4,there existtwoedgesinthesingle-valuedneutrosophicdualgraphofG.Truth-membership,indeterminacy-membership andfalsity-membershipvaluesoftheseedgesaregivenby

TB′ (x2x4) = TB(cd) = 0.7, TB′ (x2x4) = TB(ad) = 0.55, IB′ (x2x4) = IB(cd) = 0.7, IB′ (x2x4) = IB(ad) = 0.55,

FB′ (x2x4) = FB(cd) = 0 01, FB′ (x2x4) = FB(ad) = 0 01

Thetruth-membership,indeterminacy-membershipandfalsity-membershipvaluesofotheredgesofthesingle-valued neutrosophicdualgrapharecalculatedas

TB′ (x1x3) = TB(bc) = 0.45, TB′ (x1x2) = TB(ac) = 0.4, TB′ (x1x4) = TB(ab) = 0.5, TB′ (x3x4) = T′ B(bc) = 0.6, IB′ (x1x3) = IB(bc) = 0 45, IB′ (x1x2) = IB(ac) = 0 4, IB′ (x1x4) = IB(ab) = 0 5, IB′ (x3x4) = I′ B(bc) = 0 6, F′ B(x1x3) = TB(bc) = 0 01, FB′ (x1x2) = FB(ac) = 0 01, FB′ (x1x4) = FB(ab) = 0 01, F′ B(x3x4) = FB(bc) = 0 01

Thustheedgesetofsingle-valuedneutrosophicdualgraphis

B′ = {(x1x3, 0 45, 0 45, 0 01), (x1x2, 0 4, 0 4, 0 01), (x1x4, 0 5, 0 5, 0 01), (x3x4, 0.6, 0.6, 0.01), (x2x4, 0.7, 0.7, 0.01), (x2x4, 0.55, 0.55, 0.01)}.

InFig. 4,thesingle-valuedneutrosophicdualgraphG′ = (V′ , A′ , B′) ofGisdrawnbydottedline.

Weakedgesinplanargraphsarenotconsideredforanycalculationinsingle-valuedneutrosophicdual graphs.WestatethefollowingTheoremswithouttheirproofs.

Theorem3.21. LetGbeasingle-valuedneutrosophicplanargraphwhosenumberofvertices,numberofsinglevaluedneutrosophicedgesandnumberofstrongfacesaredenotedbyn,p,m,respectively.LetG′ bethesingle-valued neutrosophicdualgraphofG.Then:

(i)thenumberofverticesofG′ isequaltom, (ii)numberofedgesofG′ isequaltop,

(iii)numberofsingle-valuedneutrosophicfacesofG′ isequalton.

Theorem3.22. LetG = (V, A, B) beasingle-valuedneutrosophicplanargraphwithoutweakedgesandthesinglevaluedneutrosophicdualgraphofGbeG′ = (V′ , A′ , B′).Thetruth-membershipindeterminacy-membershipand falsity-membershipvaluesofsingle-valuedneutrosophic edgesofG′ areequaltotruth-membershipindeterminacymembershipandfalsity-membershipvaluesofthesingle-valuedneutrosophicedgesofG.

4.Conclusion

Asingle-valuedneutrosophicgraphisageneralizationofintuitionisticfuzzygraphthatisveryuseful tosolvereallifeproblems.Inthisresearcharticle,wehavepresentedsingle-valuedneutrosophicplanar graphs.Weareextendingourworkto(1)Bipolarneutrosophicplanargraphs,(2)Intuitionisticneutrosophic planargraphs,(3)Interval-valuedneutrosophicgraphs,and(4)single-valuedneutrosophicsoftplanar graphs.

5.Acknowledgment

Theauthorsarethankfultotherefereesfortheirvaluablecommentsandsuggestionsforimprovingthe qualityofourpaper.

References

[1]N.Abdul-Jabbar,J.H.NaoomandandE.H.Ouda,Fuzzydual graph,JournalofAl-NahrainUniversity,12(4)(2009)168-171.

[2]M.AkramandW.A.Dudek,Intuitionisticfuzzyhypergraphswithapplications,InformationSciences,218(2013)182-193.

[3]M.AkramandB.Davvaz,Strongintuitionisticfuzzygraphs,Filomat,26(2012)177-196.

[4]M.AkramandN.O.Al-Shehri,Intuitionisticfuzzycyclesandintuitionisticfuzzytrees,TheScientificWorldJournal,2014(2014) ArticleID305836,11pages.

