Research Article Application of Bipolar Fuzzy Sets in Graph Structures

14 Pages·2016·2 MB·English

Hindawi Publishing Corporation
Applied Computational Intelligence and So Computing
Volume 2016, Article ID 5859080, 13 pages
http://dx.doi.org/10.1155/2016/5859080
Research Article
Application of Bipolar Fuzzy Sets in Graph Structures
MuhammadAkramandRabiaAkmal
DepartmentofMathematics,UniversityofthePunjab,NewCampus,Lahore54590,Pakistan
CorrespondenceshouldbeaddressedtoMuhammadAkram;[email protected]
Received27November2015;Revised25December2015;Accepted28December2015
AcademicEditor:BaodingLiu
Copyright©2016M.AkramandR.Akmal.ThisisanopenaccessarticledistributedundertheCreativeCommonsAttribution
License,whichpermitsunrestricteduse,distribution,andreproductioninanymedium,providedtheoriginalworkisproperly
cited.
Agraphstructureisausefultoolinsolvingthecombinatorialproblemsindifferentareasofcomputerscienceandcomputational
intelligencesystems.Inthispaper,weapplytheconceptofbipolarfuzzysetstographstructures.Weintroducecertainnotions,
includingbipolarfuzzygraphstructure(BFGS),strongbipolarfuzzygraphstructure,bipolarfuzzy𝑁𝑖-cycle,bipolarfuzzy𝑁𝑖-tree,
bipolarfuzzy𝑁𝑖-cutvertex,andbipolarfuzzy𝑁𝑖-bridge,andillustratethesenotionsbyseveralexamples.Westudy𝜙-complement,
self-complement,strongself-complement,andtotallystrongself-complementinbipolarfuzzygraphstructures,andweinvestigate
someoftheirinterestingproperties.
1.Introduction notionofafuzzygraphstructureanddiscussedsomerelated
properties.Akrametal.[9–13]haveintroducedbipolarfuzzy
Conceptsofgraphtheoryhaveapplicationsinmanyareasof graphs,regularbipolarfuzzygraphs,irregularbipolarfuzzy
computer science including data mining, image segmenta- graphs, antipodal bipolar fuzzy graphs, and bipolar fuzzy
tion, clustering, image capturing, and networking. A graph hypergraphs.Inthispaper,weintroducethecertainnotions
structure,introducedbySampathkumar[1],isageneraliza- includingbipolarfuzzygraphstructure(BFGS),strongbipo-
tion of undirected graph which is quite useful in studying lar fuzzy graph structure, bipolar fuzzy 𝑁𝑖-cycle, bipolar
somestructuresincludinggraphs,signedgraphs,andgraphs fuzzy𝑁𝑖-tree,bipolarfuzzy𝑁𝑖-cutvertex,andbipolarfuzzy
inwhicheveryedgeislabeledorcolored.Agraphstructure 𝑁𝑖-bridge and illustrate these notions by several examples.
helps to study the various relations and the corresponding We present 𝜙-complement, self-complement, strong self-
edgessimultaneously. complement, and totally strong self-complement in bipolar
Afuzzyset,introducedbyZadeh[2],givesthedegreeof fuzzy graph structures, and we investigate some of their
membership of an object in a given set. Zhang [3] initiated interestingproperties.
the concept of a bipolar fuzzy set as a generalization of a Wehaveusedstandarddefinitionsandterminologiesin
fuzzy set. A bipolar fuzzy set is an extension of fuzzy set this paper. For other notations, terminologies, and applica-
whosemembershipdegreerangeis[−1,1].Inabipolarfuzzy tionsnotmentionedinthepaper,thereadersarereferredto
set, the membership degree 0 of an element means that [1,5,7,14–18].
theelementisirrelevanttothecorrespondingproperty,the
membership degree (0,1] of an element indicates that the 2.Preliminaries
elementsomewhatsatisfiestheproperty,andthemembership
degree [−1,0) of an element indicates that the element Inthissection,wereviewsomedefinitionsthatarenecessary
somewhat satisfies the implicit counterproperty. Kauffman forthispaper.
defined in [4] a fuzzy graph. Rosenfeld [5] described the A graph structure 𝐺∗ = (𝑈,𝐸1,𝐸2,...,𝐸𝑘) consists of a
structureoffuzzygraphsobtaininganalogsofseveralgraph nonemptyset𝑈togetherwithrelations𝐸1,𝐸2,...,𝐸𝑘 on𝑈,
theoretical concepts. Bhattacharya [6] gave some remarks which are mutually disjoint such that each 𝐸𝑖 is irreflexive
on fuzzy graphs. Several concepts on fuzzy graphs were and symmetric. If (𝑢,V) ∈ 𝐸𝑖 for some 𝑖, 1 ≤ 𝑖 ≤ 𝑘,
introducedbyMordesonetal.[7].Dinesh[8]introducedthe we call it an 𝐸𝑖-edge and write it as “𝑢V.” A graph structure
2 AppliedComputationalIntelligenceandSoftComputing
𝐺∗ = (𝑈,𝐸1,𝐸2,...,𝐸𝑘)iscomplete,if(i)eachedge𝐸𝑖, 1 ≤ Definition 3 (see [8]). Let 𝐺 = (],𝜌1,𝜌2,...,𝜌𝑘) be a fuzzy
𝑖 ≤ 𝑘,appearsatleastoncein𝐺∗;(ii)betweeneachpairof graphstructureofagraphstructure𝐺∗ =(𝑈,𝐸1,𝐸2,...,𝐸𝑘).
vertices𝑢Vin𝑈,𝑢Visan𝐸𝑖-edgeforsome𝑖, 1 ≤ 𝑖 ≤ 𝑘.A Then 𝐹 = (],𝜏1,𝜏2,...,𝜏𝑘) is a partial fuzzy spanning
graph structure 𝐺∗ = (𝑈,𝐸1,𝐸2,...,𝐸𝑘) is connected, if the subgraphstructureof𝐺if𝜏𝑖 ⊆𝜌𝑖for𝑖=1,2,...,𝑘.
bowunenhtldiwyceher𝐸ely𝑖cn-ioenntdgwsgieogsstrvsafeoporrhftiosciosnemslcyo𝑢en𝐸𝑖na𝑖,-neeacddntegdVde,s.siiIfsmontrihalsaeogrmrplyaa,ept𝐸hh𝑖.𝑖-wsActryhugcirlcceahtupisrhceot,snht𝐸rseui𝑖s-cctptysauctorlheef ibDseesfiaanifdiuttziooznybg4era(asp𝜌eh𝑖e-es[dt8rg]u)e.cotLufer𝐺et.𝐺of∗𝐺b∗e.aIfg𝑥r𝑦ap∈hssutrpupc(t𝜌u𝑖r)e,tahnednl“e𝑥t𝑦𝐺”
tisshtraeutucrtneuedrieefrilityniidsnucgocgenrdnapebchyteis𝐸da𝑖a-tnerddegece.os𝐺ni∗tsaiiasnastnrne𝐸oe.𝑖c-Styricemlee,ioliafrrtelhyqe,us𝐺iuv∗bagliersnatpalnhy aDfeufiznziytiognra5ph(sseteru[8c]t)u.reTh𝐺eisst⋀re𝑛𝑗n=g1𝜌th𝑖(𝑥of𝑗−a1𝑥𝜌𝑖𝑗-)pfaotrh𝑖𝑥=0𝑥11,2⋅⋅,⋅.𝑥..𝑛,o𝑘f.
A𝐸in1d𝐸gur2ca⋅ep⋅dh⋅𝐸bs𝑗yt-rtu𝐸rce𝑖t-eue,driegfe𝐺iss∗isainas a𝐸fon𝑖r-ef𝐸sot𝑖r,-ettsrhte,aetiffiostr,hieefaisctuhhbg𝑗a,rsan1poh≤𝐸st𝑖r𝑗-ucyc≤ctuler𝑘se.. D𝜌𝑖e∘fi𝜌n𝑖i(t𝑥io𝑦n)6=(s⋁ee𝑧{[𝜌8𝑖]()𝑥.𝑧I)n∧a𝜌f𝑖u(𝑧z𝑦zy)}g,r𝜌a𝑖𝑗p(h𝑥𝑦s)tr=uc(t𝜌u𝑖𝑗r−e1𝐺∘,𝜌𝜌𝑖𝑖2)((𝑥𝑥𝑦𝑦))==
Let𝑆 ⊆ 𝑈;thenthesubgraphstructure⟨𝑆⟩inducedby𝑆has ⋁𝑧{𝜌𝑖𝑗−1(𝑥𝑧)∧𝜌𝑖(𝑧𝑦)},𝑗 = 2,3,...,𝑚,forany𝑚 ≥ 2.Also
vertexset𝑆,wheretwovertices𝑢andVin⟨𝑆⟩arejoinedby 𝜌∞(𝑥𝑦)=⋁{𝜌𝑗(𝑥𝑦), 𝑗=1,2,...}.
