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InternationalJ.Math.Combin.Vol.4(2010),46-52TheUpperMonophonicNumberof - PPT Presentation

1ReceivedAugust62010AcceptedDecember162010 TheUpperMonophonicNumberofaGraph47geodeticsetofcardinalitygGisaminimumgeodeticsetofGThegeodeticnumberofagraphisintroducedin2andfurtherstudiedin3N ID: 518476

1ReceivedAugust6 2010.AcceptedDecember16 2010. TheUpperMonophonicNumberofaGraph47geodeticsetofcardinalityg(G)isaminimumgeodeticsetofG.Thegeodeticnumberofagraphisintroducedin[2]andfurtherstudiedin[3].N

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InternationalJ.Math.Combin.Vol.4(2010),46-52TheUpperMonophonicNumberofaGraphJ.JohnDepartmentofMathematicsofGovernmentCollegeofEngineering,Tirunelveli-627007,IndiaS.PanchaliDepartmentofMathematicsofCapeInstituteofTechnology,Levengipuram-627114,IndiaEmail:johnramesh1971@yahoo.co.in,panchalis64@gmail.comAbstract 1ReceivedAugust6,2010.AcceptedDecember16,2010. TheUpperMonophonicNumberofaGraph47geodeticsetofcardinalityg(G)isaminimumgeodeticsetofG.Thegeodeticnumberofagraphisintroducedin[2]andfurtherstudiedin[3].N(v)=fu2V(G):uv2E(G)giscalledtheneighborhoodofthevertexvinG.ForanysetSofverticesofG,theinducedsubgraphS&#x-282;&#x.181;isthemaximalsubgraphofGwithvertexsetS.AvertexvisanextremevertexofagraphGifN(v)&#x-282;&#x.183;iscomplete.Achordofapathu0;u1;u2;:::;uhisanedgeuiuj,withji+2.Anuvpathiscalledamonophonicpathifitisachordlesspath.ASmarandachelyk-monophonicsetofGisasetMV(G)suchthateveryvertexofGiscontainedinapathwithlessorequalkchordsjoiningsomepairofverticesinM.TheSmarandachelyk-monophonicnumbermkS(G)ofGistheminimumorderofitsSmarandachelyk-monophonicsets.Particularly,aSmarandachely0-monophonicpath,aSmarandachely0-monophonicnumberisabbrevatedtomonophonicpath,monophonicnumberm(G)ofGrespectively.Thus,amonophonicsetofGisasetMVsuchthateveryvertexofGiscontainedinamonophonicpathjoiningsomepairofverticesinM.Themonophonicnumberm(G)ofGistheminimumorderofitsmonophonicsetsandanymonophonicsetoforderm(G)isaminimummonophonicsetorsimplyamsetofG.ItiseasilyobservedthatnocutvertexofGbelongstoanyminimummonophonicsetofG.Themonophonicnumberofagraphisstudiedin[4,5,6].ForthegraphGgiveninFigure1.1,S1=fv2;v4;v5g,S2=fv2;v4;v6garetheonlyminimumgeodeticsetsofGsothatg(G)=3.Also,M1=fv2;v4g;M2=fv4;v6g;M3=fv2;v5garearetheonlyminimummonophonicsetsofGsothatm(G)=2. Figure1:Gx2:TheUpperMonophonicNumberofaGraphDenition2:1AmonophonicsetMinaconnectedgraphGiscalledaminimalmonophonicsetifnopropersubsetofMisamonophonicsetofG.Theuppermonophonicnumberm+(G)ofGisthemaximumcardinalityofaminimalmonophonicsetofG.Example2:2ForthegraphGgiveninFigure1.1,M4=fv1;v3;v5gandM5=fv1;v3;v6gareminimalmonophonicsetsofGsothatm+(G)3.ItiseasilyveriedthatnofourelementsubsetsorveelementsubsetsofV(G)isaminimalmonophonicsetofGandsom+(G)=3.Remark2:3EveryminimummonophonicsetofGisaminimalmonophonicsetofGandtheconverseisnottrue.ForthegraphGgiveninFigure1.1,M4=fv1;v3;v5gisaminimal 