Certain Competition Graphs Based on Intuitionistic Neutrosophic Environment

Article

CertainCompetitionGraphsBasedonIntuitionistic NeutrosophicEnvironment

MuhammadAkram* ID andMaryamNasir

DepartmentofMathematics,UniversityofthePunjab,NewCampus,Lahore54590,Pakistan; maryamnasir912@gmail.com

* Correspondence:m.akram@pucit.edu.pk;Tel.:+92-42-99231241

Received:7September2017;Accepted:19October2017;Published:24October2017

Abstract: Theconceptofintuitionisticneutrosophicsetsprovidesanadditionalpossibilitytorepresent imprecise,uncertain,inconsistentandincompleteinformation,whichexistsinrealsituations.Thisresearch articlefirstpresentsthenotionofintuitionisticneutrosophiccompetitiongraphs.Then, p-competition intuitionisticneutrosophicgraphsand m-stepintuitionisticneutrosophiccompetitiongraphsarediscussed. Further,applicationsofintuitionisticneutrosophiccompetitiongraphsinecosystemandcareercompetition aredescribed.

Keywords: intuitionisticneutrosophiccompetitiongraphs;intuitionisticneutrosophicopen-neighborhood graphs; p-competitionintuitionisticneutrosophicgraphs; m-stepintuitionisticneutrosophiccompetition graphs

MSC: 03E72;68R10;68R05

1.Introduction

Euler[1]introducedtheconceptofgraphtheoryin 1736,whichhasapplicationsinvariousfields, includingimagecapturing,datamining,clusteringandcomputerscience[2–5].Agraphisalsousedto developaninterconnectionbetweenobjectsinaknownsetofobjects.Everyobjectcanbeillustratedby avertex,andinterconnectionbetweenthemcanbeillustratedbyanedge.Thenotionofcompetition graphswasdevelopedbyCohen[6]in 1968,dependingonaprobleminecology.Thecompetition graphshavemanyutilizationsinsolvingdailylifeproblems,includingchannelassignment,modelingof complexeconomic,phytogenetictreereconstruction,codingandenergysystems.

Fuzzysettheoryandintuitionisticfuzzysetstheoryareusefulmodelsfordealingwithuncertainty andincompleteinformation.However,theymaynotbesufficientinmodelingofindeterminateand inconsistentinformationencounteredintherealworld.Inordertocopewiththisissue,neutrosophic settheorywasproposedbySmarandache[7]asageneralizationoffuzzysetsandintuitionisticfuzzy sets.However,sinceneutrosophicsetsareidentifiedbythreefunctionscalledtruth-membership (t), indeterminacy-membership (i) andfalsity-membership ( f ),whosevaluesaretherealstandardor non-standardsubsetofunitinterval ]0 ,1+ [.Therearesomedifficultiesinmodelingofsomeproblems inengineeringandsciences.Toovercomethesedifficulties,Smarandachein1998[8]andWangetal.[9] in2010definedtheconceptofsingle-valuedneutrosophicsetsandtheiroperationsasageneralizationof intuitionisticfuzzysets.Yangetal.[10]introducedtheconceptofthesingle-valuedneutrosophicrelation basedonthesingle-valuedneutrosophicset.Theyalsodevelopedkernelsandclosuresofasingle-valued neutrosophicset.Theconceptofthesingle-valuedintuitionisticneutrosophicsetwasproposedby BhowmikandPal[11,12].

Information 2017, 8,132;doi:10.3390/info8040132

www.mdpi.com/journal/information

Thevaluablecontributionoffuzzygraphandgeneralizedstructureshasbeenstudiedbyseveral researchers[13–22].Smarandache[23]proposedthenotionoftheneutrosophicgraphandseparatedthem intofourmaincategories.Wu[24]discussedfuzzydigraphs.Fuzzy m-competitionand p-competition graphswereintroducedbySamantaandPal[25].Samantaetal.[26]introduced m-stepfuzzycompetition graphs.Dhavaseelanetal.[27]definedstrongneutrosophicgraphs.AkramandShahzadi[28]introduced thenotionofasingle-valuedneutrosophicgraphandstudiedsomeofitsoperations.Theyalsodiscussed thepropertiesofsingle-valuedneutrosophicgraphsbylevelgraphs.AkramandShahzadi[29]introduced theconceptofneutrosophicsoftgraphswithapplications.Broumietal.[30]proposedsingle-valued neutrosophicgraphsanddiscussedsomeproperties.Ye[31–33]haspresentedseveralnovelconcepts ofneutrosophicsetswithapplications.Inthispaper,wefirstintroducetheconceptofintuitionistic neutrosophiccompetitiongraphs.Wethendiscuss m-stepintuitionisticneutrosophiccompetitiongraphs. Further,wedescribeapplicationsofintuitionisticneutrosophiccompetitiongraphsinecosystemand careercompetition.Finally,wepresentourdevelopedmethodsbyalgorithms.

Ourpaperisdividedintothefollowingsections:InSection 2,weintroducecertaincompetition graphsusingtheintuitionisticneutrosophicenvironment.InSection 3,wepresentapplicationsof intuitionisticneutrosophiccompetitiongraphsinecosystemandcareercompetition.Finally,Section 4 providesconclusionsandfutureresearchdirections.

2.IntuitionisticNeutrosophicCompetitionGraphs

Wehaveusedstandarddefinitionsandterminologiesinthispaper.Forothernotations,terminologies andapplicationsnotmentionedinthepaper,thereadersarereferredto[34–44].

Definition1. [38]Let X beafixedset.Ageneralizedintuitionisticfuzzyset I of X isanobjecthavingthe form I={(u, µ I (u), νI (u))|u ∈ U},wherethefunctions µ I (u) :→ [0,1] and νI (u) :→ [0,1] definethedegreeof membershipanddegreeofnon-membershipofanelementu ∈ X,respectively,suchthat:

min{µ I (u), νI (u)}≤ 0.5,forallu ∈ X.

Thisconditioniscalledthegeneralizedintuitionisticcondition.

Definition2. [11]Anintuitionisticneutrosophicset(IN-set)isdefinedas ˘ A =(w,tA(w),iA(w),fA(w)),where: t ˘ A (w) ∧ f ˘ A (w) ≤ 0.5, tA (w) ∧ iA (w) ≤ 0.5, i ˘ A (w) ∧ f ˘ A (w) ≤ 0.5, forall,w ∈ X,suchthat: 0 ≤ t ˘ A (w)+ i ˘ A (w)+ f ˘ A (w) ≤ 2.

Definition3. [12]Anintuitionisticneutrosophicrelation(IN-relation)isdefinedasanintuitionisticneutrosophic subsetofX × Y,whichhasoftheform: R = {((w,z),tR (w,z),iR (w,z),fR (w,z)) : w ∈ X,z ∈ Y}, wheretR,iR andfR areintuitionisticneutrosophicsubsetsofX × Ysatisfyingtheconditions: 1. oneofthesetR (w,z),iR (w,z) andfR (w,z) isgreaterthanorequalto 0.5, 2. 0 ≤ tR (w, z)+ iR (w, z)+ fR (w, z) ≤ 2

Definition4. Anintuitionisticneutrosophicgraph(IN-graph) G =(X, h, k) (inshort G)on X (vertexset)is atripletsuchthat:

1. tk (w, z) ≤ th (w) ∧ th (z),ik (w, z) ≤ ih (w) ∧ ih (z),fk (w, z) ≤ fh (w) ∨ fh (z), 2. tk (w, z) ∧ ik (w, z) ≤ 0.5,tk (w, z) ∧ fk (w, z) ≤ 0.5,ik (w, z) ∧ fk (w, z) ≤ 0.5, 3. 0 ≤ tk (w, z)+ ik (w, z)+ fk (w, z) ≤ 2,forallw, z ∈ X, where, th,ih andfh → [0,1]

denotethetruth-membership,indeterminacy-membershipandfalsity-membershipofanelementw ∈ Xand: tk,ik andfk → [0,1]

denotethetruth-membership,indeterminacy-membershipandfalsity-membershipofanelement (w, z) ∈ E (edgeset).

Wenowillustratethiswithanexample.

Example1. ConsiderIN-graph G onnon-emptysetX,asshowninFigure 1. a(0 1, 0 4, 0 5)

(0. 1, 0. 2, 0. 4) (0 .5 , 0 .2 , 0 .1) (0.1, 0.2, 0.3) (0 . 5, 0 . 2, 0 . 2) (0 5, 0 2, 0 1)

b(0 6, 0 3, 0 2) c(0 8, 0 3, 0 4) d(0 7, 0 4, 0 2)

Figure1. Intuitionisticneutrosophicgraph(IN-graph).