an 𝐸𝑖-edge, 1 ≤ 𝑖 ≤ 𝑘, if and only if, they are joined by 𝑖 𝑖
an 𝐸𝑖-edge in 𝐺∗. For some 𝑖, 1 ≤ 𝑖 ≤ 𝑘, the 𝐸𝑖-subgraph Definition 7 (see [8]). Let 𝑥𝑦 be a 𝜌𝑖-edge of 𝐺 =
ionfd𝐺u∗ce,djobinyi𝑆ngisvdeerntiocteesdibny𝑆𝐸.𝑖I-f⟨𝑆𝑇⟩.iIstahassuobnselyt tohfoesdeg𝐸e𝑖-seedtgiens s(]p,a𝜌n1n,i𝜌n2g,.s.u.b,𝜌g𝑛r)a.phLesttru(c]t,u𝜌r1󸀠e,𝜌o2󸀠b,t.a.i.n,e𝜌d𝑛󸀠)bybdeelaetipnagrt“i𝑥a𝑦l”fuwzizthy
𝐺∗,thensubgraphstructure⟨𝑇⟩inducedby𝑇hasthevertex 𝜌𝑖󸀠(𝑥𝑦) = 0and𝜌𝑖󸀠(𝑥1𝑦1) = 𝜌𝑖(𝑥1𝑦1) ∀𝜌𝑖-edges(𝑥1𝑦1)other
s𝐺ger∗at,p=“hths(te𝑈rue1cn,t𝐸du1rv,ee𝐸s.r2tTh,ic.e.e.sn,i𝐸𝐺n𝑚∗𝑇)a”na,ndwd𝐻h𝐻o∗s∗aeree=disg(oe𝑈ms2oa,r𝐸rep1󸀠h,tih𝐸co2,󸀠,sif.e(.ii.n),𝑚𝐸𝑇𝑛󸀠.=)Lb𝑛eet, 𝑥th𝑦anis(a𝑥𝜌,𝑖𝑦-)b.rIifd𝜌g𝑖∞e.(𝑢V)>𝜌𝑖󸀠∞(𝑢V)forsome𝑢V∈supp(𝜌𝑖),then
(𝜙ii):th{e𝐸r1e,𝐸ex2i,s.t..a,𝐸bi𝑛j}ect→ion 𝑓{𝐸1󸀠:,𝐸𝑈2󸀠1,..→.,𝐸𝑛󸀠𝑈},2saanyd𝐸a𝑖 b→ijecti𝐸o𝑗󸀠n, Dfuezfiznyistiuobng8ra(psehes[t8r]u)c.tuLreet𝐺ob󸀠t=ain(]e,d𝜌b1󸀠,y𝜌d2󸀠,e.le.t.i,n𝜌g𝑛󸀠)vberettehxe𝑤paorfti𝐺al,
1 ≤ 𝑖,𝑗 ≤ 𝑛, such that for all 𝑢,V ∈ 𝑈1, 𝑢V ∈ 𝐸𝑖 implies thatis,]󸀠(𝑤)=0and]󸀠(V)=](V) ∀V≠𝑤, 𝜌𝑖󸀠(V𝑤)=0 ∀V∈
that𝑓(𝑢)𝑓(V)∈𝐸𝑗󸀠. 𝑈𝑟 and 𝜌𝑖󸀠(𝑢V) = 𝜌𝑖(𝑢V) ∀𝑢V ≠ 𝑤V, 𝑖 = 1,2,...,𝑘. Then a
(𝑈,𝐸T1󸀠w,o𝐸2󸀠g,r.a.p.h,𝐸st𝑘󸀠r)u,cotnurtehse𝐺s∗am=e(v𝑈e,r𝐸te1x,𝐸se2t,.𝑈..,,a𝐸re𝑘)idaenndti𝐻ca∗l,=if v𝑢e,rVtewxit𝑤ho𝑢f,V𝐺=is̸𝑤a.𝜌𝑖-cutvertexif𝜌𝑖∞(𝑢V)>𝜌𝑖󸀠∞(𝑢V)forsome
thereexistsabijection𝑓:𝑈 → 𝑈,suchthatforall𝑢andVin
𝑈,𝑢Visan𝐸𝑖-edgein𝐺∗,then𝑓(𝑢)𝑓(V)isan𝐸𝑖󸀠-edgein𝐻∗, Definition 9 (see [8]). 𝐺 = (],𝜌1,𝜌2,...,𝜌𝑘) is a 𝜌𝑖-cycle if
where1 ≤ 𝑖 ≤ 𝑘and𝐸𝑖 ≃ 𝐸𝑖󸀠 ∀𝑖.Let𝜙beapermutationon andonlyif(supp(]),supp(𝜌1),supp(𝜌2),...,supp(𝜌𝑘))isa𝐸𝑖-
{𝐸1,𝐸2,...,𝐸𝑘}.Thenthe𝜙-cycliccomplementof𝐺∗,denoted cycle.
pbLyeert(m𝐺𝐺u∗∗t)a𝜙t=𝑐i,oins(𝑈oo,nb𝐸t{a1𝐸i,n1𝐸,e2𝐸d,2.b,..y..,r.𝐸e,𝑘p𝐸)l𝑘ab}c;eitnhageng𝐸r𝑖abpyh𝜙st(r𝐸u𝑖c)t,u1re≤an𝑖d≤𝜙𝑘a. iDcsyeacfinlen𝐸iift𝑖i-aocnnydc1lo0ena(lnysedieft(h[s8eu]rp)ep.e(𝐺x]i)s,t=ssunp(o]p,u(𝜌𝜌n11i,)q𝜌,us2eu,.p“.𝑥p.𝑦(,𝜌”𝜌2𝑘i)n),.sis.u.pa,psfu(u𝜌pz𝑖p)z(ys𝜌u𝑘𝜌c)𝑖h-)
(i)𝐺∗ is 𝜙-self complementary, if 𝐺∗ is isomorphic to that𝜌𝑖(𝑥𝑦)=⋀{𝜌𝑖(𝑢V)|𝑢V∈supp(𝜌𝑖)}.
(c𝐺om∗)p𝜙l𝑐e;mthenet𝜙,i-fcy𝜙cl=ic̸idcoemntpitleympeenrmtoufta𝐺ti∗onan.d𝐺∗ isself- Definition 11 (see [8]). 𝐺 = (],𝜌1,𝜌2,...,𝜌𝑘) is a fuzzy 𝜌𝑖-
tree if it has a partial fuzzy spanning subgraph structure,
(ii)𝐺∗isstrong𝜙-selfcomplementary,if𝐺∗isidenticalto 𝐹𝑖 =(],𝜏1,𝜏2,...,𝜏𝑘),whichisa𝜏𝑖-treewhereforall𝜌𝑖-edges
(𝐺∗)𝜙𝑐;the𝜙-complementof𝐺∗and𝐺∗isstrongself- notin𝐹𝑖,𝜌𝑖(𝑥𝑦)<𝜏𝑖∞(𝑥𝑦).
complement,if𝜙≠identitypermutation.
Definition 12 (see [8]). Let 𝐺∗ = (𝑈,𝐸1,𝐸2,...,𝐸𝑘) be a
Definition 1 (see [2]). A fuzzy subset 𝜇 on a set 𝑋 is a map graph structure and let ],𝜌1,𝜌2,...,𝜌𝑘 be the fuzzy subsets
𝜇:𝑋 → [0,1].Afuzzybinaryrelationon𝑋isafuzzysubset of𝑈,𝐸1,𝐸2,...,𝐸𝑘,respectively,suchthat
𝜇on𝑋×𝑋.Byafuzzyrelationwemeanafuzzybinaryrelation
0≤𝜌 (𝑥𝑦)≤𝜇(𝑥)∧𝜇(𝑦)
givenby𝜇:𝑋×𝑋 → [0,1]. 𝑖
(2)
∀𝑥,𝑦∈𝑉, 𝑖=1,2,...,𝑘.
gDreafipnhitsiotrnuc2tu(rseeean[d8]l)e.tL]e,t𝜌1𝐺,𝜌∗2,.=..,(𝜌𝑈𝑘,b𝐸e1,t𝐸he2,f.u.z.z,y𝐸𝑘s)ubbseetas Then𝐺=(],𝜌1,𝜌2,...,𝜌𝑘)isafuzzygraphstructureof𝐺∗.
of𝑈,𝐸1,𝐸2,...,𝐸𝑘,respectively,suchthat Definition 13 (see [3]). Let 𝑋 be a nonempty set. A bipolar
0≤𝜌 (𝑥𝑦)≤𝜇(𝑥)∧𝜇(𝑦) fuzzyset𝐵in𝑋isanobjecthavingtheform
𝑖
(1)
∀𝑥,𝑦∈𝑈, 𝑖=1,2,...,𝑘. 𝐵={(𝑥,𝜇𝑃(𝑥),𝜇𝑁(𝑥))|𝑥∈𝑋}, (3)
𝐵 𝐵
Then𝐺=(],𝜌1,𝜌2,...,𝜌𝑘)isafuzzygraphstructureof𝐺∗. where𝜇𝐵𝑃 :𝑋 → [0,1]and𝜇𝐵𝑁 :𝑋 → [−1,0]aremappings.
AppliedComputationalIntelligenceandSoftComputing 3
Weusethepositivemembershipdegree𝜇𝐵𝑃(𝑥)todenote If𝐻̌𝑏 =(𝑀󸀠,𝑁1󸀠,𝑁2󸀠,...,𝑁𝑛󸀠)isabipolarfuzzygraphstructure
the satisfaction degree of an element 𝑥 to the property of𝐺∗suchthat
correspondingtoabipolarfuzzyset𝐵andthenegativemem-
bershipdegree𝜇𝑁(𝑥)todenotethesatisfactiondegreeofan 𝜇𝑀𝑃󸀠(𝑥)≤𝜇𝑀𝑃 (𝑥),
𝐵
element 𝑥 to some implicit counterproperty corresponding 𝜇𝑁 (𝑥)≥𝜇𝑁(𝑥)
toabipolarfuzzyset𝐵.If𝜇𝑃(𝑥) ≠ 0and𝜇𝑁(𝑥) = 0,itisthe 𝑀󸀠 𝑀
𝐵 𝐵
situationthat𝑥isregardedashavingonlypositivesatisfaction ∀𝑥∈𝑈,
fdoore𝐵s.nIoft𝜇s𝐵𝑃at(i𝑥s)fy=th0eapnrodp𝜇e𝐵𝑁rt(y𝑥o)f=𝐵̸ 0b,uittsiosmtheewshitautastaitoinsfitehsatth𝑥e 𝜇𝑁𝑃󸀠(𝑥𝑦)≤𝜇𝑁𝑃 (𝑥𝑦), (7)
counter property of 𝐵. It is possible for an element 𝑥 to be 𝑖 𝑖
suchthat𝜇𝐵𝑃(𝑥) ≠ 0and𝜇𝐵𝑁(𝑥) ≠ 0whenthemembership 𝜇𝑁𝑁𝑖󸀠(𝑥𝑦)≥𝜇𝑁𝑁𝑖(𝑥𝑦)
functionofthepropertyoverlapsthatofitscounterproperty
oversomeportionof𝑋. ∀𝑥𝑦∈𝐸𝑖, 𝑖=1,2,...,𝑛,
For the sake of simplicity, we will use the symbol 𝐵 =
(𝜇𝑃,𝜇𝑁)forthebipolarfuzzyset: then𝐻̌𝑏 iscalledabipolarfuzzysubgraphstructureofBFGS
𝐵 𝐵 𝐺̌𝑏.