48J.JohnandS.PanchalimonophonicsetbutnotaminimummonophonicsetofG.Theorem2:4EachextremevertexofGbelongstoeverymonophonicsetofG.ProofLetMbeamonophonicsetofGandvbeanextremevertexofG.Letfv1;v2;:::;vkgbetheneighborsofvinG.Supposethatv=2M.ThenvliesonamonophonicpathP:x=x1;x2;:::;vi;v;vj;:::;xm=y,wherex;y2M.SincevivjisachordofPandsoPisnotamonophonicpath,whichisacontradiction.Henceitfollowsthatv2M.Theorem2:5LetGbeaconnectedgraphwithcut-verticesandSbeamonophonicsetofG.Ifvisacut-vertexofG,theneverycomponentofGvcontainsanelementofS.ProofSupposethatthereisacomponentG1ofGvsuchthatG1containsnovertexofS.ByTheorem2.4,G1doesnotcontainanyend-vertexofG.ThusG1containsatleastonevertex,sayu.SinceSisamonophonicset,thereexistsverticesx;y2SsuchthatuliesonthexymonophonicpathP:x=u0;u1;u2;:::;u;:::;ut=yinG.LetP1beaxusubpathofPandP2beauysubpathofP.Sincevisacut-vertexofG,bothP1andP2containvsothatPisnotapath,whichisacontradiction.ThuseverycomponentofGvcontainsanelementofS.Theorem2:6ForanyconnectedgraphG,nocut-vertexofGbelongstoanyminimalmono-phonicsetofG.ProofLetMbeaminimalmonophonicsetofGandv2Mbeanyvertex.WeclaimthatvisnotacutvertexofG.SupposethatvisacutvertexofG.LetG1;G2;:::;Gr(r2)bethecomponentsofGv.ByTheorem2.5,eachcomponentGi(1ir)containsanelementofM.WeclaimthatM1=MfvgisalsoamonophonicsetofG.LetxbeavertexofG.SinceMisamonophonicset,xliesonamonophonicpathPjoiningapairofverticesuandvofM.Assumewithoutlossofgeneralitythatu2G1.SincevisadjacenttoatleastonevertexofeachGi(1ir),assumethatvisadjacenttozinGk,k=1.SinceMisamonophonicset,zliesonamonophonicpathQjoiningvandavertexwofMsuchthatwmustnecessarilybelongstoGk.Thusw=v.Now,sincevisacutvertexofG,P[QisapathjoininguandwinMandthusthevertexxliesonthismonophonicpathjoiningtwoverticesuandwofM1.ThuswehaveprovedthateveryvertexthatliesonamonophonicpathjoiningapairofverticesuandvofMalsoliesonamonophonicpathjoiningtwoverticesofM1.HenceitfollowsthateveryvertexofGliesonamonophonicpathjoiningtwoverticesofM1,whichshowsthatM1isamonophonicsetofG.SinceM1(M,thiscontradictsthefactthatMisaminimalmonophonicsetofG.Hencev=2MsothatnocutvertexofGbelongstoanyminimalmonophonicsetofG.Corollary2:7Foranynon-trivialtreeT,themonophonicnumberm+(T)=m(T)=k,wherekisnumberofendverticesofT.ProofThisfollowsfromTheorems2:4and2:6. TheUpperMonophonicNumberofaGraph49Corollary2:8ForthecompletegraphKp(p2);m+(Kp)=m(Kp)=p.ProofSinceeveryvertexofthecompletegraph,Kp(p2)isanextremevertex,thevertexsetofKpistheuniquemonophonicsetofKp.Thusm+(Kp)=m(Kp)=p.Theorem2:9ForacycleG=Cp(p4);m+(G)=2=m(G).ProofLetx;ybetwoindependentverticesofG.ThenM=fx;ygisamonophonicsetofGsothatm(G)=2.Weshowthatm+(G)=2.Supposethatm+(G)�2.ThenthereexistsaminimalmonophonicsetM1suchthatjM1j3.NowitisclearthatM(M1,whichisacontradictiontoM1aminimalmonophonicsetofG.Therefore,m+(G)=2.Theorem2:10ForaconnectedgraphG,2m(G)m+(G)p.ProofAnymonophonicsetneedsatleasttwoverticesandsom(G)2.Sinceeverymin-imalmonophonicsetisamonophonicset,m(G)m+(G).Also,sinceV(G)isamonophonicsetofG,itisclearthatm+(G)p.Thus2m(G)m+(G)p.ThefollowingTheoremisprovedin[3].TheoremALetGbeaconnectedgraphwithdiameterd.Theng(G)pd+1:Theorem2:11LetGbeaconnectedgraphwithdiameterd.Thenm(G)pd+1.ProofSinceeverygeodeticsetofGisamonophonicsetofG,theassertionfollowsfromTheorem2.10andTheoremA.Theorem2:12Foranon-completeconnectedgraphG,m(G)pk(G),wherek(G)isvertexconnectivityofG.ProofSinceGisnoncomplete,itisclearthat1k(G)p2.LetU=fu1;u2;:::;ukgbeaminimumcutsetofverticesofG.LetG1;G2;:::;Gr(r2)bethecomponentsofGUandletM=V(G)U.Theneveryvertexui(1ik)isadjacenttoatleastonevertexofGj(1jr).Thenitfollowsthatthevertexuiliesonthemonophonicpathx;ui;y,wherex;y2MsothatMisamonophonicset.Thusm(G)pk(G).ThefollowingTheorems2:13and2:15characterizegraphsforwhichm+(G)=pandm+(G)=p1respectively.Theorem2:13ForaconnectedgraphGoforderp,thefollowingareequivalent:(i)m+(G)=p;(ii)m(G)=p;(iii)G=Kp:Proof(i))(ii).Letm+(G)=p.ThenM=V(G)istheuniqueminimalmonophonicsetofG.SincenopropersubsetofMisamonophonicset,itisclearthatMistheuniqueminimummonophonicsetofGandsom(G)=p.(ii))(iii).Letm(G)=p.IfG=Kp,then 50J.JohnandS.PanchalibyTheorem2.11,m(G)p1,whichisacontradiction.ThereforeG=Kp.(ii))(iii).LetG=Kp.ThenbyCorollary2.8,m+(G)=p.Theorem2:14LetGbeanoncompleteconnectedgraphwithoutcutvertices.Thenm+(G)p2.ProofSupposethatm+(G)p1.ThenbyTheorem2.13,m+(G)=p1.LetvbeavertexofGandletM=V(G)fvgbeaminimalmonophonicsetofG.ByTheorem2.4,visnotanextremevertexofG.Thenthereexistsx;y2N(v)suchthatxy62E(G).SincevisnotacutvertexofG,Gv&#x-330;&#x.363;isconnected.Letx;x1;x2;:::;xn;ybeamonophonicpathinGv&#x-270;&#x.138;.ThenM1=Mfx1;x2;:::;xngisamonophonicsetofG.SinceM1(M,M1isnotaminimalmonophonicsetofG,whichisacontradiction.Thereforem+(G)p2:Theorem2:15ForaconnectedgraphGoforderp,thefollowingareequivalent:(i)m+(G)=p1;(ii)m(G)=p1;(iii)G=K1+SmjKj,Pmj2:Proof(i))(ii).Letm+(G)=p1.ThenitfollowsfromTheorem2.13thatGisnon-complete.HencebyTheorem2:14,Gcontainsacutvertex,sayv.Sincem+(G)=p1,henceitfollowsfromTheorem2.6thatM=VfvgistheuniqueminimalmonophonicsetofG.Weclaimthatm(G)=p1.Supposethatm(G)p1.ThenthereexistsaminimummonophonicsetM1suchthatjM1jp1.Itisclearthatv=2M1.ThenitfollowsthatM1(M,whichisacontradiction.Thereforem(G)=p1.