[5]M.Akram,S.SamantaandM.Pal,Applicationofbipolarfuzzysetsinplanargraphs,InternationalJournalofApplied and ComputationalMathematics,2016,doi:10.1007/s40819-016-0132-4.

[6]M.AkramandS.Shahzadi,Neutrosophicsoftgraphswithapplication,JournalofIntelligentandFuzzySysytems,DOI: 10.3233/JIFS-16090,(2016)1-18.

[7]M.AkramandS.Shahzadi,Representationofgraphsusing intuitionisticneutrosophicsoftsets,JournalofMathematicalAnalysis, 7(6)(2016)1-23.

[8]M.Akram,S.ShahzadiandA.BorumandSaeid,Single-valuedneutrosophichypergraphs,TWMSJournalofAppliedand EngineeringMathematics,2016(Inpress).

[9]M.AkramandG.Shahzadi,Operationsonsingle-valuedneutrosophicgraphs,JournalofUncertainSystem,11(2017)1-26.

[10]N.AlshehriandM.Akram,Intuitionisticfuzzyplanargraphs,DiscreteDynamicsinNatureandSociety,2014(2014),ArticleID 397823,9pages.

[11]K.T.Atanassov,Intuitionisticfuzzysets, VIIITKR’sSession,DeposedinCentralforScience-TechnicalLibraryofBulgarianAcademyof Sciences,1697/84,1983,Sofia,Bulgaria,June,(Bulgarian).

[12]P.Bhattacharya,Someremarksonfuzzygraphs,Pattern RecognitionLetters,6(5))1987)297-302.

[13]S.Broumi,M.Talea,A.BakaliandF.Smarandache,Single-valuedneutrosophicgraphs,JournalofNewTheory,10(2016)86-101.

[14]R.Dhavaseelan,R.VikramaprasadandV.Krishnaraj,Certaintypesofneutrosophicgraphs,InternationalJournal ofMathematical SciencesandApplications,5(2)(2015)333-339.

[15]A.Kauffman, IntroductionalaTheoriedesSous-emsemblesFlous,MassonetCie,Vol.1,1973.

[16]J.M.MordesonandP.Chang-Shyh,Operationsonfuzzygraphs,InformationSciences,79(1994)59-170.

[17]A.Pal,S.SamantaandM.Pal,Conceptoffuzzyplanargraphs,ProceedingsofScienceandInformationConference2013,October 7-9,2013,London,UK,557-563,2013.

[18]A.Rosenfeld,Fuzzygraphs:FuzzySetsandTheirApplications,AcademicPress,NewYork,(1975)77-95.

[19]S.Samanta,M.PalandA.Pal,Newconceptsoffuzzyplanargraph,InternationalJournalofAdvancedResearchinArtificial Intelligence,3(1)(2014)52-59.

[20]F.Smarandache,AUnifyingfieldinlogicsneutrosophy: Neutrosophicprobability,setandlogic,Rehoboth:AmericanResearch Press,1998.

[21]N.ShahandA.Hussain,Neutrosophicsoftgraphs,NeutrosophicSetandSystems,11(2016)31-44.

[22]H.Wang,F.Smarandache,Y.Zhang,andR.Sunderraman,Single-valuedneutrosophicsets,MultispaceandMultistructure, 4(2010)410-413..

[23]H.-L.Yang,Z.-L.Guo,Y.SheandX.Liao,Onsinglevaluedneutrosophicrelations,JournalofIntelligentFuzzySystems, 30(2016)1045-1056.

[24]R.R.Yager,Onthetheoryofbags,Int.J.GeneralSystems,13(1986)23-37.

[25]J.Ye,Amulticriteriadecision-makingmethodusingaggregationoperatorsforsimplifiedneutrosophicsets,JournalofIntelligent &FuzzySystems,26(5)(2014)2459-2466.

[26]S.YeandJ.Ye,Dicesimilaritymeasurebetweensinglevaluedneutrosophicmultisetsanditsapplicationinmedicaldiagnosis, NeutrosophicSetsandSystems,6(2014)48-52.

[27]L.A.Zadeh,Fuzzysets,InformationandControl,8(1965)338-353.

[28]L.A.Zadeh,Similarityrelationsandfuzzyorderings, InformationSciences,3(2)(1971)177-200.