𝐵={(𝑥,𝜇𝐵𝑃(𝑥),𝜇𝐵𝑁(𝑥))|𝑥∈𝑋}. (4) BFGS 𝐻̌𝑏 = (𝑀󸀠,𝑁1󸀠,𝑁2󸀠,...,𝑁𝑛󸀠) is a bipolar fuzzy
Definition14(see[3]). Let𝑋beanonemptyset.Thenwecall iansduubcseedts𝑊ubogfra𝑈phifstructureof𝐺̌𝑏 = (𝑀,𝑁1,𝑁2,...,𝑁𝑛),by
amapping𝐴=(𝜇𝑃,𝜇𝑁):𝑋×𝑋 → [0,1]×[−1,0]abipolar
fuzzyrelationon𝑋𝐴suc𝐴hthat𝜇𝐴𝑃(𝑥,𝑦)∈[0,1]and𝜇𝐴𝑁(𝑥,𝑦)∈ 𝜇𝑀𝑃󸀠(𝑥)=𝜇𝑀𝑃 (𝑥),
[−1,0].
𝜇𝑁 (𝑥)=𝜇𝑁(𝑥)
𝑀󸀠 𝑀
Definition15(see[9]). Abipolarfuzzygraph𝐺 = (𝑉,𝐴,𝐵) ∀𝑥∈𝑊,
is a nonempty set 𝑉 together with a pair of functions 𝐴 =
[(0𝜇,𝐴𝑃1,]𝜇×𝐴𝑁)[−:1𝑉,0→]su[c0h,1th]a×t[fo−r1a,0ll]𝑥a,n𝑦d∈𝐵𝑉=,(𝜇𝐵𝑃,𝜇𝐵𝑁):𝑉×𝑉 → 𝜇𝑁𝑃𝑖󸀠(𝑥𝑦)=𝜇𝑁𝑃𝑖(𝑥𝑦), (8)
𝜇𝑁 (𝑥𝑦)=𝜇𝑁 (𝑥𝑦)
𝜇𝐵𝑃(𝑥,𝑦)≤min(𝜇𝐴𝑃(𝑥),𝜇𝐴𝑃(𝑦)), 𝑁𝑖󸀠 𝑁𝑖
(5) ∀𝑥,𝑦∈𝑊, 𝑖=1,2,...,𝑛.
𝜇𝑁(𝑥,𝑦)≥max(𝜇𝑁(𝑥),𝜇𝑁(𝑦)).
𝐵 𝐴 𝐴
Similarly, BFGS 𝐻̌𝑏 is a bipolar fuzzy spanning subgraph
Noticethat𝜇𝐵𝑃(𝑥,𝑦)>0,𝜇𝐵𝑁(𝑥,𝑦)<0for(𝑥,𝑦)∈𝑉×𝑉, structureof𝐺̌𝑏if𝑀󸀠 =𝑀and
𝜇𝑃(𝑥,𝑦)=𝜇𝑁(𝑥,𝑦)=0for(𝑥,𝑦)∉𝑉×𝑉,and𝐵issymmetric
𝐵 𝐵 𝜇𝑃 ≤𝜇𝑃 ,
relation. 𝑁󸀠 𝑁
𝑖 𝑖
𝜇𝑁 ≥𝜇𝑁, (9)
3.BipolarFuzzyGraphStructures 𝑁󸀠 𝑁
𝑖 𝑖
Definition 16. 𝐺̌𝑏 = (𝑀,𝑁1,𝑁2,...,𝑁𝑛) is called a bipolar 𝑖=1,2,...,𝑛.
fa(u𝑈nzd,z𝐸yf1og,rr𝐸ae2pa,ch.h.s.t𝑖,r𝐸u=c𝑛t)1u,irf2e𝑀,(.B..=F,G𝑛(𝜇;S𝑁𝑀𝑃)o,𝑖f𝜇=𝑀a𝑁(g)𝜇ri𝑁𝑃asp𝑖a,h𝜇b𝑁𝑁sipt𝑖r)ouliacsrtaufurbezipz(oyGlsaSer)tf𝐺oun∗zz𝑈=y {sE𝑎ux3ca𝑎hm4,tp𝑎hl1ae𝑎t41𝑈8}.. C=o{n𝑎s1i,d𝑎e2r,𝑎a3,g𝑎r4a}p,h𝐸s1tru=ct{u𝑎r1e𝑎2𝐺,𝑎∗2𝑎=4},(a𝑈n,d𝐸1𝐸,2𝐸2=)
seton𝐸𝑖suchthat (i) Let 𝑀,𝑁1, and 𝑁2 be bipolar fuzzy subsets of 𝑈,𝐸1,
𝜇𝑃 (𝑥𝑦)≤𝜇𝑃 (𝑥)∧𝜇𝑃 (𝑦), and𝐸2,respectively,suchthat
𝑁 𝑀 𝑀
𝑖 𝑀={(𝑎 ,0.5,−0.2),(𝑎 ,0.7,−0.3),(𝑎 ,0.4,−0.3),
1 2 3
𝜇𝑁 (𝑥𝑦)≥𝜇𝑁(𝑥)∨𝜇𝑁(𝑦) (6)
𝑁 𝑀 𝑀
𝑖 (𝑎 ,0.7,−0.3)},
4
∀𝑥𝑦∈𝐸 ⊂𝑈×𝑈. (10)
𝑖 𝑁 ={(𝑎 𝑎 ,0.5,−0.2),(𝑎 𝑎 ,0.7,−0.3)},
1 1 2 2 4
Note that 𝜇𝑁𝑃 (𝑥𝑦) = 0 = 𝜇𝑁𝑁(𝑥𝑦) for all 𝑥𝑦 ∈ 𝑈 × 𝑈 − 𝐸𝑖 𝑁 ={(𝑎 𝑎 ,0.3,−0.2),(𝑎 𝑎 ,0.3,−0.1)}.
and0 < 𝜇𝑁𝑃 (𝑖𝑥𝑦) ≤ 1,−1 ≤ 𝑖𝜇𝑁𝑁(𝑥𝑦) < 0 ∀𝑥𝑦 ∈ 𝐸𝑖,where 2 3 4 1 4
𝑈and𝐸𝑖 (𝑖 𝑖= 1,2,...,𝑛)arecall𝑖edunderlyingvertexsetand Then, by direct calculations, it is easy to see that 𝐺̌𝑏 =
underlying𝑖-edgesetof𝐺̌𝑏,respectively. (𝑀,𝑁1,𝑁2)isaBFGSof𝐺∗asshowninFigure1.
(ii) Consider 𝑀1 = {(𝑎1,0.4,−0.1),(𝑎2,0.5,−0.3),(𝑎3,
Definition17. Let𝐺̌𝑏 =(𝑀,𝑁1,𝑁2,...,𝑁𝑛)beabipolarfuzzy 0.4,−0.2),(𝑎4,0.1,−0.3)}, 𝑁11 = {(𝑎1𝑎2,0.4,−0.1),(𝑎2𝑎4,0.1,
graphstructureofagraphstructure𝐺∗ = (𝑈,𝐸1,𝐸2,...,𝐸𝑛). −0.2)}, and 𝑁12 = {(𝑎3𝑎4,0.1,−0.2),(𝑎1𝑎4,0.1,−0.0)}. Then,
4 AppliedComputationalIntelligenceandSoftComputing
a1(0.5,−0.2) a2(0.7,−0.3) ABFGS𝐺̌𝑏 = (𝑀,𝑁1,𝑁2,...,𝑁𝑛)issaidtobestrong ifitis
N(0.5,−0.2) 𝑁𝑖-strongBFGSforall𝑖∈{1,2,3,...,𝑛}.
1
Example22. ConsiderBFGS𝐺̌𝑏 = (𝑀,𝑁1,𝑁2)asshownin
1) Figure3.
(0.3,−0. (0.7,−0.3) stroThng.en 𝐺̌𝑏 is a strong BFGS since it is both 𝑁1- and 𝑁2-
N2 N 1
Definition 23. A BFGS 𝐺̌𝑏 = (𝑀,𝑁1,𝑁2,...,𝑁𝑛) with
N2(0.3,−0.2) underlyingvertexset𝑈issaidtobecompleteor𝑁1𝑁2⋅⋅⋅𝑁𝑛-
a (0.7,−0.3) complete,ifthefollowingaretrue:
4 a (0.4,−0.3)
3
Figure1:𝐺̌𝑏=(𝑀,𝑁1,𝑁2). (i)𝐺̌𝑏aisstrongBFGS.
(ii)supp(𝑁𝑖)≠0 ∀𝑖=1,2,3,...,𝑛.
a1(0.4,−0.1) a2(0.5,−0.3) (iii)Foreachpairofvertices𝑥,𝑦 ∈ 𝑈, 𝑥𝑦isan𝑁𝑖-edge
forsome𝑖.
N (0.4,−0.1)
11
N(0.1,−0.0)12 N 1(10.1,−0.2) {csE𝑎taxr2luac𝑎Mmuc3t}lpuao,tlarreieenoodn𝐺2vs4𝐸e∗,.r2i,t=Ls=ieust(p{e𝑈𝑎pa𝐺1,(s𝑎̌𝐸𝑏𝑁y21,t1,o𝑎)=𝐸1s=𝑎2e)̸3e0}(s,t𝑀aushsuca,hspt𝑁hp𝐺to1(ȟ𝑏w,𝑁a𝑁itn2s2)𝑈ia)n=s̸Fb=t0rieo,g{anu𝑎Bngr1Fed,BG𝑎4eF2S.v,GBe𝑎rSoy3y.}fr,pogau𝐸rita1ripno=hef
verticesbelongingto𝑈iseitheran𝑁1-edgeoran𝑁2-edge.