(ii))(iii).Letm(G)=p1.ThenbyTheorem2:11,d2.Ifd=1,thenG=Kp,whichisacontradiction.Therefored=2.IfGhasnocutvertex,thenbyTheorem2:12,m(G)p2,whichisacontradiction.ThereforeGhasauniquecut-vertex,sayv.SupposethatG=K1+SmjKj.Thenthereexistsacomponent,sayG1ofGvsuchthatG1&#x-282;&#x.181;isnoncomplete.HencejV(G1)j3.ThereforeG1&#x-306;&#x.273;containsachordlesspathPoflengthatleasttwo.LetybeaninternalvertexofthepathPandletM=V(G)fv;yg.ThenMisamonophonicsetofGsothatm(G)p2,whichisacontradiction.ThusG=K1+SmjKj.(iii))(i)).LetG=K1+SmjKj.ThenbyTheorems2:4and2:6,m+(G)=p1.IntheviewofTheorem2:10,wehavethefollowingrealizationresult.Theorem2:16Foranypositiveintegers2ab,thereexistsaconnectedgraphGsuchthatm(G)=aandm+(G)=b.ProofLetGbeagraphgiveninFigure2.1obtainedfromthepathonthreeverticesP:u1;u2;u3byaddingthenewverticesv1;v2;:::;vba+1andw1;w2;:::;wa1andjoiningeachvi(1iba+1)toeachvj(1jba+1),i=j,andalsojoiningeachwi(1ia1)withu1andu2.Firstweshowthatm(G)=a.LetMbeamonophonicsetofGandletW=fw1;w2;:::;wa1g.ByTheorem2.4,WM.ItiseasilyseenthatWisnotamonophonicsetofG.However,W[fu3gisamonophonicsetofGandsom(G)=a.Nextweshowthatm+(G)=b.LetM1=W[fv1;v2;:::;vba+1g.ThenM1isamonophonic TheUpperMonophonicNumberofaGraph51 Figure2:GsetofG.IfM1isnotaminimalmonophonicsetofG,thenthereisapropersubsetTofM1suchthatTisamonophonicsetofG.Thenthereexistsv2M1suchthatv=2T.ByTheorem2.4,v=wi(1ia1).Thereforev=viforsomei(1iba+1).Sincevivj(1i;jba+1),i=jisachord,vidoesnotlieonamonophonicpathjoiningsomeverticesofTandsoTisnotamonophonicsetofG,whichisacontradiction.ThusM1isaminimalmonophonicsetofGandsom+(G)b.LetT0beaminimalmonophonicsetofGwithjT0jb+1.ByTheorem2.4,WT0.SinceW[fu3gisamonophonicsetofG,u362T0.SinceM1isamonophonicsetofG,thereexistsatleastonevisuchthatvi62T0.Withoutlossofgeneralityletusassumethatv162T0.SincejT0jb+1,thenu1;u2mustbelongtoT0.Nowitisclearthatv1doesnotlieonamonophonicpathjoiningapairofverticesofT0,itfollowsthatT0isnotamonophonicsetofG,whichisacontradiction.Thereforem+(G)=b.References[1]F.BuckleyandF.Harary,DistanceinGraphs,Addition-Wesley,RedwoodCity,CA,1990.[2]F.Buckley,F.Harary,L.V.Quintas,Extremalresultsonthegeodeticnumberofagraph,ScientiaA2,(1988),17-26.[3]G.Chartrand,F.Harary,Zhang,OntheGeodeticNumberofagraph,Networks,39(1), 52J.JohnandS.Panchali(2002)1-6.[4]CarmenHernando,TaoJiang,MerceMora,Ignacio.M.PelayoandCarlosSeara,OntheSteiner,geodeticandhullnumberofgraphs,DiscreteMathematics,293(2005),139-154.[5]EsamelM.paluga,SergioR.Canoy,Jr.,MonophonicnumbersofthejoinandCompositionofconnectedgraphs,DiscreteMathematics,307(2007),1146-1154.[6]MitreC.Dourado,FabioProttiandJayme.L.Szwarcter,AlgorithmicAspectsofMono-phonicConvexity,ElectronicNotesinDiscreteMathematics,30(2008),177-1822.