Definition5. Let −→ G beanintuitionisticneutrosophicdigraph(IN-digraph),thenintuitionisticneutrosophic out-neighborhoods(IN-out-neighborhoods)ofavertexwareanIN-set: N+ (w)=(X+ w ,t+ w ,i+ w ,f + w ), where, X+ w = {z|k1 −−−→ (w, z) > 0,k2 −−−→ (w, z) > 0,k3 −−−→ (w, z) > 0}, suchthat t+ w : X+ w → [0,1] definedby t+ w (z)= k1 −−−→ (w, z), i+ w : X+ w → [0,1] definedby i+ w (z)= k2 −−−→ (w, z) and f + z : X+ z → [0,1] definedbyf + w (z)= k3 −−−→ (w, z)

Definition6. Let −→ G beanIN-digraph,thentheintuitionisticneutrosophicin-neighborhoods(IN-in-neighborhoods) ofavertexwareanIN-set:

N (w)=(Xw ,tw ,iw ,fw ), where, Xw = {z|k1 −−−→ (z, w) > 0,k2 −−−→ (z, w) > 0,k3 −−−→ (z, w) > 0}, suchthat tw : Xw → [0,1] definedby tw (z)= k1 −−−→ (z, w), iw : Xw → [0,1] definedby iw (z)= k2 −−−→ (z, w) and fw : Xw → [0,1] definedbyfw (z)= k3 −−−→ (z, w)

2017, 8,132

Example2. Consider −→ G =(X, h, k) tobeanIN-digraph,suchthat, X = {a, b, c, d, e}, h = {(a, 0.5, 0.3, 0.1), (b, 0.6, 0.4, 0.2), (c, 0.8, 0.3, 0.1), (d, 0.1, 0.9, 0.4), (e, 0.4, 0.3, 0.6)} andk = {(−→ ab, 0.3, 0.3, 0.1), (−→ ae, 0.3, 0.2, 0.4), (−→ bc, 0.5, 0.2, 0.1), (−→ ed, 0.1, 0.2, 0.5), (−→ dc, 0.1, 0.2, 0.3), (−→ bd, 0.1, 0.3, 0.3)},asshowninFigure 2

(0 . 3 , 0 . 2 , 0 4) (0 3, 0 3, 0 1) (0 .5 , 0.2 , 0.1) (0 . 1 , 0 . 3 , 0 . 3) (0 1, 0 2, 0 5) (0. 1,0. 2,0. 3)

a(0 5, 0.3, 0 1) b(0 6, 0 4, 0 2) c(0 8, 0 3, 0 1) d(0 1, 0 9, 0 4) e(0 4, 0 3, 0 6)

Figure2. IN-digraph. Then, N+ (a)= {(b, 0.3, 0.3, 0.1), (e, 0.3, 0.2, 0.4)}, N+ (c)= ∅, N+ (d)= {(c, 0.1, 0.2, 0.3)}, and N (b)= {(a, 0.3, 0.3, 0.1)}, N (c)= {(b, 0.5, 0.2, 0.1), (d, 0.1, 0.2, 0.3)}.Similarly,wecancalculate IN-outandin-neighborhoodsoftheremainingvertices.

Definition7. TheheightofanIN-set ˘ A =(w,tA,iA,f A ) isdefinedas: H( ˘ A)=(sup w∈X t ˘ A (w), sup w∈X i ˘ A (w), inf w∈X f ˘ A (w))=(H1( ˘ A),H2( ˘ A),H3( ˘ A))

Forexample,theheightofanIN-set ˘ A = {(a, 0.5, 0.7, 0.2), (b, 0.1, 0.2, 1), (c, 0.3, 0.5, 0.3)} in X = {a,b,c} isH( ˘ A)=(0.5, 0.7, 0.2).

Definition8. Anintuitionisticneutrosophiccompetitiongraph(INC-graph) C(−→ G ) ofanIN-digraph −→ G =(X,h,k) isanundirectedIN-graph G =(X, h, k),whichhasthesameintuitionisticneutrosophicsetofverticesasin −→ G and hasanintuitionisticneutrosophicedgebetweentwovertices w, z ∈ X in C(−→ G ) ifandonlyif N+(w) ∩ N+(z) is anon-emptyIN-setin −→ G .Thetruth-membership,indeterminacy-membershipandfalsity-membershipvaluesofedge (w,z) in C(−→ G ) are: tk (w, z)=(th (w) ∧ th (z))H(N+ (w) ∩ N+ (z)), ik (w, z)=(ih (w) ∧ ih (z))H(N+ (w) ∩ N+ (z)), fk (w, z)=( fh (w) ∨ fh (z))H(N+ (w) ∩ N+ (z)), respectively.

Example3. Consider −→ G =(X, h, k) tobeanIN-digraph,suchthat, X = {a, b, c, d}, h = {(a, 0.1, 0.4, 0.5), (b, 0.6, 0.3, 0.2), (c, 0.8, 0.3, 0.4), (d, 0.7, 0.4, 0.2)} and k = {(−→ ab, 0.1, 0.2, 0.4), (−→ ac , 0.1, 0.2, 0.3), (−→ bc , 0.5, 0.2, 0.2), (−→ bd, 0.5, 0.2, 0.1), (−→ cd, 0.5, 0.2, 0.1)},asshowninFigure 3.

a(0 1, 0 4, 0 5)

(0. 1, 0. 2, 0. 4) (0 .5 , 0 .2 , 0 .1) (0 1, 0 2, 0 3) (0 5, 0 . 2, 0 . 2) (0 5, 0 2, 0 1)

b(0 6, 0 3, 0 2) c(0 8, 0 3, 0 4) d(0 7, 0 4, 0 2)

Figure3. IN-digraph.

Bydirectcalculations,wehaveTables 1 and 2 representingIN-outandin-neighborhoods,respectively.

Table1. IN-out-neighborhoods. w N+ (w) a{(b,0.1,0.2,0.4),(c,0.1,0.2,0.3)} b{(d,0.5,0.2,0.1)} c{(b,0.5,0.2,0.2),(d,0.5,0.2,0.1)} d ∅

Table2. IN-in-neighborhoods. w N (w) a ∅ b{(a,0.1,0.2,0.4),(c,0.1,0.2,0.3)} c{(a,0.1,0.2,0.3)} d{(b,0.5,0.2,0.1),(c,0.5,0.2,0.1)}

TheINC-graphofFigure 3 isshowninFigure 4

a(0 1, 0 4, 0 5)

(0 3, 0 06, 0 16) (0 01, 0 06, 0 2)

b(0 6, 0 3, 0 2) c(0 8, 0 3, 0 4) d(0.7, 0.4, 0.2)

Figure4. Intuitionisticneutrosophiccompetitiongraph(INC-graph). Therefore,thereisanedgebetweentwoverticesinINC-graph C(−→ G ),whosetruth-membership, indeterminacy-membershipandfalsity-membershipvaluesaregivenbytheaboveformula.

Definition9. ForanIN-graph G =(X, h, k),where h =(h1, h2, h3) and k =(k1, k2, k3),thenanedge (w, z), w,z ∈ Xiscalledindependentstrongif: 1 2 [h1(w) ∧ h1(z)] < k1(w, z), 1 2 [h2(w) ∧ h2(z)] > k2(w, z), 1 2 [h3(w) ∨ h3(z)] > k3(w, z)

Otherwise,itiscalledweak.

Theorem1. Suppose −→ G isanIN-digraph.If N+ (w) ∩ N+ (z) containsonlyoneelementof −→ G ,thentheedge (w,z) of C(−→ G ) isindependentstrongifandonlyif:

|[N+ (w) ∩ N+ (z)]|t > 0.5, |[N+ (w) ∩ N+ (z)]|i < 0.5, |[N+ (w) ∩ N+ (z)]| f < 0.5.

Proof. Suppose, −→ G isanIN-digraph.Suppose N+ (w) ∩ N+ (z)=(a, ˘ p, q, r),where ˘ p, q and r arethe truth-membership,indeterminacy-membershipandfalsity-membershipvaluesofeithertheedge (w, a) ortheedge (z, a),respectively.Here,

|[N+ (w) ∩ N+ (z)]|t = ˘ p = H1(N+ (w) ∩ N+ (z)), |[N+ (w) ∩ N+ (z)]|i = q = H2(N+ (w) ∩ N+ (z)), |[N+ (w) ∩ N+ (z)]| f = r = H3(N+ (w) ∩ N+ (z))

Then, k1(w, z)= ˘ p × [h1(w) ∧ h1(z)], k2(w, z)= q × [h2(w) ∧ h2(z)], k3(w, z)= r × [h3(w) ∨ h3(z)]

Therefore,theedge (w, z) in C(−→ G ) isindependentstrongifandonlyif ˘ p > 0.5, q < 0.5 and r < 0.5 Hence,theedge (w, z) of C(−→ G ) isindependentstrongifandonlyif:

|[N+ (w) ∩ N+ (z)]|t > 0.5, |[N+ (w) ∩ N+ (z)]|i < 0.5, |[N+ (w) ∩ N+ (z)]| f < 0.5.

WeillustratethetheoremwithanexampleasshowninFigure 5.

a(0 7, 0 5, 0 4)

b(0 8, 0 4, 0 5)

a(0 7, 0 5, 0 4)

b(0 8, 0 4, 0 5)

c(0 8, 0 5, 0 4) d(0 3, 0 4, 0 5)

(0 6, 0 4, 0 4) (0 . 7 , 0 3 , 0 3) (0 2, 0 3, 0 4) (0 . 2 , 0 3 , 0 . 3) (b) (a)

(0 .42 , 0 .15 , 0 .12)

d(0 3, 0 4, 0 5) c(0 8, 0 5, 0 4)

Figure5. INC-graph.(a)IN-digraph;(b)correspondingINC-graph. Theorem2. IfalltheedgesofanIN-digraph −→ G areindependentstrong,then:

k1(w, z) (h1(w) ∧ h1(z))2 > 0.5, k2(w, z) (h2(w) ∧ h2(z))2 < 0.5, k3(w, z) (h3(w) ∨ f3(z))2 < 0.5 foralledges (w,z) in C(−→ G )

Proof. SupposealltheedgesofIN-digraph −→ G areindependentstrong.Then:

1 2 [h1(w) ∧ h1(z)] < k1 −−−→ (w, z), 1 2 [h2(w) ∧ h2(z)] > k2 −−−→ (w, z), 1 2 [h3(w) ∨ h3(z)] > k3 −−−→ (w, z),

foralltheedges (w, z) in −→ G .LetthecorrespondingINC-graphbe C(−→ G ).