N12(0.1,−0.2) So𝐺̌𝑏isacompleteBFGS,thatis,𝑁1𝑁2-completeBFGS.
a4(0.1,−0.3) a3(0.4,−0.2) Definition25. Let𝐺̌𝑏 = (𝑀,𝑁1,𝑁2,...,𝑁𝑛)beaBFGSwith
Figure2:Bipolarfuzzysubgraphstructure𝐾̌𝑏=(𝑀1,𝑁11,𝑁12). underlyingvertexset𝑈.Thenpositiveandnegativestrengths
ofa𝑁𝑖-path“𝑃𝑁 =𝑎1𝑎2⋅⋅⋅𝑎𝑚”arecalledgainandlossofthat
𝑁𝑖-path and den𝑖oted by 𝐺.𝑃𝑁 and 𝐿.𝑃𝑁, respectively, such
by routine calculations, it is easy to see that 𝐾̌𝑏 = that 𝑖 𝑖
(𝑀1,𝑁11,𝑁12)isthebipolarfuzzysubgraphstructureof𝐺̌𝑏 𝑚
𝐺.𝑃 =⋀[𝜇𝑃 (𝑎 𝑎 )],
asshowninFigure2. 𝑁 𝑁 𝑗−1 𝑗
𝑖 𝑖
𝑗=2
DThgreaefipnnhi𝑥tsi𝑦tornu∈c1t9𝐸u.r𝑖eLiseotcfa𝐺aľ𝑏lger=dapa(𝑀hbis,pt𝑁orul1ac,rt𝑁ufur2e,z.z𝐺.y.∗,𝑁𝑁=𝑖-𝑛e()𝑈dbg,ee𝐸ao1b,r𝐸isp2iom,l.ap.r.lyf,u𝐸𝑁z𝑛z𝑖)y-. 𝐿.𝑃𝑁𝑖 =󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨𝑗⋁𝑚=2𝜇𝑁𝑁𝑖(𝑎𝑗−1𝑎𝑗)󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨. (14)
edge,if
𝜇𝑁𝑃𝑖(𝑥𝑦)>0 or iEnxaFmigpulree246..WCeonnositdeetrhaatB𝑃F𝑁GS=𝐺̌𝑎𝑏1𝑎=3𝑎(4𝑀𝑎1,𝑁is1a,n𝑁𝑁2)2a-psasthho.wSno
(11) 2
𝜇𝑁 (𝑥𝑦)<0. 𝐺.𝑃𝑁 =𝜇𝑁𝑃 (𝑎3𝑎1)∧𝜇𝑁𝑃 (𝑎1𝑎2)=0.5∧0.4=0.4.Consider
𝑁 2 2 2
𝑖
Thensupportof𝑁𝑖,𝑖=1,2,...,𝑛,consequently,is 𝐿.𝑃𝑁 =󵄨󵄨󵄨󵄨󵄨𝜇𝑁𝑁 (𝑎3𝑎1)∨𝜇𝑁𝑁 (𝑎1𝑎2)󵄨󵄨󵄨󵄨󵄨=|−0.4∨−0.4|
2 2 2
(15)
supp(𝑁𝑖)={𝑥𝑦∈𝐸𝑖 :𝜇𝑁𝑃 (𝑥𝑦)>0, 𝜇𝑁𝑁 (𝑥𝑦)<0}. (12) =|−0.4|=0.4.
𝑖 𝑖
Definition20. 𝑁𝑖-pathinaBFGS𝐺̌𝑏 = (𝑀,𝑁1,𝑁2,...,𝑁𝑛) Definition27. Let𝐺̌𝑏 = (𝑀,𝑁1,𝑁2,...,𝑁𝑛)beaBFGSwith
of a graph structure 𝐺∗ = (𝑈,𝐸1,𝐸2,...,𝐸𝑛) is a sequence underlyingvertexset𝑈.Then
2𝑎in1,,3𝑈𝑎,2.,,.s..u.,.c𝑚,h𝑎.𝑚thoatfd𝑎𝑗i−st1i𝑎n𝑗ctisvaerbtiicpeosla(rexfcuezpztyt𝑁he𝑖-cehdogeicfeo𝑎r𝑚al=l 𝑗𝑎1=) (i)b𝑁y𝑖-𝜇g𝑁a∞in,+(o𝑥f𝑦c)on=ne⋁ct𝑗e≥d1n{𝜇es𝑁𝑗s,+b(𝑥e𝑦tw)}e,esnuc𝑥hatnhdat𝑦𝜇i𝑁𝑗s,+d(e𝑥fi𝑦n)e=d
𝑖 𝑖 𝑖
(𝜇𝑗−1,+ ∘ 𝜇1,+)(𝑥𝑦) for 𝑗 ≥ 2 and 𝜇2,+(𝑥𝑦) = (𝜇1,+ ∘
Du{1ne,dfi2en,r3ilt,yi.oi.nn.g,2𝑛v1}e.riftAefxorBseaFtlGl𝑈𝑥S𝑦i𝐺s∈̌s𝑏asuidp=pto(𝑁b(𝑀𝑖e)𝑁,𝑁𝑖-s1t,r𝑁on2g,.f.o.r,𝑁so𝑛m) ew𝑖ith∈ 𝜇𝜇𝑁𝑁1𝑃𝑁,+𝑖𝑖,𝑖)(∀𝑥𝑖𝑦.) 𝑁=𝑖 ⋁𝑧{𝜇𝑁1,+𝑖 (𝑥𝑧) ∧ 𝜇𝑁1,+𝑖 (𝑧𝑦𝑁)}𝑖, where 𝜇𝑁1,𝑁+𝑖 𝑖 =
𝜇𝑃 (𝑥𝑦)=𝜇𝑃 (𝑥)∧𝜇𝑃 (𝑦), (ii)𝑁𝑖-loss of connectedness between 𝑥 and 𝑦 is defined
𝑁𝑖 𝑀 𝑀 by𝜇∞,−(𝑥𝑦) = ⋁ {𝜇𝑗,−(𝑥𝑦)},suchthat𝜇𝑗,−(𝑥𝑦) =
(13) 𝑁𝑖 𝑗≥1 𝑁𝑖 𝑁𝑖
𝜇𝑁 (𝑥𝑦)=𝜇𝑁(𝑥)∨𝜇𝑁(𝑦). (𝜇𝑗−1,− ∘ 𝜇1,−)(𝑥𝑦) for 𝑗 ≥ 2 and 𝜇2,−(𝑥𝑦) = (𝜇1,− ∘
𝑁 𝑀 𝑀 𝑁 𝑁 𝑁 𝑁
𝑖 𝑖 𝑖 𝑖 𝑖
AppliedComputationalIntelligenceandSoftComputing 5
a1(0.3,−0.6) N(0.3,−0.6) a2(0.3,−0.7)
2
a4(0.4,−0.4) NN1(0.(30,.−2,0−.40).4) a3(0.2,−0.6) N2(0.2, −0.6)
1
4)
0.3,−0.4) N(0.1,−0.6)1 (0.2,−0.6)N1 N(0.2,−0.2
(
a (0.3,−0N.25) N2(0.1a,8−(00..51),−a06.8(0).6,−0.6) N1N(01.(10.,2−,0−.40.)4) a7(0.2,−0.4)
5 N(0.3,−0.5)
2
Figure3:BFGS𝐺̌𝑏=(𝑀,𝑁1,𝑁2).
a (0.5,−0.4) ∨[𝜇1,+(𝑎 𝑎 )∧𝜇1,+(𝑎 𝑎 )]
1 𝑁 2 5 𝑁 5 3
1 1
N 2(0.5,−0.4) N2(0.4,−0.4) 𝜇𝑁2,+(𝑎2𝑎4)==[(0𝜇.10,+∧𝑛𝑁0.5∘]𝜇∨𝑁1,[+0).4(𝑎∧2𝑎04.0)]=0,
1 1 1
=[𝜇2,+(𝑎 𝑎 )∧𝜇1,+(𝑎 𝑎 )]
a (0.5,−0.7) N(0.4,−0.7) a (0.4,−0.7) 𝑁1 2 3 𝑁1 3 4
3 1 2
Figure4:𝐺̌𝑏 =(𝑀,𝑁1,𝑁2). ∨[𝜇𝑁1,+1 (𝑎2𝑎5)∧𝜇𝑁1,+1 (𝑎5𝑎4)]
=[0.3∧0.5]∨[0.4∧0.3]=0.3,
a4(0.6,−0.9) N1(0.5,−0.7) a3(0.5,−0.7) 𝜇2,+(𝑎 𝑎 )=(𝜇1,+𝑛 ∘𝜇1,+)(𝑎 𝑎 )
a5(0.4,−0.5) N(10.3,−0.3) N(0.1,−0.4)2 N(0.2,−0.4)2 N1(0.3,−0.5)a2(0.4,−0.7) 𝑁1 2 5 =[∨𝜇𝑁1[,𝜇+1𝑁1(,𝑁𝑎+121(𝑎𝑎32)𝑎𝑁∧4)1𝜇∧𝑁1,𝜇+1𝑁21(,𝑎+153(𝑎𝑎54)𝑎]5)]
N2(0.1,−0.4) N1(0.4,−0.4) N(20.2,−0.6) 𝜇𝑁2,+1 (𝑎3𝑎4)==[(0𝜇.13,+∧𝑛𝑁01.0∘]𝜇∨𝑁1,[+10).0(𝑎∧3𝑎04.3)]=0,
a6(0.3,−0.4) N1(0.3,−0.2) a1(0.3,−0.6) =[𝜇1,+(𝑎 𝑎 )∧𝜇1,+(𝑎 𝑎 )]
Figure5:𝐺̌𝑏=(𝑀,𝑁1,𝑁2). 𝑁1 3 2 𝑁1 2 4
∨[𝜇1,+(𝑎 𝑎 )∧𝜇1,+(𝑎 𝑎 )]
𝑁 3 5 𝑁 5 4
1 1
𝜇1,−)(𝑥𝑦) = ⋁ {𝜇1,−(𝑥𝑧) ∧ 𝜇1,−(𝑧𝑦)}, where 𝜇1,− = =[0.3∧0.0]∨[0.0∧0.3]=0,
𝑁 𝑧 𝑁 𝑁 𝑁
𝑖 𝑖 𝑖 𝑖
|𝜇𝑁|, ∀𝑖. 𝜇2,+(𝑎 𝑎 )=(𝜇1,+𝑛 ∘𝜇1,+)(𝑎 𝑎 )
𝑁𝑖 𝑁1 3 5 𝑁1 𝑁1 3 5
Example28. Let𝐺̌𝑏 = (𝑀,𝑁1,𝑁2)beBFGSofgraphstruc- =[𝜇𝑁1,+(𝑎3𝑎2)∧𝜇𝑁1,+(𝑎2𝑎5)]
ture𝐺 = (𝑈,𝐸1,𝐸2)suchthat𝑈 = {𝑎1,𝑎2,𝑎3,𝑎4,𝑎5,𝑎6},𝐸1 = 1 1
{𝑎1𝑎6,𝑎2𝑎3,𝑎2𝑎5,𝑎3𝑎4,𝑎4𝑎5},and𝐸2 ={𝑎1𝑎3,𝑎1𝑎2,𝑎4𝑎6,𝑎5𝑎6}, ∨[𝜇𝑁1,+(𝑎3𝑎4)∧𝜇𝑁1,+(𝑎4𝑎5)]
1 1
asisshowninFigure5.
Since𝜇𝑁1,+(𝑎2𝑎3) = 0.3,𝜇𝑁1,+(𝑎2𝑎4) = 0.0,𝜇𝑁1,+(𝑎2𝑎5) = 0.4, =[0.3∧0.4]∨[0.5∧0.3]=0.3,
1 1 1
𝜇𝑁1,+(𝑎3𝑎4) = 0.5, 𝜇𝑁1,+(𝑎5𝑎3) = 0.0, 𝜇𝑁1,+(𝑎4𝑎5) = 0.3, and 𝜇2,+(𝑎 𝑎 )=(𝜇1,+𝑛 ∘𝜇1,+)(𝑎 𝑎 )
𝜇𝑁1,+1(𝑎1𝑎6)=0.3,there1fore 1 𝑁1 4 5 𝑁1 𝑁1 4 5
1 =[𝜇1,+(𝑎 𝑎 )∧𝜇1,+(𝑎 𝑎 )]
𝑁 4 2 𝑁 2 5
1 1
𝜇2,+(𝑎 𝑎 )=(𝜇1,+𝑛 ∘𝜇1,+)(𝑎 𝑎 )
𝑁1 2 3 𝑁1 𝑁1 2 3 ∨[𝜇𝑁1,+(𝑎4𝑎3)∧𝜇𝑁1,+(𝑎3𝑎5)]
1 1
=[𝜇𝑁1,+(𝑎2𝑎4)∧𝜇𝑁1,+(𝑎4𝑎3)] =[0.0∧0.4]∨[0.5∧0.0]=0,
1 1
6 AppliedComputationalIntelligenceandSoftComputing
𝜇2,+(𝑎 𝑎 )=(𝜇1,+𝑛 ∘𝜇1,+)(𝑎 𝑎 )=0, 𝜇4,+(𝑎 𝑎 )=(𝜇1,+𝑛 ∘𝜇1,+)(𝑎 𝑎 )=0,
𝑁 1 6 𝑁 𝑁 1 6 𝑁 3 4 𝑁 𝑁 3 4
1 1 1 1 1 1
𝜇𝑁3,+1 (𝑎2𝑎3)=(𝜇2,+𝑛𝑁1 ∘𝜇𝑁1,+1)(𝑎2𝑎3) 𝜇𝑁4,+1 (𝑎3𝑎5)=(𝜇1,+𝑛𝑁1 ∘𝜇𝑁1,+1)(𝑎3𝑎5)=0.3,
=[𝜇2,+(𝑎 𝑎 )∧𝜇1,+(𝑎 𝑎 )]
𝑁 2 4 𝑁 4 3 𝜇4,+(𝑎 𝑎 )=(𝜇1,+𝑛 ∘𝜇1,+)(𝑎 𝑎 )=0,
1 1 𝑁 4 5 𝑁 𝑁 4 5
1 1 1
∨[𝜇2,+(𝑎 𝑎 )∧𝜇1,+(𝑎 𝑎 )]
𝑁1 2 5 𝑁1 5 3 𝜇4,+(𝑎 𝑎 )=(𝜇1,+𝑛 ∘𝜇1,+)(𝑎 𝑎 )=0.
𝑁 1 6 𝑁 𝑁 1 6
=[0.3∧0.5]∨[0.0∧0.0]=0.3, 1 1 1
(17)
𝜇3,+(𝑎 𝑎 )=(𝜇2,+𝑛 ∘𝜇1,+)(𝑎 𝑎 ) Thisimpliesthat
𝑁 2 4 𝑁 𝑁 2 4
1 1 1
𝜇∞,+(𝑎 𝑎 )=∨{0.3,0.0,0.3,0.0}=0.3,
=[𝜇2,+(𝑎 𝑎 )∧𝜇1,+(𝑎 𝑎 )] 𝑁1 2 3
𝑁 2 3 𝑁 3 4
1 1
𝜇∞,+(𝑎 𝑎 )=∨{0.0,0.3,0.0,0.3}=0.3,
∨[𝜇2,+(𝑎 𝑎 )∧𝜇1,+(𝑎 𝑎 )] 𝑁 2 4
𝑁 2 5 𝑁 5 4 1
1 1
=[0.0∧0.5]∨[0.0∧0.3]=0.0, 𝜇∞,+(𝑎 𝑎 )=∨{0.4,0.0,0.3,0.0}=0.4,
𝑁 2 5
1
𝜇𝑁3,+1 (𝑎2𝑎5)=(𝜇2,+𝑛𝑁1 ∘𝜇𝑁1,+1)(𝑎2𝑎5) 𝜇𝑁∞1,+(𝑎3𝑎4)=∨{0.5,0.0,0.3,0.0}=0.5, (18)
=[𝜇𝑁2,+(𝑎2𝑎3)∧𝜇𝑁1,+(𝑎3𝑎5)] 𝜇∞,+(𝑎 𝑎 )=∨{0.0,0.3,0.0,0.3}=0.3,
1 1 𝑁 3 5
1
∨[𝜇2,+(𝑎 𝑎 )∧𝜇1,+(𝑎 𝑎 )]
𝑁 2 4 𝑁 4 5 𝜇∞,+(𝑎 𝑎 )=∨{0.3,0.0,0.3,0.0}=0.3,
1 1 𝑁 4 5
1
=[0.0∧0.0]∨[0.3∧0.3]=0.3,
𝜇∞,+(𝑎 𝑎 )=∨{0.3,0.0,0.0,0.0}=0.3.
𝑁 1 6
𝜇3,+(𝑎 𝑎 )=(𝜇2,+𝑛 ∘𝜇1,+)(𝑎 𝑎 ) 1
𝑁1 3 4 𝑁1 𝑁1 3 4 Since
=[𝜇𝑁2,+1 (𝑎3𝑎2)∧𝜇𝑁1,+1 (𝑎2𝑎4)] 𝜇𝑁1,−1 (𝑎2𝑎3)=0.5,
∨[𝜇2,+(𝑎 𝑎 )∧𝜇1,+(𝑎 𝑎 )] 𝜇1,−(𝑎 𝑎 )=0.0,
𝑁 3 5 𝑁 5 4 𝑁 2 4
1 1 1
=[0.0∧0.0]∨[0.3∧0.3]=0.3, 𝜇1,−(𝑎 𝑎 )=0.4,
𝑁 2 5
1
𝜇3,+(𝑎 𝑎 )=(𝜇2,+𝑛 ∘𝜇1,+)(𝑎 𝑎 )
𝑁1 3 5 𝑁1 𝑁1 3 5 𝜇1,−(𝑎 𝑎 )=0.7, (19)
𝑁 3 4
1
=[𝜇2,+(𝑎 𝑎 )∧𝜇1,+(𝑎 𝑎 )]
𝑁 3 2 𝑁 2 5
1 1 𝜇1,−(𝑎 𝑎 )=0.0,
𝑁 5 3
∨[𝜇2,+(𝑎 𝑎 )∧𝜇1,+(𝑎 𝑎 )] 1
𝑁 3 4 𝑁 4 5
1 1 𝜇1,−(𝑎 𝑎 )=0.3,
=[0.0∧0.4]∨[0.0∧0.3]=0.0, 𝑁1 4 5
𝜇1,−(𝑎 𝑎 )=0.2,
𝜇3,+(𝑎 𝑎 )=(𝜇2,+𝑛 ∘𝜇1,+)(𝑎 𝑎 ) 𝑁 1 6
𝑁 4 5 𝑁 𝑁 4 5 1
1 1 1
wehave
=[𝜇2,+(𝑎 𝑎 )∧𝜇1,+(𝑎 𝑎 )]
𝑁 4 2 𝑁 2 5 𝜇2,−(𝑎 𝑎 )=(𝜇1,−𝑛 ∘𝜇1,−)(𝑎 𝑎 )
1 1 𝑁 2 3 𝑁 𝑁 2 3
1 1 1
∨[𝜇2,+(𝑎 𝑎 )∧𝜇1,+(𝑎 𝑎 )]
𝑁 4 3 𝑁 3 5 =[𝜇1,−(𝑎 𝑎 )∧𝜇1,−(𝑎 𝑎 )]
1 1 𝑁 2 4 𝑁 4 3
1 1
=[0.3∧0.4]∨[0.0∧0.0]=0.3,
∨[𝜇1,−(𝑎 𝑎 )∧𝜇1,−(𝑎 𝑎 )]
𝑁 2 5 𝑁 5 3
𝜇3,+(𝑎 𝑎 )=(𝜇2,+𝑛 ∘𝜇1,+)(𝑎 𝑎 )=0. 1 1
𝑁1 1 6 𝑁1 𝑁1 1 6 =[0.0∧0.7]∨[0.4∧0.0]=0.0,
(16)
𝜇2,−(𝑎 𝑎 )=(𝜇1,−𝑛 ∘𝜇1,−)(𝑎 𝑎 )
𝑁 2 4 𝑁 𝑁 2 4
Similarly, 1 1 1
𝜇𝑁4,+1 (𝑎2𝑎3)=(𝜇1,+𝑛𝑁1 ∘𝜇𝑁1,+1)(𝑎2𝑎3)=0, =[𝜇𝑁2,−1 (𝑎2𝑎3)∧𝜇𝑁1,−1 (𝑎3𝑎4)]
𝜇4,+(𝑎 𝑎 )=(𝜇1,+𝑛 ∘𝜇1,+)(𝑎 𝑎 )=0.3, ∨[𝜇1,−(𝑎 𝑎 )∧𝜇1,−(𝑎 𝑎 )]
𝑁 2 4 𝑁 𝑁 2 4 𝑁 2 5 𝑁 5 4
1 1 1 1 1
𝜇4,+(𝑎 𝑎 )=(𝜇1,+𝑛 ∘𝜇1,+)(𝑎 𝑎 )=0, =[0.5∧0.7]∨[0.4∧0.3]=0.5,
𝑁 2 5 𝑁 𝑁 2 5
1 1 1
AppliedComputationalIntelligenceandSoftComputing 7
𝜇2,−(𝑎 𝑎 )=(𝜇1,−𝑛 ∘𝜇1,−)(𝑎 𝑎 ) 𝜇3,−(𝑎 𝑎 )=(𝜇2,−𝑛 ∘𝜇1,−)(𝑎 𝑎 )
𝑁 2 5 𝑁 𝑁 2 5 𝑁 3 4 𝑁 𝑁 3 4
1 1 1 1 1 1
=[𝜇𝑁1,−1 (𝑎2𝑎3)∧𝜇𝑁1,−1 (𝑎3𝑎5)] =[𝜇𝑁2,−1 (𝑎3𝑎2)∧𝜇𝑁1,−1 (𝑎2𝑎4)]
∨[𝜇2,−(𝑎 𝑎 )∧𝜇1,−(𝑎 𝑎 )]
∨[𝜇1,−(𝑎 𝑎 )∧𝜇1,−(𝑎 𝑎 )] 𝑁 3 5 𝑁 5 4
𝑁 2 4 𝑁 4 5 1 1
1 1
=[0.0∧0.0]∨[0.4∧0.3]=0.3,
=[0.5∧0.0]∨[0.0∧0.3]=0.0,
𝜇3,−(𝑎 𝑎 )=(𝜇2,−𝑛 ∘𝜇1,−)(𝑎 𝑎 )
𝜇2,−(𝑎 𝑎 )=(𝜇1,−𝑛 ∘𝜇1,−)(𝑎 𝑎 ) 𝑁1 3 5 𝑁1 𝑁1 3 5
𝑁 3 4 𝑁 𝑁 3 4
1 1 1 =[𝜇2,−(𝑎 𝑎 )∧𝜇1,−(𝑎 𝑎 )]
𝑁 3 2 𝑁 2 5
=[𝜇1,−(𝑎 𝑎 )∧𝜇1,−(𝑎 𝑎 )] 1 1
𝑁 3 2 𝑁 2 4
1 1 ∨[𝜇2,−(𝑎 𝑎 )∧𝜇1,−(𝑎 𝑎 )]
𝑁 3 4 𝑁 4 5
∨[𝜇1,−(𝑎 𝑎 )∧𝜇1,−(𝑎 𝑎 )] 1 1
𝑁 3 5 𝑁 5 4
1 1 =[0.0∧0.4]∨[0.0∧0.3]=0.0,
=[0.5∧0.0]∨[0.0∧0.3]=0.0,
𝜇3,−(𝑎 𝑎 )=(𝜇2,−𝑛 ∘𝜇1,−)(𝑎 𝑎 )
𝑁 4 5 𝑁 𝑁 4 5
1 1 1
𝜇2,−(𝑎 𝑎 )=(𝜇1,−𝑛 ∘𝜇1,−)(𝑎 𝑎 )
𝑁 3 5 𝑁 𝑁 3 5 =[𝜇2,−(𝑎 𝑎 )∧𝜇1,−(𝑎 𝑎 )]
1 1 1 𝑁 4 2 𝑁 2 5
1 1
=[𝜇1,−(𝑎 𝑎 )∧𝜇1,−(𝑎 𝑎 )] ∨[𝜇2,−(𝑎 𝑎 )∧𝜇1,−(𝑎 𝑎 )]
𝑁1 3 2 𝑁1 2 5 𝑁1 4 3 𝑁1 3 5
∨[𝜇1,−(𝑎 𝑎 )∧𝜇1,−(𝑎 𝑎 )] =[0.5∧0.4]∨[0.0∧0.0]=0.4,
𝑁 3 4 𝑁 4 5
1 1
𝜇3,−(𝑎 𝑎 )=(𝜇2,−𝑛 ∘𝜇1,−)(𝑎 𝑎 )=0.