Case(1):When N+ (w) ∩ N+ (z)= ∅ forall w, z ∈ X,thentheredoesnotexistanyedgein C(−→ G ) between w and z.Thus,wehavenothingtoproveinthiscase.

Case(2):When N+(w) ∩ N+(z) = ∅,let N+(w) ∩ N+(z)= {(a1, m1, n1, ˘ p1), (a2, m2, n2, ˘ p2),..., (al, ml, nl, ˘ pl )},where mi, ni and ˘ pi arethetruth-membership,indeterminacy-membershipand falsity-membershipvaluesofeither −−−→ (w, ai) or −−−→ (z, ai) for i = 1,2,..., l,respectively.Therefore, mi =[k1 −−−→ (w, ai ) ∧ k1 −−−→ (z, ai )], ni =[k2 −−−→ (w, ai ) ∧ k2 −−−→ (z, ai )], ˘ pi =[k3 −−−→ (w, ai ) ∨ k3 −−−→ (z, ai )], fori = 1,2,..., l

Suppose, H1(N+ (w) ∩ N+ (z))= max{mi, i = 1,2,..., l} = mmax, H2(N+ (w) ∩ N+ (z))= max{ni, i = 1,2,..., l} = nmax, H3(N+ (w) ∩ N+ (z))= min{ ˘ pi, i = 1,2,..., l} = ˘ pmin

Obviously, mmax > k1 −−−→ (w, z) and nmax < k2 −−−→ (w, z) and ˘ pmin < k3 −−−→ (w, z) foralledges −−−→ (w, z) showthat:

mmax h1(w) ∧ h1(z) > k1 −−−→ (w, z) h1(w) ∧ h1(z) > 0.5, nmax h2(w) ∧ h2(z) < k2 −−−→ (w, z) h2(w) ∧ h2(z) < 0.5, pmin h3(w) ∨ h3(z) < k3 −−−→ (w, z) h3(w) ∧ h3(z) < 0.5, therefore,

k1(w, z)=(h1(w) ∧ h1(z))H1(N+ (w) ∩ N+ (z)), k1(w, z)=[h1(w) ∧ h1(z)] × mmax, k1(w, z) (h1(w) ∧ h1(z)) = mmax, k1(w, z) (h1(w) ∧ h1(z))2 = mmax (h1(w) ∧ h1(z)) > 0.5,

k2(w, z)=(h2(w) ∧ h2(z))H2(N+ (w) ∩ N+ (z)), k2(w, z)=[h2(w) ∧ h2(z)] × nmax, k2(w, z) (h2(w) ∧ h2(z)) = nmax, k2(w, z) (h2(w) ∧ h2(z))2 = nmax (h2(w) ∧ h2(z)) < 0.5, and: k3(w, z)=(h3(w) ∨ h3(z))H3(N+ (w) ∩ N+ (z)), k3(w, z)=[h3(w) ∨ h3(z)] × ˘ pmin, k3(w, z) (h3(w) ∨ h3(z)) = ˘ pmin, k3(w, z) (h3(w) ∨ h3(z))2 = ˘ pmin (h3(w) ∨ h3(z)) < 0.5. Hence, k1(w,z) (h1(w)∧h1(z))2 > 0.5, k2(w,z) (h2(w)∧h2(z))2 < 0.5,and k3(w,z) (h3(w)∨h3(z))2 < 0.5 foralledges (w, z) in C(−→ G ) Theorem3. Let C(−→

2. th1 h2 = th1 (w1) ∧ th2 (w2),ih1 h2 = ih1 (w1) ∧ ih2 (w2),fh1 h2 = fh1 (w1) ∨ fh2 (w2). 3. tk ((w1, w2)(w1, z2))=[th1 (w1) ∧ th2 (w2) ∧ th2 (z2)] ×∨x2 {th1 (w1) ∧ t−→ l2 (w2 x2) ∧ t−→ l2 (z2 x2)}, (w1, w2)(w1, z2) ∈ EC(−→ G1)∗ EC(−→ G2)∗ , x2 ∈ (N+ (w2) ∩ N+ (z2))∗

4. ik ((w1, w2)(w1, z2))=[ih1 (w1) ∧ ih2 (w2) ∧ ih2 (z2)] ×∨x2 {ih1 (w1) ∧ i−→ l2 (w2 x2) ∧ i−→ l2 (z2 x2)}, (w1, w2)(w1, z2) ∈ EC(−→ G1)∗ EC(−→ G2)∗ , x2 ∈ (N+ (w2) ∩ N+ (z2))∗

5. fk ((w1, w2)(w1, z2))=[ fh1 (w1) ∨ fh2 (w2) ∨ fh2 (z2)] ×∨x2 { fh1 (w1) ∨ f−→ l2 (w2 x2) ∨ f−→ l2 (z2 x2)}, (w1, w2)(w1, z2) ∈ EC(−→ G1)∗ EC(−→ G2)∗ , x2 ∈ (N+ (w2) ∩ N+ (z2))∗ .

6. tk ((w1, w2)(z1, w2))=[th1 (w1) ∧ th1 (z1) ∧ th2 (w2)] ×∨x1 {th2 (w2) ∧ t−→ l1 (w1 x1) ∧ t−→ l1 (z1 x1)}, (w1, w2)(z1, w2) ∈ EC(−→ G1)∗ EC(−→ G2)∗ , x1 ∈ (N+ (w1) ∩ N+ (z1))∗ .

7. ik ((w1, w2)(z1, w2))=[ih1 (w1) ∧ ih1 (z1) ∧ ih2 (w2)] ×∨x1 {ih2 (w2) ∧ i−→ l1 (w1 x1) ∧ i−→ l1 (z1 x1)}, (w1, w2)(z1, w2) ∈ EC(−→ G1)∗ EC(−→ G2)∗ , x1 ∈ (N+ (w1) ∩ N+ (z1))∗ .

8. fk ((w1, w2)(z1, w2))=[ fh1 (w1) ∨ fh1 (z1) ∨ fh2 (w2)] ×∨x1 { fh2 (w2) ∨ f−→ l1 (w1 x1) ∨ t−→ l1 (z1 x1)}, (w1, w2)(z1, w2) ∈ EC(−→ G1)∗ EC(−→ G2)∗ , x1 ∈ (N+ (w1) ∩ N+ (z1))∗ .

9. tk ((w1, w2)(z1, z2))=[th1 (w1) ∧ th1 (z1) ∧ th2 (w2) ∧ th2 (z2)] × [th1 (w1) ∧ t−→ l1 (z1w1) ∧ th2 (z2) ∧ t−→ l2 (w2z2)], (w1, z1)(w2, z2) ∈ E

10. ik ((w1, w2)(z1, z2))=[ih1 (w1) ∧ ih1 (z1) ∧ ih2 (w2) ∧ ih2 (z2)] × [ih1 (w1) ∧ i−→ l1 (z1w1) ∧ ih2 (z2) ∧ i−→ l2 (w2z2)], (w1, z1)(w2, z2) ∈ E 11. fk ((w1, w2)(z1, z2))=[ fh1 (w1) ∨ fh1 (z1) ∨ fh2 (w2) ∨ fh2 (z2)] × [ fh1 (w1) ∨ f−→ l1 (z1w1) ∨ fh2 (z2) ∨ f−→ l2 (w2z2)], (w1, z1)(w2, z2) ∈ E .

Proof. UsingsimilarargumentsasinTheorem2.1.[39],itcanbeproven. Example4. Consider −→ G1 =(X1, h1, l1) and −→ G2 =(X2, h2, l2) tobetwoIN-digraphs,respectively,asshown inFigure 6.Theintuitionisticneutrosophicoutandin-neighborhoodsof −→ G1 and −→ G2 aregiveninTables 3 and 4 TheINC-graphs C(−→ G1) and C(−→ G2) aregiveninFigure 7

Table3. IN-outandin-neighborhoodsof −→ G1 w ∈ X1 N+ (w) N (w) w1 {w2(0.2,0.2,0.3)} ∅ w2 ∅ {w1(0.2,0.2,0.3), w3(0.3,0.1,0.1)} w3 {w2(0.3,0.2,0.1)}{w4(0.3,0.1,0.1)} w4 {w3(0.3,0.1,0.1)} ∅

Table4. IN-outandin-neighborhoodsof −→ G2 w ∈ X2 N+ (w) N (w) z1 {z3(0.3,0.2,0.2)} ∅ z2 {z3(0.3,0.1,0.1)} ∅ z3 ∅ {z1(0.3,0.2,0.2), z2(0.3,0.1,0.1)}

z1 (0 4, 0 3, 0 2)

w1 (0 3, 0 4, 0 5)

w2 (0 4, 0 3, 0 1) w3 (0 5, 0 2, 0 1) w4 (0 4, 0 3, 0 1)

z2 (0 4, 0 3, 0 5)

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

z3 (0 7, 0 2, 0 3)

→G1 →G2 (0 .2 ,0 .2 ,0 . 3) (0 . 3, 0. 2, 0. 1) (0 .2 ,0 .2 ,0 . 3)

Figure6. IN-digraphs.

w1 (0.3, 0.4, 0.5) w2 (0 4, 0 3, 0 1) w3 (0 5, 0 2, 0 1) w4 (0 4, 0 3, 0 1)

z1 (0 4, 0 3, 0 2)

C ( →G2 )

C ( →G1 ) (0 . 12 , 0 03 , 0 . 1) (0 . 06, 0. 04, 0. 15)

z2 (0 4, 0 3, 0 5)

z3 (0 7, 0 2, 0 3)

Figure7. INC-graphsof −→ G1 and −→ G2

WenowconstructtheINC-graph GC(−→ G1)∗ C(−→ G2)∗ ∪ G =(w, k),where w =(tw, iw, fw) and k =(tk,ik,fk), from C(−→ G1)∗ and C(−→ G2)∗ usingTheorem 2.14.WeobtainedtwosetsofedgesbyusingCondition (1)

EC(−→ G1)∗ EC(−→ G2)∗ ={(w1, z1)(w1, z2), (w2, z1)(w2, z2), (w3, z1)(w3, z2), (w4, z1)(w4, z2), (w1, z1)(w3, z1), (w1, z2)(w3, z2), (w1, z3)(w3, z3)}, E ={(w2, z1)(w1, z3), (w2, z1)(w3, z3), (w2, z2)(w1, z3) (w2, z2)(w3, z3), (w3, z1)(w4, z3), (w3, z2)(w4, z3)}.

Thetruth-membership,indeterminacy-membershipandfalsity-membershipofedgescanbecalculatedbyusing Conditions (3) to (11) as, k((w1, z1)(w1, z2))=(th1 (w1) ∧ th2 (z1) ∧ th2 (z2), ih1 (w1) ∧ ih2 (z1) ∧ ih2 (z2), fh1 (w1) ∨ fh2 (z1) ∨ fh2 (z2)) ×(th1 (w1) ∧ tl2 (z1z3) ∧ tl2 (z2z3), ih1 (w1) ∧ il2 (z1z3) ∧ il2 (z2z3), fh1 (w1) ∨ fl2 (z1z3) ∨ fl2 (z2z3) =(0.3,0.3,0.5) × (0.3,0.1,0.5) =(0.09,0.03,0.25), k((w2, z1)(w1, z3))=(th1 (w2) ∧ th2 (z1) ∧ th1 (w1) ∧ th2 (z3), ih1 (w2) ∧ ih2 (z1) ∧ ih1 (w1) ∧ ih2 (z3), fh1 (w2) ∨ fh2 (z1) ∨ fh1 (w1) ∨ fh2 (z3)) ×(th1 (w2) ∧ tl1 (w1w2) ∧ tl2 (z3) ∧ tl2 (z1z3), ih1 (w2) ∧ il1 (w1w2) ∧ il2 (z3) ∧ il2 (z1z3), fh1 (w2) ∨ fl1 (w1w2) ∨ fl2 (z3) ∨ fl2 (z1z3)) =(0.3,0.2,0.5) × (0.2,0.2,0.3) =(0.06,0.04,0.15)

Allthetruth-membership,indeterminacy-membershipandfalsity-membershipdegreesofadjacentedgesof GC(−→ G1)∗ C(−→ G2)∗ and G aregiveninTable 5.

Table5. Adjacentedgesof GC(−→ G1 )∗ C(−→ G2 )∗ ∪ G .

(w, ´ w)(z, ´ z) k (w, ´ w)(z, ´ z)

(w1, z1)(w1, z2)(0.09,0.03,0.25) (w2, z1)(w2, z2)(0.12,0.03,0.1) (w3, z1)(w3, z2)(0.12,0.02,0.1) (w4, z1)(w4, z2)(0.12,0.03,0.1) (w1, z1)(w3, z1)(0.06,0.04,0.15)

(w1, z3)(w3, z3)(0.06,0.04,0.15)

(w2, z1)(w1, z3)(0.06,0.04,0.15)

(w2, z1)(w3, z3)(0.12,0.04,0.09)

(w2, z2)(w1, z3)(0.06,0.02,0.15)

(w2, z2)(w3, z3)(0.12,0.02,0.15)

(w3, z1)(w4, z3)(0.12,0.02,0.09)

(w3, z2)(w4, z3)(0.12,0.02,0.15) (w1, z2)(w3, z2)(0.06,0.04,0.25)

TheINC-graphobtainedbyusingthismethodisgiveninFigure 8 wheresolidlinesindicatepartofINC-graph obtainedfrom GC(−→ G1)∗ C(−→ G2)∗ ,andthedottedlinesindicatethepartof G . TheCartesianproduct −→ G1 −→ G2 ofIN-digraphs −→ G1 and −→ G2 isshowninFigure 9.TheIN-out-neighborhoods of −→ G1 −→ G2 arecalculatedinTable 6.TheINC-graphsof −→ G1 −→ G2 areshowninFigure 10

w1 w2 w3 w4

w1 w2 w3 w4

(0 09, 0 03, 0 25) (0 06, 0 04, 0 15) (0 06, 0 04, 0 25)

(0 12, 0 02, 0 1) (0 12, 0 03, 0 1)

(0. 06,0. 02,0. 15) (0. 06, 0. 02, 0. 15) (0 12, 0 03, 0 1) (0 .12, 0.04 , 0.09) (0 .12 , 0 .02 , 0 .15) (0 .12, 0.02, 0.09)

z1 z2 z3 (0 3, 0 3, 0 5) (0 4, 0 3, 0 2) (0.4, 0.2, 0.2) (0 4, 0 3, 0 2)

(0 3, 0 3, 0 5) (0 4, 0 3, 0 5) (0 4, 0 2, 0 5) (0 4, 0 3, 0 5)

(0 .12 , 0 .02 , 0 .15)

(0.3, 0.2, 0.5) (0 4, 0 2, 0 3) (0 5, 0 2, 0 3) (0 4, 0 2, 0 3)

(0 . 06 , 0 . 04 , 0 . 15)

z1 z2 z3

(0 2, 0 2, 0 3) (0 3, 0 2, 0 2) (0.3, 0.1, 0.2)

(0.3,0.2,0.5) (0.3,0.2,0.2) (0 3,0.2,0.2) (0.3,0.2,0.2)

Figure8. GC(−→ G1 )∗ C(−→ G2 )∗ ∪ G (0 3, 0 3, 0 5) (0 4, 0 3, 0 2) (0 4, 0 2, 0 2) (0 4, 0 3, 0 2)

(0 2, 0 2, 0 5)

(0 4, 0 3, 0 5)

(0.3, 0.2, 0.5)

(0.4, 0.2, 0.5)

(0 3, 0 1, 0 5)

(0.3, 0.3, 0.5) (0 3, 0 1, 0 5) (0.3, 0.1, 0.1) (0 3, 0 1, 0 1) (0 3, 0 1, 0 1)

Figure9. −→ G1 −→ G2

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

(0 3, 0 2, 0 5) (0 4, 0 2, 0 3) (0 5, 0 2, 0 3) (0 4, 0 2, 0 3) (0.4, 0.3, 0.5)

Table6. IN-out-neighborhoodsof −→ G1 −→ G2

(w, z) N+ (w, z)

(w1, z1) {((w2, z1),0.2,0.2,0.3), ((w1, z3),0.3,0.2,0.5)} (w1, z2) {((w1, z3),0.3,0.1,0.5), ((w2, z2),0.2,0.2,0.5)}

(w1, z3) {((w2, z3),0.2,0.2,0.3)}

(w2, z1) {((w2, z3),0.3,0.2,0.2)}

(w2, z2) {((w2, z3),0.3,0.1,0.1)}

(w2, z3) ∅

(w3, z1) {((w3, z3),0.3,0.2,0.2), ((w2, z1),0.3,0.2,0.2)} (w3, z2) {((w2, z2),0.3,0.2,0.5), ((w3, z3),0.3,0.1,0.1)}

(w3, z3) {((w2, z3),0.3,0.2,0.3)}

(w4, z1) {((w4, z3),0.3,0.2,0.2), ((w3, z1),0.3,0.1,0.2)} (w4, z2) {((w4, z3),0.3,0.1,0.1), ((w3, z2),0.3,0.1,0.5)} (w4, z2) {((w3, z3),0.3,0.1,0.3)} w1 w2 w3 w4

(0 09, 0 03, 0 25) (0 06, 0 04, 0 15) (0 06, 0 04, 0 25)

(0. 06,0. 02,0. 15) (0. 06, 0. 02, 0. 15) (0 12, 0 03, 0 1) (0 .12, 0.04 , 0.09) (0 .12 , 0 .02 , 0 .15) (0 .12, 0.02, 0.09)

z1 z2 z3 (0 3, 0 3, 0 5) (0 4, 0 3, 0 2) (0 4, 0 2, 0 2) (0 4, 0 3, 0 2)

(0 3, 0 3, 0 5) (0 4, 0 3, 0 5) (0 4, 0 2, 0 5) (0 4, 0 3, 0 5)

(0 12, 0 02, 0 1) (0 12, 0 03, 0 1)

(0 .12 , 0 .02 , 0 .15)

(0 3, 0 2, 0 5) (0 4, 0 2, 0 3) (0 5, 0 2, 0 3) (0 4, 0 2, 0 3)