=[0.5∧0.4]∨[0.7∧0.3]=0.4, 𝑁 1 6 𝑁 𝑁 1 6
1 1 1
(20)
𝜇2,−(𝑎 𝑎 )=(𝜇1,−𝑛 ∘𝜇1,−)(𝑎 𝑎 )
𝑁1 4 5 𝑁1 𝑁1 4 5 Similarly,
=[𝜇1,−(𝑎 𝑎 )∧𝜇1,−(𝑎 𝑎 )] 𝜇4,−(𝑎 𝑎 )=(𝜇1,−𝑛 ∘𝜇1,−)(𝑎 𝑎 )=0,
𝑁1 4 2 𝑁1 2 5 𝑁1 2 3 𝑁1 𝑁1 2 3
∨[𝜇1,−(𝑎 𝑎 )∧𝜇1,−(𝑎 𝑎 )] 𝜇4,−(𝑎 𝑎 )=(𝜇1,−𝑛 ∘𝜇1,−)(𝑎 𝑎 )=0.5,
𝑁1 4 3 𝑁1 3 5 𝑁1 2 4 𝑁1 𝑁1 2 4
=[0.0∧0.4]∨[0.7∧0.0]=0.0, 𝜇4,−(𝑎 𝑎 )=(𝜇1,−𝑛 ∘𝜇1,−)(𝑎 𝑎 )=0,
𝑁 2 5 𝑁 𝑁 2 5
1 1 1
𝜇𝑁2,−1 (𝑎1𝑎6)=(𝜇1,−𝑛𝑁1 ∘𝜇𝑁1,−1)(𝑎1𝑎6)=0, 𝜇𝑁4,−(𝑎3𝑎4)=(𝜇1,−𝑛𝑁 ∘𝜇𝑁1,−)(𝑎3𝑎4)=0, (21)
1 1 1
𝜇3,−(𝑎 𝑎 )=(𝜇2,−𝑛 ∘𝜇1,−)(𝑎 𝑎 ) 𝜇4,−(𝑎 𝑎 )=(𝜇1,−𝑛 ∘𝜇1,−)(𝑎 𝑎 )=0.4,
𝑁1 2 3 𝑁1 𝑁1 2 3 𝑁1 3 5 𝑁1 𝑁1 3 5
=[𝜇2,−(𝑎 𝑎 )∧𝜇1,−(𝑎 𝑎 )] 𝜇4,−(𝑎 𝑎 )=(𝜇1,−𝑛 ∘𝜇1,−)(𝑎 𝑎 )=0,
𝑁1 2 4 𝑁1 4 3 𝑁1 4 5 𝑁1 𝑁1 4 5
∨[𝜇2,−(𝑎 𝑎 )∧𝜇1,−(𝑎 𝑎 )] 𝜇4,−(𝑎 𝑎 )=(𝜇1,−𝑛 ∘𝜇1,−)(𝑎 𝑎 )=0.
𝑁1 2 5 𝑁1 5 3 𝑁1 1 6 𝑁1 𝑁1 1 6
=[0.5∧0.7]∨[0.0∧0.0]=0.5, Thisimpliesthat
𝜇3,−(𝑎 𝑎 )=(𝜇2,−𝑛 ∘𝜇1,−)(𝑎 𝑎 ) 𝜇𝑁∞,−(𝑎2𝑎3)=∨{0.5,0.0,0.5,0.0}=0.5,
𝑁 2 4 𝑁 𝑁 2 4 1
1 1 1
𝜇∞,−(𝑎 𝑎 )=∨{0.0,0.5,0.0,0.5}=0.5,
=[𝜇2,−(𝑎 𝑎 )∧𝜇1,−(𝑎 𝑎 )] 𝑁 2 4
𝑁 2 3 𝑁 3 4 1
1 1
𝜇∞,−(𝑎 𝑎 )=∨{0.4,0.0,0.3,0.0}=0.4,
∨[𝜇𝑁2,−(𝑎2𝑎5)∧𝜇𝑁1,−(𝑎5𝑎4)] 𝑁1 2 5
1 1
=[0.0∧0.7]∨[0.0∧0.3]=0.0, 𝜇𝑁∞,−(𝑎3𝑎4)=∨{0.7,0.0,0.3,0.0}=0.7, (22)
1
𝜇∞,−(𝑎 𝑎 )=∨{0.0,0.4,0.0,0.4}=0.4,
𝜇3,−(𝑎 𝑎 )=(𝜇2,−𝑛 ∘𝜇1,−)(𝑎 𝑎 ) 𝑁1 3 5
𝑁 2 5 𝑁 𝑁 2 5
1 1 1
𝜇∞,−(𝑎 𝑎 )=∨{0.3,0.0,0.4,0.0}=0.4,
=[𝜇2,−(𝑎 𝑎 )∧𝜇1,−(𝑎 𝑎 )] 𝑁1 4 5
𝑁 2 3 𝑁 3 5
∨[𝜇12,−(𝑎 𝑎 )∧𝜇11,−(𝑎 𝑎 )] 𝜇𝑁∞1,−(𝑎1𝑎6)=∨{0.3,0.0,0.0,0.0}=0.2.
𝑁 2 4 𝑁 4 5
1 1 Foralltheremainingpairsofvertices,𝑁1-lossand𝑁1-gain
=[0.0∧0.0]∨[0.5∧0.3]=0.3, ofconnectednessarezero.
8 AppliedComputationalIntelligenceandSoftComputing
(Dgsrueafippnhpi(t𝑀siotrn)u,cs2tuu9pr.peA(𝐺𝑁∗B1)F,Gs=uSp(p𝐺𝑈(̌𝑁𝑏,𝐸21)=,,𝐸..2.,(,.𝑀s.u.p,,𝑁𝐸p(𝑛1𝑁),𝑁𝑛is)2),ai.sn.a.𝑁,n𝑁𝑖𝐸-𝑛c𝑖)-yccyolecfleia.f a4(0.6−,0.−3)0.9) N1󳰀(0.5,−0.7) a3(0.5,−0.7)
Definition30. ABFGS𝐺̌𝑏 = (𝑀,𝑁1,𝑁2,...,𝑁𝑛)ofagraph 󳰀N(10.3, −0.4) −0.4)
fsotrruscotumree𝑖𝐺i∗f = (𝑈,𝐸1,𝐸2,...,𝐸𝑛)isabipolarfuzzy𝑁𝑖-cycle a5(0.4,−0.5)N󳰀 󳰀N(0.1,2 󳰀N(0.2,2
2(0.
(i)𝐺̌𝑏isan𝑁𝑖-cycle; 1,−0.
(ii)t𝜇h𝑁𝑃er(e𝑢Vi)s=nomiunn{i𝜇q𝑁𝑃ue(𝑥𝑦𝑁)𝑖-:ed𝑥g𝑦e ∈𝑢V𝐸i𝑖n=𝐺š𝑏upspu(c𝑁h𝑖)t}hoart a6(0.3,−4)0.4) N1󳰀(0.3,−0.2) a1(0.3,−0.6)
𝜇𝑁𝑁𝑖(𝑢V)=max{𝜇𝑁𝑁(𝑖𝑥𝑦):𝑥𝑦∈𝐸𝑖 =supp(𝑁𝑖)}. Figure6:Bipolarfuzzysubgraphstructure(𝑀\{𝑎2},𝑁1󸀠,𝑁2󸀠).