(0 06 , 0 04 , 0 15) Figure10. C(−→ G1 −→ G2). Itcanbeseenthat C(−→ G1 −→ G2) ∼ = GC(−→ G1)∗ C(

G fromFigures 8 and 10 Definition10. Theintuitionisticneutrosophicopen-neighborhoodofavertex w ofanIN-graph G =(X, h, k) is IN-set N(w)=(Xw,tw,iw,fw ),where, Xw

{z|k1(w, z) > 0,k2(w, z) > 0,k3(w, z) > 0}, and tw : Xw → [0,1] definedby tw (z)= k1(w, z), iw : Xw → [0,1] definedby iw (z)= k2(w, z) and fz : Xw → [0,1] definedby fw (z)= k3(w, z).Foreveryvertex w ∈ X,theintuitionisticneutrosophic singletonset, Aw =(w, h1, h2, h3),suchthat: h1 : {w}→ [0,1], h2 : {w}→ [0,1], h3 : {w}→ [0,1] definedby h1(w)= h1(w), h2(w)= h2(w) and h3(w)= h3(w),respectively.Theintuitionisticneutrosophic closed-neighborhoodofavertexwis N[w]= N(w) ∪ Aw

Definition11. Suppose G =(X, h, k) isanIN-graph.Thesingle-valuedintuitionisticneutrosophic open-neighborhoodgraphof G isanIN-graph N(G)=(X,h,k ),whichhasthesameintuitionisticneutrosophic setofverticesin G andhasanintuitionisticneutrosophicedgebetweentwovertices w, z ∈ X in N(G) if andonlyif N(w) ∩ N(z) isanon-emptyIN-setin G.Thetruth-membership,indeterminacy-membershipand falsity-membershipvaluesoftheedge (w,z) aregivenby:

k1(w, z)=[h1(w) ∧ h1(z)]H1(N(w) ∩ N(z)), k2(w, z)=[h2(w) ∧ h2(z)]H2(N(w) ∩ N(z)), k3(w, z)=[h3(w) ∨ h3(z)]H3(N(w) ∩ N(z)), respectively

Definition12. Suppose G =(X, h, k) isanIN-graph.Thesingle-valuedintuitionisticneutrosophic closed-neighborhoodgraphof G isanIN-graph N(G)=(X, h, k ),whichhasthesameintuitionisticneutrosophic setofverticesin G andhasanintuitionisticneutrosophicedgebetweentwovertices w, z ∈ X in N[G] if andonlyif N[w] ∩ N[z] isanon-emptyIN-setin G.Thetruth-membership,indeterminacy-membershipand falsity-membershipvaluesoftheedge (w,z) aregivenby: k1(w, z)=[h1(w) ∧ h1(z)]H1(N[w] ∩ N[z]), k2(w, z)=[h2(w) ∧ h2(z)]H2(N[w] ∩ N[z]), k3(w, z)=[h3(w) ∨ h3(z)]H3(N[w] ∩ N[z]), respectively

Example5. Consider G =(X, h, k) tobeanIN-graph,suchthat X = {a, b, c, d}, h = {(a, 0.5, 0.4, 0.3), (b, 0.6, 0.3, 0.1), (c, 0.7, 0.3, 0.1), (d, 0.5, 0.6, 0.3)},and k = {(ab, 0.3, 0.2, 0.2), (ad, 0.4, 0.3, 0.2), (bc, 0.5, 0.2, 0.1), (cd, 0.4, 0.2, 0.2)},asshowninFigure 11.Then,correspondingintuitionisticneutrosophicopenand closed-neighborhoodgraphsareshowninFigure 12. a(0 5, 0 4, 0 3) b(0 6, 0 3, 0 1) c(0 7, 0 3, 0 1) d(0 5, 0 6, 0 3)

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

(0 3, 0 2, 0 2) (0 . 5 , 0 . 2 , 0 . 1) (0 4, 0 2, 0 2)

Figure11. IN-digraph.

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a(0 5, 0 4, 0 3)

b(0 6, 0 3, 0 1)

d(0 5, 0 6, 0 3)

(0 2, 0 06, 0 06) (0 2, 0 06, 0 06) (0 15, 0 06, 0 06)

c(0 7, 0 3, 0 1)

a(0 5, 0 4, 0 3)

(a) (b)

b(0 6, 0 3, 0 1)

(0 .2 , 0 .06 , 0 .06)

c(0 7, 0 3, 0 1) d(0 5, 0 6, 0 3)

(0 2 , 0 12 , 0 09) (0 2, 0 06, 0 06)

(0 . 3 , 0 06 , 0 01) (0 2, 0 06, 0 06)

Figure12. (a) N(G); (b) N[G]

Theorem4. ForeachedgeofanIN-graph G,thereexistsanedgein N[G] Proof. Suppose (w, z) isanedgeofanIN-graph G =(V, h, k).Suppose N[G]=(V, h, k ) isthe correspondingclosedneighborhoodofanIN-graph.Suppose w, z ∈ N[w] and w, z ∈ N[z].Then, w, z ∈ N[w] ∩ N[z].Hence, H1(N[w] ∩ N[z]) = 0, H2(N[w] ∩ N[z]) = 0, H3(N[w] ∩ N[z]) = 0. Then, k1(w, z)=[h1(w) ∧ h1(z)]H1(N[w] ∩ N[z]) = 0, k2(w, z)=[h2(w) ∧ h2(z)]H2(N[w] ∩ N[z]) = 0, k3(w, z)=[h3(w) ∨ h3(z)]H3(N[w] ∩ N[z]) = 0.

Thus,foreachedge (w, z) inIN-graph G,thereexistsanedge (w, z) in N[G]. Definition13. ThesupportofanIN-set ˘ A =(w,tA,iA,f A ) inXisthesubset ˆ AofXdefinedby: ˆ A = {w ∈ X:t ˘ A (w) = 0,i ˘ A (w) = 0,f ˘ A (w) = 1} and |supp( ˆ A)| isthenumberofelementsintheset.

Wenowdiscuss p-competitionintuitionisticneutrosophicgraphs. Suppose p isapositiveinteger.Then, p-competitionIN-graph Cp (−→ G ) oftheIN-digraph −→ G =(X, h, k) isanundirectedIN-graph G =(X, h, k),whichhasthesameintuitionisticneutrosophic setofverticesasin −→ G andhasanintuitionisticneutrosophicedgebetweentwovertices w, z ∈ X in

Cp (−→ G ) ifandonlyif |supp(N+ (w) ∩ N+ (z))|≥ p.Thetruth-membershipvalueofedge (w, z) in Cp (−→ G ) is t(w, z)= (i p)+1 i [h1(w) ∧ h1(z)]H1(N+ (w) ∩ N+ (z));theindeterminacy-membershipvalueofedge (w, z) in Cp (−→ G ) is i(w, z)= (i p)+1 i [h2(w) ∧ h2(z)]H2(N+ (w) ∩ N+ (z));andthefalsity-membership valueofedge (w, z) in Cp (−→ G ) is f (w, z)= (i p)+1 i [h3(w) ∨ h3(z)]H3(N+ (w) ∩ N+ (z)) where i = |supp(N+ (w) ∩ N+ (z))|

Thethree-competitionIN-graphisillustratedbythefollowingexample. Example6. Consider −→ G =(X, h, k) tobeanIN-digraph,suchthat X = {w1, w2, w3, z1, z2, z3}, h = {(w1, 0.5, 0.1, 0.2), (w2, 0.1, 0.6, 0.3), (w3, 0.1, 0.2, 0.5), (z1, 0.7, 0.2, 0.1), (z2, 0.5, 0.2, 0.3), (z3, 0.3, 0.7, 0.2)} and k = {( −−−−→ (w1, z1),0.4, 0.1, 0.1), ( −−−−→ (w1, z2),0.5, 0.1, 0.3), ( −−−−→ (w1, z3),0.2, 0.1, 0.1), ( −−−−→ (w2, z1),0.1, 0.1, 0.2), ( −−−−→ (w2, z2),0.1, 0.1, 0.2), ( −−−−→ (w2, z3),0.1, 0.5, 0.2), ( −−−−→ (w3, z1),0.1, 0.1, 0.1)( −−−−→ (w3, z2),0.1, 0.1, 0.2)},asshownin Figure 13.Then, N+ (w1)= {(z1, 0.4, 0.1, 0.1), (z2, 0.5, 0.1, 0.3), (z3, 0.2, 0.1, 0.1)}, N+ (w2)= {(z1, 0.1, 0.1, 0.2), (z2, 0.1, 0.1, 0.2), (z3, 0.1, 0.5, 0.2)} and N+ (w3)= {(z1, 0.1, 0.1, 0.1), (z2, 0.1, 0.1, 0.2)} Therefore, N+ (w1) ∩ N+ (w2)= {(z1, 0.1, 0.1, 0.2), (z2, 0.1, 0.1, 0.3), (z3, 0.1, 0.1, 0.2)}, N+ (w1) ∩ N+ (w3)= {(z1, 0.1, 0.1, 0.1), (z2, 0.1, 0.1, 0.3)} and N+ (w2) ∩ N+ (w3)= {(z1, 0.1, 0.1, 0.2), (z2, 0.1, 0.1, 0.2)}

Now, i = |supp(N+ (w1) ∩ N+ (w2))| = 3.For p = 3, t(w1, w2)= 0.003, i(w1, w2)= 0.003 and f (w1,w2)= 0.02.AsshowninFigure 14

w1 (0 5, 0 1, 0 2)

z1 (0 7, 0 2, 0 1)

w2 (0 1, 0 6, 0 3) w3 (0 1, 0 2, 0 5)

Figure13. IN-digraph.

z2 (0 5, 0 2, 0 3)

z3 (0 3, 0 7, 0 2)

w1 (0 5, 0 1, 0 2)

z1 (0 7, 0 2, 0 1)

(0 003 , 0 . 003 , 0 02)

w2 (0 1, 0 6, 0 3)

z2 (0 5, 0 2, 0 3)

w3 (0 1, 0 2, 0 5)

z3 (0 3, 0 7, 0 2)

Figure14. Three-competitionIN-graph.