𝑖 𝑖
Example 31. Consider BFGS 𝐺̌𝑏 = (𝑀,𝑁1,𝑁2) as shown
in Figure3. Then 𝐺̌𝑏 is an 𝑁1-cycle as well as bipolar fuzzy Then𝑎2isabipolarfuzzy𝑁1-𝑁cutvertexsince
𝑁1-cycle,since(supp(𝑀),supp(𝑁1),supp(𝑁2))isan𝐸1-cycle 𝜇∞,−(𝑎 𝑎 )=0.7=𝜇∞,−(𝑎 𝑎 ),
and there are two 𝑁1-edges with minimum positive degree 𝑁1 3 4 𝑁1󸀠 3 4
oanfdallm𝑁o1r-eetdhgaens.one𝑁1-edgewithmaximumnegativedegree 𝜇𝑁∞1,−(𝑎3𝑎5)=0.4>0.3=𝜇𝑁∞1,−(𝑎3𝑎5),
(25)
gDLreeatfip(nh𝑀itsi󸀠to,rn𝑁uc31󸀠t2,u𝑁.reL2󸀠,𝐺e.t∗.𝐺.=,̌𝑏𝑁(=𝑈𝑛󸀠),(b𝐸𝑀e1,,a𝐸𝑁b21i,p,.o𝑁.l.a2,r,𝐸.f𝑛.u).z,ay𝑁nsd𝑛u)b𝑥bgareavapehBrtFsetxGruoScfto𝐺ufř𝑏ae. 𝜇𝜇𝑁𝑁∞∞11,,−−((𝑎𝑎41𝑎𝑎56))==00..42>=0𝜇.𝑁∞31,−=(𝜇𝑎𝑁∞1𝑎1󸀠,−6)(.𝑎4𝑎5),
of𝐺̌𝑏inducedby𝑈\{𝑥}suchthat Definition 34. Let 𝐺̌𝑏 = (𝑀,𝑁1,𝑁2,...,𝑁𝑛) be a BFGS of
𝜇𝑃 (𝑥)=0=𝜇𝑁 (𝑥), a graph structure 𝐺∗ = (𝑈,𝐸1,𝐸2,...,𝐸𝑛) and let 𝑥𝑦 be an
𝑀󸀠 𝑀󸀠 𝑁𝑖-edge.Let(𝑀,𝑁1󸀠,𝑁2󸀠,...,𝑁𝑛󸀠)beabipolarfuzzyspanning
𝜇𝑁𝑃󸀠(𝑥V)=0=𝜇𝑁𝑁󸀠(𝑥V) subgraphstructureof𝐺̌𝑏,obtainedbytaking
𝑖 𝑖
∀edges𝑥V∈𝐺̌𝑏, 𝜇𝑁𝑃𝑖󸀠(𝑥𝑦)=0=𝜇𝑁𝑁𝑖󸀠(𝑥𝑦),
𝜇𝑃 (V)=𝜇𝑃 (V), 𝜇𝑃 (𝑢V)=𝜇𝑃 (𝑢V),
𝑀󸀠 𝑀 𝑁𝑖󸀠 𝑁𝑖
(26)
𝜇𝑁 (V)=𝜇𝑁(V), (23) 𝜇𝑁 (𝑢V)=𝜇𝑁 (𝑢V)
𝑀󸀠 𝑀 𝑁󸀠 𝑁
𝑖 𝑖
∀V≠𝑥,
∀edges𝑢V≠𝑥𝑦.
𝜇𝑁𝑃𝑖󸀠(𝑢V)=𝜇𝑁𝑃𝑖(𝑢V), Then𝑥𝑦isabipolarfuzzy𝑁𝑖-bridgeif
𝜇𝑁 (𝑢V)=𝜇𝑁 (𝑢V) 𝜇∞,+(𝑢V)>𝜇∞,+(𝑢V),
𝑁𝑖󸀠 𝑁𝑖 𝑁𝑖 𝑁𝑖󸀠
∀𝑖, such that 𝑢≠𝑥, V≠𝑥. 𝜇∞,−(𝑢V)>𝜇∞,−(𝑢V) (27)
𝑁𝑖 𝑁𝑖󸀠
Then𝑥isabipolarfuzzy𝑁𝑖-cutvertexforsome𝑖,if for some 𝑢,V∈𝑈.
𝜇𝜇𝑁𝑁∞∞𝑖𝑖,,+−((𝑢𝑢VV))>>𝜇𝜇𝑁𝑁∞∞𝑖𝑖󸀠󸀠,,+−((𝑢𝑢VV)), (24) csEeodcngodenitd𝑥io𝑦cnonihsdoialtdniosn𝑁anh𝑖-doPladnbs.i𝑁po𝑖l-aNrbfiupzozlyarbfruizdzgyebirfidognelyiftohnelyfitrhset
for some 𝑢,V∈𝑈\{𝑥}. Example35. ConsidertheBFGS𝐺̌𝑏 =(𝑀,𝑁1,𝑁2)asshown
in Figure6 and let 𝐺̌󸀠 = (𝑀,𝑁󸀠,𝑁󸀠) be bipolar fuzzy
cAtEhoxnenaddsm,eitcp𝑥ioloenins3d3hac.nooClnd𝑁odsn𝑖i-atsPinioddnbeirahpBoo𝑁llFad𝑖S-rsGN.fu𝐺bži𝑏zpy=olca(ur𝑀tfuv,𝑁ezrz1tye,x𝑁cu2itf)vaoesnrcltyoexntshiifedoefirnrelsdyt s𝜇0epd.𝑁∞3ag1,ne+=n((𝑎i𝜇𝑎n2𝑁∞2𝑎g𝑎󸀠5,5−s))u(=.𝑎b2Thg0𝑎r.5a4e)pn,>ha𝑎n0s2dt.𝑎3ru5a𝑏=lcsitsou𝜇r𝑁∞aaeb1󸀠,b+oipi(fp𝑎o𝐺o2la𝑎̌l𝑏ar5ro)f1ubafztunazzdiy2zn𝜇y𝑁e𝑁∞d𝑁11,-b−1Ny(-𝑎bdb2rre𝑎iidl5de)gtgei=en,,g0ssii.𝑁nn4cc1>ee-
i𝑎n2,Ethxaemrepsluel2ti8nganbdipsohlaorwfnuziznyFsiugbugrrea5p;hasftterrucdteulreetiwngillvbereteaxs 𝜇0.𝑁∞31,−=(𝑎𝜇3∞𝑎15,−)(𝑎=𝑎0.4).>0.3=𝜇𝑁∞1󸀠,−(𝑎3𝑎5)and𝜇𝑁∞1,−(𝑎4𝑎5)=0.4>
showninFigure6. 𝑁󸀠 4 5
1
AppliedComputationalIntelligenceandSoftComputing 9
a (0.3,−0.6)
1
2)
0.
−
3,
0.
N (20.2,−0.4) N 2(0.1,− 0.4N) (N(02.3,−a06(.30).3N,2−(00..14,)−0.4)N2(0.2,−0.6)
1
N 1(0.5, − 0.7) a4(0.6,−0.9) a5(0.4,−0.5) N1(0.2,−0.1)
a3(0.5,−0.7) N1(0.3,−0.5) a2(0.4,−0.7)
Figure7:𝐺̌𝑏=(𝑀,𝑁1,𝑁2).
a (0.3,−0.3) b(0.3,−0.3)
1 1
N 1(0.2,− 0.2) N1(0.3,−0.2) N󳰀2(0.2,− 0.2) N2󳰀(0.3,−0.2)
a2(0.5,−0.5) N2(0.3,−0.4) a3(0.3,−0.8) b2(0.5,−0.5) N1󳰀(0.3,−0.4) b3(0.3,−0.8)
(BFGSǦ ) (BFGSǦ )
b1 b2
Figure8:Isomorphicbipolarfuzzygraphstructures.
Definition 36. A BFGS 𝐺̌𝑏 = (𝑀,𝑁1,𝑁2,...,𝑁𝑛) of a (𝑈2,𝐸21,𝐸22,...,𝐸2𝑛)ifthereexistsabijection𝑓: 𝑈1 → 𝑈2
graph structure 𝐺∗ = (𝑈,𝐸1,𝐸2,...,𝐸𝑛) is an 𝑁𝑖-tree if andapermutation𝜙ontheset{1,2,...,𝑛}suchthat
(supp(𝐴),supp(𝑁1),supp(𝑁2),...,supp(𝑁𝑛))isan𝐸𝑖-tree.In
otherwords,𝐺̌𝑏isan𝑁𝑖-treeifasubgraphof𝐺̌𝑏,inducedby 𝜇𝑀𝑃 (𝑢1)=𝜇𝑀𝑃 (𝑓(𝑢1)),
supp(𝑁𝑖),formsatree. 1 2
𝜇𝑁 (𝑢 )=𝜇𝑁 (𝑓(𝑢 )) (29)
𝑀 1 𝑀 1
Definition37. ABFGS𝐺̌𝑏 = (𝑀,𝑁1,𝑁2,...,𝑁𝑛)ofagraph 1 2
structure 𝐺∗ = (𝑈,𝐸1,𝐸2,...,𝐸𝑛) is a bipolar fuzzy 𝑁𝑖-tree ∀𝑢1 ∈𝑈1
i(f𝐴𝐺,̌𝐶𝑏1h,a𝐶s2a,.b..ip,𝐶ol𝑛a)rsfuuczhzythspaatn𝐻ňi𝑏nigssaub𝐶g𝑖r-atrpehesatnrudct𝜇u𝑁𝑃re(𝑥𝐻𝑦̌𝑏) =< andfor𝜙(𝑖)=𝑗
𝑖
𝜇𝐶∞𝑖,+In(𝑥m𝑦)oarendco|n𝜇c𝑁𝑁e𝑖(r𝑥n𝑦ed)|v<ie𝜇w𝐶∞,𝑖𝐺,−̌𝑏(𝑥i𝑦s)a∀bi𝑁po𝑖-leadrgfeusznzyot𝑁in𝑖-P𝐻̌t𝑏r.eeif 𝜇𝑁𝑃1𝑖(𝑢1𝑢2)=𝜇𝑁𝑃2𝑗(𝑓(𝑢1)𝑓(𝑢2)),
onlythefirstconditionholdsandabipolarfuzzy𝑁𝑖-Ntreeif 𝜇𝑁𝑁 (𝑢1𝑢2)=𝜇𝑁𝑁 (𝑓(𝑢1)𝑓(𝑢2)) (30)
1𝑖 2𝑗
onlythesecondconditionholds.
∀𝑢 𝑢 ∈𝐸 , 𝑖=1,2,...,𝑛.
1 2 1𝑖
Example38. ConsiderBFGS𝐺̌𝑏 = (𝑀,𝑁1,𝑁2)asshownin
Figure7,whichisan𝑁2-tree.Itisnotan𝑁1-treebutabipolar Example40. Let𝐺̌𝑏1 = (𝑀,𝑁1,𝑁2)and𝐺̌𝑏2 = (𝑀󸀠,𝑁1󸀠,𝑁2󸀠)
sfduterzluezctyitnu𝑁gr1e𝑁-t(r1𝑀e-ee,ds𝑁gine1󸀠c,𝑎e𝑁2𝑎i2󸀠t5)hfaarossmaanb𝐺ǐp𝑁𝑏oa1ln-atdrrefeu,zzwyhsipchaninsinogbtsauinbgedrapbhy (b𝑈e󸀠t,wH𝐸oe1󸀠r,Be𝐸F𝐺2󸀠G)̌𝑏,S1rseissopfisegocrtmaivpoehrlyps,htariusccs(thnuoorwetsnid𝐺ien1∗nFt=iicg(au𝑈l)r,et𝐸o81.,𝐺𝐸̌𝑏22)uanndde𝐺r2∗th=e
𝜇𝑁𝑃1(𝑎2𝑎5)=0.2<0.3=𝜇𝑁∞1󸀠,+(𝑎2𝑎5), 𝑓m(a𝑎p3p)i=ng𝑏3𝑓,:an𝑈d→ape𝑈rm󸀠,duetafitnioend𝜙byg𝑓iv(e𝑎n1)by=𝜙𝑏(11,)𝑓=(𝑎22),𝜙=(𝑏22),=an1d,
󵄨󵄨󵄨󵄨󵄨𝜇𝑁𝑁1(𝑎2𝑎5)󵄨󵄨󵄨󵄨󵄨=0.1<0.3=𝜇𝑁∞1󸀠,−(𝑎2𝑎5). (28) suchth𝜇a𝑃t (𝑎)=𝜇𝑃 (𝑓(𝑎)),
𝑀 𝑖 𝑀󸀠 𝑖
Definition 39. A BFGS 𝐺̌𝑠1 = (𝑀1,𝑁11,𝑁12,...,𝑁1𝑛) of 𝜇𝑁(𝑎)=𝜇𝑁 (𝑓(𝑎))
graph structure 𝐺1∗ = (𝑈1,𝐸11,𝐸12,...,𝐸1𝑛) is isomor- 𝑀 𝑖 𝑀󸀠 𝑖
phic to a BFGS 𝐺̌𝑠2 = (𝑀2,𝑁21,𝑁22,...,𝑁2𝑛) of 𝐺2∗ = ∀𝑎𝑖 ∈𝑈,
10 AppliedComputationalIntelligenceandSoftComputing
b(0.2,−0.5)
6
−0.0) N1(0.2, a5(0.4,−0.4) N2󳰀(0.1,−0.2) a4(0.1,−0.2)
N(10.2, −0.1) N2󳰀(0.4,−0.4) N󳰀2(0.1,− 0.2)
a (0.6,−0.6)
b1(0.4,−0.4) N2(0.1,−0.2) b5(0.1,−0.2) 4) 6
0.
−
N1(0.2,−0.0) 2,
0.
b(0.9,−0.0) b(0.2,−0.1) (
2 4) 4 󳰀N1 a1(0.2,−0.5)
N2(0.4,−0. N(0.2,−0.1 (0.1,−0.2) N󳰀1(0.2,0.0) N1󳰀(0.2,0.0)N1󳰀(0.2,−0.1)
4) N2 a (0.9,−0.0) a (0.2,−0.1)
2 3
b(0.6,−0.6)
3
(Ǧ ) (Ǧ )
b1 b2
Figure9:Identicalbipolarfuzzygraphstructures.
𝜇𝑃 (𝑎𝑎 )=𝜇𝑃 (𝑓(𝑎)𝑓(𝑎 )), 𝜇𝑃 (𝑎𝑎 )=𝜇𝑃 (𝑓(𝑎)𝑓(𝑎 )),
𝑁 𝑖 𝑗 𝑁 𝑖 𝑗 𝑁 𝑖 𝑗 𝑁󸀠 𝑖 𝑗
𝑘 𝜙(𝑘) 𝑘 𝑘
𝜇𝑁 (𝑎𝑎 )=𝜇𝑁 (𝑓(𝑎)𝑓(𝑎 )) 𝜇𝑁 (𝑎𝑎 )=𝜇𝑁 (𝑓(𝑎)𝑓(𝑎 ))
𝑁 𝑖 𝑗 𝑁 𝑖 𝑗 𝑁 𝑖 𝑗 𝑁󸀠 𝑖 𝑗
𝑘 𝜙(𝑘) 𝑘 𝑘
∀𝑎𝑎 ∈𝐸 , 𝑘=1,2. ∀𝑎𝑎 ∈𝐸 , 𝑘=1,2.
𝑖 𝑗 𝑘 𝑖 𝑗 𝑘
(31) (33)
D𝐺t(h𝑀1∗eefir2en,=𝑁ietxi2oi1(ns𝑈,t𝑁,4a𝐸21b.21,i1jA.,e.𝐸c.Bt1,i2Fo𝑁,Gn.2.𝑛S𝑓.),𝐺:o𝐸̌𝑠𝑈f11𝑛G)=→Sis(𝐺𝑀𝑈i2∗d,1es,=nu𝑁tci1(ch1𝑈a,tl,𝑁h𝐸tao12t21,,a.𝐸.B2.2,F,𝑁G..1S.𝑛,)𝐺𝐸ǒ𝑠2f2𝑛)G=iSf DtoahneGefi{S𝑁snei𝐺1tti,∗o{𝑁𝐸n=214,,.(3𝐸.𝑈.2.,,,L𝐸.𝑁e.1t.𝑛,,}𝐺𝐸𝐸;̌2𝑏t𝑛,h}.a=.at.ni,(sd𝐸𝑀,𝜙𝑛t)h,(.𝑁𝑁eL𝑖1ce),ot𝑁=r𝜙r2𝑁e,bs.𝑗ep.io.af,nna𝑁dyni𝑛dnp)geobrnpmeleyauritmfBa𝜙tuFi(otG𝐸antS𝑖i)ooo=nnf
𝜇𝑃 (𝑢)=𝜇𝑃 (𝑓(𝑢)), 𝐸𝑗 ∀𝑖.
𝑀1 𝑀2 If𝑥𝑦∈𝑁𝑟forsome𝑟and
𝜇𝑁 (𝑢)=𝜇𝑁 (𝑓(𝑢))
𝑀 𝑀
1 2 𝜇𝑃 (𝑥𝑦)=𝜇𝑃 (𝑥)∧𝜇𝑃 (𝑦)−⋁𝜇𝑃 (𝑥𝑦),
∀𝑢∈𝑈, 𝑁𝑖𝜙 𝑀 𝑀 𝑗=𝑖̸ 𝜙𝑁𝑗
(32)
𝜇𝑃 (𝑢 𝑢 )=𝜇𝑃 (𝑓(𝑢 )𝑓(𝑢 )), 𝜇𝑁 (𝑥𝑦)=𝜇𝑁(𝑥)∨𝜇𝑁(𝑦)−⋀𝜇𝑁 (𝑥𝑦), (34)
𝑁1𝑖 1 2 𝑁2𝑖 1 2 𝑁𝑖𝜙 𝑀 𝑀 𝑗=𝑖̸ 𝜙𝑁𝑗
𝜇𝑁 (𝑢 𝑢 )=𝜇𝑁 (𝑓(𝑢 )𝑓(𝑢 ))
𝑁1𝑖 1 2 𝑁2𝑖 1 2 𝑖=1,2,...,𝑛,
∀𝑢 𝑢 ∈𝐸 , 𝑖=1,2,...,𝑛.
1 2 1𝑖
then 𝑥𝑦 ∈ 𝐵𝜙, while 𝑚 is chosen such that 𝜇𝑃 (𝑥𝑦) ≥
EbexatwmoplBeF4G2.SsLoetfg𝐺ř𝑏a1p=hs(t𝑀ru,c𝑁tu1r,e𝑁s2𝐺)1∗an=d(𝑈𝐺̌,𝑏2𝐸1=,𝐸(𝑀2)a󸀠,n𝑁d1󸀠𝐺,𝑁2∗2󸀠=) 𝜇𝑁𝑃𝑖𝜙(𝑥𝑦)and𝜇𝑁𝑁𝑚𝑚𝜙(𝑥𝑦)≤𝜇𝑁𝑁𝑖𝜙(𝑥𝑦) ∀𝑖. 𝑁𝑚𝜙
(𝑈󸀠,𝐸1󸀠,𝐸2󸀠),respectively,asshowninFigure9. AndBFGS(𝑀,𝑁𝜙,𝑁𝜙,...,𝑁𝜙),denotedby𝐺̌𝜙𝑐,iscalled
𝑈 H→ere𝑈𝐺󸀠̌,𝑏1diesfinideedntbicyal𝑓w(𝑎i1th) 𝐺=̌𝑏2𝑏u6,n𝑓d(e𝑎r2)the=m𝑏a2p,p𝑓i(n𝑎g3)𝑓=: the𝜙-complementof1BFG2S𝐺̌𝑏. 𝑛 𝑏
𝑏4,𝑓(𝑎4)=𝑏5,𝑓(𝑎5)=𝑏1,and𝑓(𝑎6)=𝑏3,suchthat Example 44. Let 𝑀 = {(𝑎1,0.3,−0.7), (𝑎2,0.5,−0.4),
𝜇𝑃 (𝑎)=𝜇𝑃 (𝑓(𝑎)), (𝑎3,0.7,−0.3)},𝑁1 = {(𝑎1𝑎3,0.3,−0.3),(𝑎2𝑎3,0.5,−0.3)},and
𝑀 𝑖 𝑀󸀠 𝑖 𝑁2 ={(𝑎1𝑎2,0.3,−0.4)}bebipolarfuzzysubsetsof𝑈,𝐸1,and
𝜇𝑁(𝑎)=𝜇𝑁 (𝑓(𝑎)) 𝐸2,respectively,sothat𝐺̌𝑏 =(𝑀,𝑁1,𝑁2)isaBFGSofgraph
𝑀 𝑖 𝑀󸀠 𝑖 structure𝐺∗ =(𝑈,𝐸1,𝐸2).Let𝜙beapermutationontheset
∀𝑎𝑖 ∈𝑈, {𝑁1,𝑁2}suchthat𝜙(𝑁1)=𝑁2and𝜙(𝑁2)=𝑁1.