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WenowdefineanotherextensionofINC-graphknownasthe m-stepINC-graph. −→ P m z,w:adirectedintuitionisticneutrosophicpathoflength m from z to w. N+ m (z):single-valuedintuitionisticneutrosophic m-stepout-neighborhoodofvertex z. Nm (z):single-valuedintuitionisticneutrosophic m-stepin-neighborhoodofvertex z. Cm −→ (G): m-stepINC-graphoftheIN-digraph −→ G

Definition14. Suppose −→ G =(X, h, k) isanIN-digraph.The m-stepIN-digraphof −→ G isdenotedby −→ G m =(X,h,k) wheretheintuitionisticneutrosophicsetofverticesof −→ G isthesameastheintuitionisticneutrosophicsetofverticesof −→ G m andhasanedgebetween z and w in −→ G m ifandonlyifthereexistsanintuitionisticneutrosophicdirectedpath −→ P m z,w in −→ G

Definition15. Theintuitionisticneutrosophic m-stepout-neighborhoodofvertex z ofanIN-digraph −→ G =(X,h,k) isIN-set: N+ m (z)=(X+ z ,t+ z ,i+ z ,f + z ),where X+ z = {w| thereexistsadirectedintuitionisticneutrosophicpathoflength m from z to w, −→ P m z,w }, t+ z : X+ z → [0, 1], i+ z : X+ z → [0, 1] and f + z : X+ z → [0, 1] aredefinedby t+ z = min{t −−−−−→ (w1,w2), (w1, w2) isanedgeof −→ P m z,w }, i+ z = min{i −−−−−→ (w1,w2), (w1, w2) isanedgeof −→ P m z,w } and f + z = max{ f −−−−−→ (w1,w2), (w1, w2) isanedgeof −→ P m z,w },respectively.

Definition16. Theintuitionisticneutrosophic m-stepin-neighborhoodofvertex z ofanIN-digraph −→ G =(X,h,k) isIN-set: Nm (z)=(Xz ,tz ,iz ,fz ),where Xz = {w| thereexistsadirectedintuitionisticneutrosophicpathoflength m from w to z, −→ P m w,z }, tz : Xz → [0, 1], iz : Xz → [0, 1] and fz : Xz → [0, 1] aredefinedby tz = min{t −−−−−→ (w1,w2), (w1, w2) isanedgeof −→ P m w,z }, iz = min{i −−−−−→ (w1,w2), (w1, w2) isanedgeof −→ P m w,z } and fz = max{ f −−−−−→ (w1,w2), (w1, w2) isanedgeof −→ P m w,z },respectively.

Definition17. Suppose −→ G =(X, h, k) isanIN-digraph.The m-stepINC-graphofIN-digraph −→ G isdenoted by Cm (−→ G )=(X, h, k),whichhasthesameintuitionisticneutrosophicsetofverticesasin −→ G andhasan edgebetweentwovertices w, z ∈ X in Cm (−→ G ) ifandonlyif (N+ m (w) ∩ N+ m (z)) isanon-emptyIN-setin −→ G .Thetruth-membershipvalueofedge (w, z) in Cm (−→ G ) is t(w, z)=[h1(w) ∧ h1(z)]H1(N+ m (w) ∩ N+ m (z)); theindeterminacy-membershipvalueofedge (w, z) in Cm (−→ G ) is i(w, z)=[h2(w) ∧ h2(z)]H2(N+ m (w) ∩ N+ m (z)); andthefalsity-membershipvalueofedge (w,z) in Cm (−→ G ) isf (w,z)=[h3(w) ∨ h3(z)]H3(N+ m (w) ∩ N+ m (z)).

Thetwo-stepINC-graphisillustratedbythefollowingexample. Example7. Consider −→ G =(X, h, k) isanIN-digraph,suchthat, X = {w1, w2, z1, z2, z3}, h = {(w1, 0.3, 0.4, 0.6), (w2, 0.2, 0.5, 0.3), (z1, 0.4, 0.2, 0.3), (z2, 0.7, 0.2, 0.1), (z3, 0.5, 0.1, 0.2), (z4, 0.6, 0.3, 0.2)}, and k = {( −−−−→ (w1,z1), 0.2, 0.1, 0.2), ( −−−−→ (w2,z4), 0.1, 0.2, 0.3), ( −−−−→ (z1,z3), 0.3, 0.1, 0.2), ( −−−−→ (z1,z2), 0.3, 0.1, 0.2), ( −−−−→ (z4,z2), 0.2, 0.1, 0.1),and ( −−−−→ (z4,z3), 0.4, 0.1, 0.4)},asshowninFigure 15 Then, N+ 2 (w1)= {(z2, 0.2, 0.1, 0.2), (z3, 0.2, 0.1, 0.2)} and N+ 2 (w2)= {(z2, 0.1, 0.1, 0.3), (z3, 0.1, 0.1, 0.4)}.Therefore, N+ 2 (w1) ∩ N+ 2 (w2)= {(z2, 0.1, 0.1, 0.3), (z3, 0.1, 0.1, 0.4)}.Thus, t(w1, w2)= 0.02, i(w1, w2)= 0.04 andf (w1, w2)= 0.18.ThisisshowninFigure 16.

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w1 (0 3, 0 4, 0 6)

w2 (0 2, 0 5, 0 3)

z1 (0 4, 0 2, 0 3)

w1 (0 3, 0 4, 0 6)

z1 (0 4, 0 2, 0 3)

z4 (0 6, 0 3, 0 2) (0 . 2 , 0 1 , 0 2) (0 . 1 , 0 . 2 , 0 . 3) (0. 2, 0. 1, 0. 1) (0 .3 , 0 .1 , 0 .2) (0 . 4, 0. 1, 0. 4) (0 .3 ,0 .1 ,0 . 2)

z2 (0 7, 0 2, 0 1)

z3 (0 5, 0 1, 0 2)

Figure15. IN-digraph.

(0 02, 0 04, 0 18)

w2 (0 2, 0 5, 0 3)

z2 (0 7, 0 2, 0 1)

z4 (0 6, 0 3, 0 2)

z3 (0 5, 0 1, 0 2)

Figure16. Two-stepINC-graph.

Definition18. Theintuitionisticneutrosophic m-stepout-neighborhoodofvertex z ofanIN-digraph −→ G =(X,h,k) isIN-set:

Nm (z)=(Xz,tz,iz,fz ),where

Xz = {w| thereexistsadirectedintuitionisticneutrosophicpathoflength m from z to w, Pm z,w }, tz : Xz → [0, 1], iz : Xz → [0, 1] and fz : Xz → [0, 1] aredefinedby tz = min{t(w1, w2), (w1, w2) isanedgeof Pm z,w }, iz = min{i(w1,w2), (w1, w2) isanedgeof Pm z,w } and fz = max{ f (w1, w2), (w1, w2) isanedgeof Pm z,w },respectively.

Definition19. Suppose G =(X, h, k) isanIN-graph.Then,the m-stepintuitionisticneutrosophic neighborhoodgraph(IN-neighborhood-graph) Nm (G) isdefinedby Nm (G)=(X, h, κ),where h =(h1, h2, h3), κ =(κ1, κ2, κ3), κ1 : X × X → [0, 1], κ2 : X × X → [0, 1] and κ3 : X × X → [0,1] aresuchthat:

κ1(w, z)= h1(w) ∧ h1(z)H1(Nm (w) ∩ Nm (z)),

κ2(w, z)= h2(w) ∧ h2(z)H2(Nm (w) ∩ Nm (z)), κ3(w, z)= h3(w) ∨ h3(z)H3(Nm (w) ∩ Nm (z)), respectively

Theorem5. IfalltheedgesofIN-digraph −→ G =(X, h, k) areindependentstrong,thenalltheedgesof Cm (−→ G ) areindependentstrong.

Proof. Suppose −→ G =(X, h, k) isanIN-digraphand Cm (−→ G )=(X, h, k) isthecorresponding m-stepINC-graph.Sincealltheedgesof −→ G areindependentstrong,then H1(N+ m (w) ∩ N+ m (z)) > 0.5, H2(N+ m (w) ∩ N+ m (z)) < 0.5 and H3(N+ m (w) ∩ N+ m (z)) < 0.5.Then, t(w, z)=(h1(w) ∧ h1(z))H1(N+ m (w) ∩ N+ m (z)),or t(w, z) > 0.5(h1(w) ∧ h1(z)),or t(w,z) (h1(w)∧h1(z)) > 0.5, i(w, z)=(h2(w) ∧ h2(z))H2(N+ m (w) ∩ N+ m (z)),or i(w, z) < 0.5(h2(w) ∧ h2(z)),or i(w,z) (h2(w)∧h2(z)) < 0.5 and f (w, z)=(h3(w) ∨ h3(z))H3(N+ m (w) ∩ N+ m (z)),or f (w, z) < 0.5(h3(w) ∨ h3(z)),or f (w,z) (h3(w)∨h3(z)) < 0.5.

Hence,theedge (w, z) isindependentstrongin Cm (−→ G ).Since, (w, z) istakentobethearbitrary edgeof Cm (−→ G ),thusalltheedgesof Cm (−→ G ) areindependentstrong.

3.Applications

Competitiongraphsareveryimportanttorepresentthecompetitionbetweenobjects.However, still,theserepresentationsareunsuccessfultodealwithallthecompetitionsofworld;forthatpurpose, INC-graphsareintroduced.Now,wediscusstheapplicationsofINC-graphstostudythecompetition alongwithalgorithms.TheINC-graphshavemanyutilizationsindifferentareas.

3.1.Ecosystem

Considerasmallecosystem:humaneatstrout;baldeagleeatstroutandsalamander;trouteats phytoplankton,mayflyanddragonfly;salamandereatsdragonflyandmayfly;snakeeatssalamanderandfrog; frogeatsdragonflyandmayfly;mayflyeatsphytoplankton;dragonflyeatsphytoplankton.Theseninespecies human,baldeagle,salamander,snake,frog,dragonfly,trout,mayflyandphytoplanktonaretakenasvertices. Letthedegreeofexistenceintheecosystemofhumanbe 60%,thedegreeofindeterminacyofexistencebe 30% andthedegreeoffalse-existencebe 10%,i.e.,thetruth-membership,indeterminacy-membershipand falsity-membershipvaluesofthevertexhumanare (0.6, 0.3, 0.1).Similarly,weassumethetruth-membership, indeterminacy-membershipandfalsity-membershipvaluesofotherverticesas (0.7, 0.3, 0.2), (0.4, 0.3, 0.5), (0.3, 0.5, 0.1), (0.3, 0.4, 0.5), (0.3, 0.5, 0.2), (0.7, 0.3, 0.2), (0.6, 0.4, 0.2) and (0.3, 0.5, 0.2).Supposethathuman likestoeattrout 20%,indeterminatetoeat 10% anddisliketoeat,say 10%.Thelikeness,indeterminacyand dislikenessofpreysforpredatorsareshowninTable 7

Itisclearthatiftroutisremovedfromthefoodcycle,thenhumanmustbelifeless,andinsuchasituation baldeagle,phytoplankton,dragonflyandmayflygrowinanundisciplinedmanner.Thus,wecanevaluate thefoodcyclewiththehelpofINC-graphs.

Table7. Likeness,indeterminacyanddislikenessofpreysandpredators.

NameofPredatorNameofPreyLiketoEatIndeterminatetoEatDisliketoEat

HumanTrout 201010

BaldeagleTrout 202020

BaldeagleSalamander 302030 SnakeSalamander 202010 SnakeFrog 302040 SalamanderDragonfly 202020 SalamanderMayfly 202040

FrogDragonfly 303030

TroutDragonfly 204010

TroutMayfly 301010

TroutPhytoplankton 201010 DragonflyPhytoplankton 101010 MayflyPhytoplankton 303020 FrogMayfly 101010

ForthisfoodwebFigure 17,wehavethefollowingTable 8 ofIN-out-neighborhoods.

Human

(0 6, 0 3, 0 1) (0 7, 0 3, 0 2) (0 4, 0 3, 0 5)

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

Trout

Baldeagle Salamander

(0 3, 0 5, 0 2) (0 6, 0 4, 0 2) (0 3, 0 5, 0 2)

Phytoplankton

Dragonfly

Snake Frog Mayfly

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

(0 3, 0 2, 0 3) (0 2, 0 2, 0 1) (0 . 3, 0. 2, 0. 4) (0 2, 0 4, 0 1) (0 2, 0 2, 0 . 2) (0 .2 ,0 .2 ,0 . 4) (0 3, 0 3, 0 2) (03 , 0 .1 , 01)

Figure17. IN-foodweb. Table8. IN-out-neighborhoods.

w ∈ X N+ (w)

(0 3, 0 3, 0 3)

(0 3, 0 5, 0 1) (0 3, 0 4, 0 5) (0 7, 0 3, 0 2)

Human {(Trout,0.2,0.1,0.1)}

Baldeagle {(Trout,0.2,0.2,0.2), (Salamander,0.3,0.2,0.3)}

Salamander {(Dragonfly,0.2,0.2,0.2), (Mayfly,0.2,0.2,0.4)} Snake {(Salamander,0.2,0.2,0.1), (Frog,0.3,0.2,0.4)} Frog {(Dragonfly,0.3,0.3,0.3), (Mayfly,0.1,0.1,0.1)} Mayfly {(Phytoplankton,0.3,0.3,0.2)}

Phytoplankton ∅

Dragonfly {(Phytoplankton,0.1,0.1,0.1)}

Trout {(Phytoplankton,0.2,0.1,0.1), (Mayfly,0.3,0.1,0.1), (Dragonfly,0.2,0.4,0.1)}

Therefore, N+(Human ∩ Baldeagle)= {(Trout, 0.2, 0.1, 0.2)}, N+(Baldeagle ∩ Snake)= {(Salamander, 0.2, 0.2, 0.3)}, N+(Salamander ∩ Frog)= {(Dragonfly, 0.2, 0.2, 0.3), (Mayfly, 0.1, 0.1, 0.4)}, N+(Salamander ∩ Trout)= {(Dragonfly, 0.2, 0.2, 0.2), (Mayfly, 0.2, 0.1, 0.4)}, N+(Trout ∩ Frog)= {(Dragonfly, 0.2, 0.3, 0.3), (Mayfly, 0.1, 0.1, 0.1)}, N+(Mayfly ∩ Trout)= {(Phytoplankton, 0.2, 0.1, 0.2)}, N+(Mayfly ∩ Dragonfly)= {(Phytoplankton, 0.1, 0.1, 0.2)} and N+(Dragonfly ∩ Trout)= {(Phytoplankton,0.1,0.1,0.1)}

Now,thereisanedgebetweenhumanandbaldeagle;snakeandbaldeagle;salamanderandtrout; salamanderandfrog;troutandfrog;troutanddragonfly;troutandmayfly;dragonflyandmayflyinthe INC-graph,whichhighlightsthecompetitionbetweenthem;andfortheotherpairofspecies,thereisno edge,whichindicatesthatthereisnocompetitionintheINC-graphFigure 18.Forexample,thereisan edgebetweenhumanandbaldeagleindicatinga 12% degreeoflikenesstopreyonthesamespecies, a3%degreeofindeterminacyanda4%degreeofnon-likenessbetweenthem.

Human

(0 6, 0 3, 0 1) (0 7, 0 3, 0 2) (0 4, 0 3, 0 5)

(0 12, 0 03, 0 04)

Trout

(0 06, 0 06, 0 06) (0 06, 0 06, 0 15) (0 03, 0 04, 0 02)

Phytoplankton

Algorithm1: Ecosystem.

(0 3, 0 5, 0 1) (0 3, 0 4, 0 5) (0 7, 0 3, 0 2)

(0 06, 0 06, 0 1) (0 06, 0 12, 0 05) (0 06, 0 05, 0 04)

(0 03, 0 04, 0 04)

(0 3, 0 5, 0 2) (0 6, 0 4, 0 2) (0 3, 0 5, 0 2)

Figure18. CorrespondingINC-graph

Wepresentourmethod,whichisusedinourecosystemapplicationinAlgorithm 1

Step1. Inputthetruth-membership,indeterminacy-membershipandfalsity-membershipvalues forsetof n species.

Step2. Ifforanytwodistinctvertices wi and wj, t(wi wj ) > 0, i(wi wj ) > 0, f (wi wj ) > 0,then (wj, t(wi wj ), i(wi wj ), f (wi wj )) ∈ N+ (wi ).

Step3. RepeatStep2forallvertices wi and wj tocalculateIN-out-neighborhoods N+ (wi )

Step4. Calculate N+ (wi ) ∩ N+ (wj ) foreachpairofdistinctvertices wi and wj.

Step5. Calculate H[N+ (wi ) ∩ N+ (wj )].

Step6. If N+ (wi ) ∩ N+ (wj ) = ∅,thendrawanedge wi wj

Step7. RepeatStep6forallpairsofdistinctvertices.

Step8. Assignmembershipvaluestoeachedge wi wj usingtheconditions:

t(wi wj )=(wi ∧ wj )H1[N+ (wi ) ∩ N+ (wj )] i(wi wj )=(wi ∧ wj )H2[N+ (wi ) ∩ N+ (wj )] f (wi wj )=(wi ∨ wj )H3[N+ (wi ) ∩ N+ (wj )]

3.2.CareerCompetition

ConsidertheIN-digraphFigure 19 representingthecompetitionbetweenapplicantsforacareer. Let {Rosaleen, Nazneen, Abner, Amara, Casper} bethesetofapplicantsfortheparticularcareers {Medicine, Pharmacy, Anatomy, Surgery}.Thetruth-membershipvalueofeachapplicantrepresentsthe degreeofloyaltyquality;theindeterminacy-valuerepresentstheindeterminatestateofloyalty;andthe false-membershipvaluerepresentsthedisloyaltyofeachapplicanttowardstheircareers.Letthedegreeof truth-membershipofNazneenofherloyaltytowardshercareerbe 30%:degreeofindeterminacyis 50%, anddegreeofdisloyaltyis 10%,i.e.,thetruth-membership,indeterminacyandfalsity-membershipvalues ofthevertexNazneenare (0.3, 0.5, 0.1).Thetruth-membershipvalueofeachdirectededgebetween anapplicantandacareerrepresentstheeligibilityforthatcareer;theindeterminacy-valuerepresents theindeterminatestateofthatcareer;andthefalse-membershipvaluerepresentsnon-eligibilityforthat particularcareer.

Abner (0 3, 0 5, 0 6)

Surgery (0 3, 0 4, 0 5)

Nazneen (0 3, 0 5, 0 1)

(0. 2, 0. 2, 0. 2) (0 .1 , 0 .4 , 0 . 2) (0 2, 0 3, 0 4) (0 2, 0 4, 0 5) (0 .3 , 0 .4 , 0 .5)

Pharmacy Medicine

(0 7, 0 3, 0 2)

(0.2, 0 4, 0.5) (0 .1 ,0 .2 ,0 . 3) (0. 1,0. 5,0. 2)

(0 .5 , 0 .3 , 0 .1)

(0 2, 0 5, 0 3) (0 3, 0 6, 0 2)

Amara (0 6, 0 5, 0 2)

Figure19. IN-digraph.

Anatomy

(0 .2 , 0 .5 , 0 . 3)

Rosaleen (0 3, 0 5, 0 2) Casper (0 1, 0 6, 0 3)

Thus,inTable 9, N+ (Nazneen) ∩ N+ (Rosaleen)= {(Surgery, 0.2, 0.2, 0.4)}, N+ (Nazneen) ∩ N+ (Amara)= {(Pharmacy, 0.1, 0.4, 0.3)}, N+ (Nazneen) ∩ N+ (Abner)= {(Pharmacy, 0.1, 0.4, 0.5)}, N+ (Nazneen) ∩ N+ (Casper)= ∅, N+ (Rosaleen) ∩ N+ (Amara)= ∅, N+ (Rosaleen) ∩ N+ (Casper)= ∅, N+ (Rosaleen) ∩ N+ (Abner)= ∅, N+ (Amara) ∩ N+ (Casper)= {(Medicine, 0.1, 0.2, 0.3)}, N+ (Amara) ∩ N+ (Abner)= {(Medicine, 0.3, 0.3, 0.5), (Pharmacy, 0.2, 0.4, 0.5)} and N+ (Casper) ∩ N+ (Abner)= {(Medicine,0.1,0.2,0.5), (Anatomy,0.1,0.4,0.5)}.

Table9. IN-out-neighborhoods.

w ∈ X N+ (w)

Nazneen {(Surgery,0.2,0.2,0.2), (Pharmacy,0.1,0.4,0.2)} Rosaleen {(Surgery,0.2,0.3,0.4)}

Amara {(Medicine,0.5,0.3,0.1), (Pharmacy,0.2,0.5,0.3)} Casper {(Medicine,0.1,0.2,0.3), (Anatomy,0.1,0.5,0.2)} Abner {(Medicine,0.3,0.4,0.5), (Anatomy,0.2,0.4,0.5), (Pharmacy,0.2,0.4,0.5)}

TheINC-graphisshowninFigure 20.Thesolidslinesindicatethestrengthofcompetitionbetween twoapplicants,anddashedlinesindicatetheapplicantcompetingfortheparticularcareer.Forexample, NazneenandRosaleenbotharecompetingforthecareer,surgery,andthestrengthofcompetition betweenthemis (0.06, 0.1, 0.08).InTable 10, W(z, c) representsthecompetitionofapplicant z forcareer c withrespecttoloyaltyquality,indeterminacyanddisloyaltytocompetewiththeothers.Thestrength tocompetewiththeotherapplicantswithrespecttoaparticularcareeriscalculatedinTable 10

FromTable 10,NazneenandRosaleenhaveequalstrengthtocompetewiththeotherforthecareer, surgery.AbnerandCasperhaveequalstrengthofcompetitionforthecareer,anatomy.Amaracompetes withtheothersforthecareer,pharmacyandmedicine.

Nazneen (0 3, 0 5, 0 1)

Abner (0 3, 0 5, 0 6)

(0 .09 , 0 .20 , 0 . 30)

(0 .01 , 0 .20 , 0 . 30)

Surgery (0 3, 0 4, 0 5)

Medicine

(0 7, 0 3, 0 2)

Casper (0 1, 0 6, 0 3)

(0. 03, 0. 20, 0. 3) (0 01, 0 10, 0 09)

Amara (0 6, 0 5, 0 2)

(0 .06, 0.1, 0.08) (0 .03 ,0 .20 ,0 . 06)

Pharmacy

(0 2, 0 5, 0 3) (0 3, 0 6, 0 2)

Rosaleen (0 3, 0 5, 0 2)

Anatomy

Figure20. CorrespondingINC-graph.

Table10. Strengthofcompetitionoftheapplicantforaparticularcareer.

(Applicant,Career)InCompetitionW(Applicant,Career)S(Applicant,Career)

(Nazneen, Surgery) Rosaleen (0.06,0.1,0.08) 0.88 (Rosaleen, Surgery) Nazneen (0.06,0.1,0.08) 0.88

(Abner, Anatomy) Casper (0.01,0.20,0.30) 0.51 (Casper, Anatomy) Abner (0.01,0.20,0.30) 0.51

(Nazneen, Pharmacy) Abner, Amara (0.03,0.20,0.18) 0.65

(Abner, Pharmacy) Amara, Nazneen (0.06,0.20,0.30) 0.56

(Amara, Pharmacy) Nazneen, Abner (0.06,0.20,0.18) 0.68

(Amara, Medicine) Abner, Casper (0.05,0.15,0.195) 0.705

(Casper, Medicine) Abner, Amara (0.01,0.15,0.195) 0.665

(Abner, Medicine) Casper, Amara (0.05,0.20,0.30) 0.55

Wepresentourmethod,whichisusedinourcareercompetitionapplicationinAlgorithm 2

Algorithm2: CareerCompetition

Step1. Inputthetruth-membership,indeterminacy-membershipandfalsity-membership valuesforsetof n applicants.

Step2. Ifforanytwodistinctvertices zi and zj, t(zi zj ) > 0, i(zi zj ) > 0, f (zi zj ) > 0,then (zj, t(zi zj ), i(zi zj ), f (zi zj )) ∈ N+ (zi )

Step3. RepeatStep2forallvertices zi and zj tocalculateIN-out-neighborhoods N+ (zi ) Step4. Calculate N+ (zi ) ∩ N+ (zj ) foreachpairofdistinctvertices zi and zj. Step5. Calculate H[N+ (zi ) ∩ N+ (zj )] Step6. If N+ (zi ) ∩ N+ (zj ) = ∅,thendrawanedge zi zj Step7. RepeatStep6forallpairsofdistinctvertices. Step8. Assignmembershipvaluestoeachedge zi zj usingtheconditions:

t(zi zj )=(zi ∧ zj )H1[N+ (zi ) ∩ N+ (zj )] i(zi zj )=(zi ∧ zj )H2[N+ (zi ) ∩ N+ (zj )] f (zi zj )=(zi ∨ zj )H3[N+ (zi ) ∩ N+ (zj )]

Step9. If z, r1, r2, r3,..., rn aretheapplicantscompetingforcareer c,thenthestrengthof competition W(z, c)=(t(z, c), i(z, c), f (z, c)) ofeachapplicant z forthecareer c is: W(z, c)= (t(zr1)+t(zr2)+ +t(zrn ),i(zr1)+i(zr2)+ +i(zrn ), f (zr1)+ f (zr2)+ + f (zrn )) n Step10. Calculate S(z, c),thestrengthofcompetitionofeachapplicant z forcareer c. S(z, c)= t(z, c) (i(z, c)+ f (z, c))+ 1.

4.Conclusions

Graphsserveasmathematicalmodelstoanalyzemanyconcretereal-worldproblemssuccessfully. Certainproblemsinphysics,chemistry,communicationscience,computertechnology,sociology andlinguisticscanbeformulatedasproblemsingraphtheory.Intuitionisticneutrosophicsettheory isamathematicaltooltodealwithincompleteandvagueinformation.Intuitionisticneutrosophicset theorydealswiththeproblemofhowtounderstandandmanipulateimperfectknowledge.Inthis researchpaper,wehavedescribedtheconceptofintuitionisticneutrosophiccompetitiongraphs.Wehave alsopresentedapplicationsofintuitionisticneutrosophiccompetitiongraphsinecosystemandcareer competition.Weaimtoextendourresearchworkoffuzzificationto(1)fuzzysoftcompetitiongraphs, (2)fuzzyroughsoftcompetitiongraphs,(3)bipolarfuzzysoftcompetitiongraphsand(4)theapplication offuzzysoftcompetitiongraphsindecisionsupportsystems.

AuthorContributions: MuhammadAkramandMaryamNasirconceivedanddesignedtheexperiments; MaryamNasirperformedtheexperiments;MuhammadAkramandMaryamNasiranalyzedthedata;MaryamNasir contributedreagents/materials/analysistools;MuhammadAkramwrotethepaper.

ConflictsofInterest: Theauthorsdeclarethattheyhavenoconflictofinterestregardingthepublicationofthis researcharticle.

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