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Connected domination in graphs

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Connected domination in graphs
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Mahalingam, Gayathri
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Connected dominating set
Algorithms
Breadth first search
Random graphs
Local optimization
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Abstract:
ABSTRACT: A connected dominating set D is a set of vertices of a graph G=(V,E) such that every vertex in V-D is adjacent to at least one vertex in D and the subgraph induced by the set D is connected. The connected domination number is the minimum of the cardinalities of the connected dominating sets of G. The problem of finding a minimum connected dominating set D is known to be NP-hard. Many polynomial time algorithms that achieve some approximation factors have been provided earlier in finding a minimum connected dominating set. In this work, we present a survey on known properties of graph domination as well as some approximation algorithms. We implemented some of these algorithms and tested them with random graphs and compared their performance in finding a minimum connected dominating set D.
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Thesis (M.A.)--University of South Florida, 2005.
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by Gayathri Mahalingam.
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by GayathriMahalingam Adissertationsubmittedinpartialfulllment oftherequirementsforthedegreeof MasterofScience DepartmentofMathematics CollegeofArtsandSciences UniversityofSouthFlorida MajorProfessor:StephenSuen,Ph.D. NatashaJonoska,Ph.D. GregoryMcColm,Ph.D. DateofApproval: June21,2005 Keywords:ConnectedDominatingset,Algorithms,BreadthFirstSearch,Randomgraphs,Localoptimization. cCopyright2005,GayathriMahalingam

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Iwouldalsoliketothanktheothercommitteemembers,Dr.NatashaJonoska,andDr.GregoryMcColmwhomonitoredmyworkandtookeortinreadingandprovidingmewithvaluablecommentsonearlierversionsofthisthesis. Itakegreatpleasureinthankingallmyfriendswhohadhelpedinmanyways.MyspecialthanksgoestoNancy,Denise,Beverly,MaryAnn,AyaandFrances,whoworkintheMathoce. Iamgratefultomybrother,sister-in-law,mymother,granny,myuncleandaunt.Averyspecialthanksgoestomysisterwhohasalwaysbeenmyfriend,philosopherandguide.Shehasalwaysbeenaboosttomymentalstrengthandability.

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ListofFiguresiv Abstractv 1Introduction1 1.1BackgroundoftheDominatingSet...................1 2BoundsontheDominationnumber5 2.1ElementaryProperties..........................5 2.2BoundsintermsofOrderandMinimumdegree............10 3AlgorithmsandtheirComplexities14 3.1NP-completenessoftheDominationProblem..............14 3.2Approximationalgorithms........................18 AlgorithmsforRegulargraphs......................19 3.3AlgorithmsforSpecialGraphs......................21 4BreadthFirstSearch25 4.1BreadthFirstSearchasaHeuristic...................25 4.2LocalOptimizationProcedures.....................30 Internal-OptProcedure..........................30 Leaf-OptProcedure............................33 4.3ExperimentalResults...........................36i

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ImplementationResults.........................38 Conclusion41 ADenitionsandNotations42 BTabulationofExperimentalresults44 CPseudoCodes47 C.1BreadthFirstSearch...........................47 C.2Internal-OptProcedure..........................48 C.3Leaf-OptProcedure............................51 DSourceCodes52 D.1RandomGraphGenerator........................52 D.2RegularGraphGenerator........................58 D.3BreadthFirstSearch...........................66 D.4Internal-Optprocedure..........................71 D.5Leaf-Optprocedure............................85 D.6CheckingtheOutput...........................93 D.7TheMainPrograms...........................95 RandomGraph..............................95 Regulargraph...............................99 References103 AbouttheAuthorEndPageii

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4.2Boundsonthec(G)ofarandomgraphG................39 4.3Theconnecteddominationnumbersoftheoptimizationprocedures..40 B.1TheconnecteddominationnumberBFSL(G)foradregulargraphG.44 B.2TheconnecteddominationnumberBFSL(G)foradregulargraphG.45 B.3TheconnecteddominationnumberBFSL(G)forarandomgraphGn;p46iii

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3.1Reductionfrom3-SATtoDOMINATINGSET.............16 4.1ExampletoshowthatBFSdoesnotworkwellwithanyvertexasroot.29 4.2ExampletoshowthatInternalOptimizationfails.(a)RandomGraph.(b)BFStree............................33iv

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GayathriMahalingam

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AsetDVofverticesinagraphG=(V;E)iscalledadominatingsetifeveryvertexv2ViseitherinthesetD,orisadjacenttoavertexinD.TheminimumofthecardinalitiesofthedominatingsetsisthedominationnumberofthegraphG,denotedby(G). ClaudeBergeinhisbook[2]denedforthersttimetheconceptofthedominationnumberofagraph.Hecalledthisnumberthe\co-ecientofexternalstability".Theterm\dominatingset"and\dominationnumber"wasrstusedbyOreinhisbook[17].In1977,CockayneandHedetniemi[6],publishedasurveyofthefewresultsknownatthattimeaboutdominatingsetsingraphs.Inthatsurveypaper,CockayneandHedetniemiwerethersttousethenotation(G)forthedominationnumberofagraph,whichsubsequentlybecametheacceptednotation. TheproblemofndingdominatingsetsinagraphGisappliedinavarietyofsituations.Theconceptofdominationismainlyusedinnetworkproblemslike,com-1

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Anotherapplicationworthmentioningistheadhocnetworks.Adhocnetworksarecommunicationsystemswithnoxedinfrastructure.Thesenetworksareusedinapplicationssuchasmobilecommerce,searchandrescue,andmilitarybattleelds.Inthesenetworks,theinformationispassedbetweenhostsinthenetwork.Theinformationiscollectedatselectedhostsinthenetworkcalled\virtualbackbone"ofthenetwork.TheproblemofndingaminimumsizebackboneinadhocnetworkscanbereducedtotheproblemofndingaminimumconnecteddominatingsetinaconnectedgraphG. Findingadominatingsetwithminimumcardinality,foranarbitrarygraphwasshowntobeNP-complete.GareyandJohnson[9]intheirbookonNP-completeness,showedthatndingaminimaldominatingsetisNP-complete.ThisisdenotedastheDOMINATINGSETproblem.Therefore,noknownpolynomialtimealgorithmexistsfordeterminingthedominationnumberofanarbitrarygraph.Ifweexpecttobeabletondapolynomialtimealgorithmtocomputethedominationnumberofagraph,thenwewouldhavetorestricttheinstancestoclassesofgraphinsteadofarbitrarygraphs.TheDOMINATINGSETproblemremainsNP-completeevenwheninstancesarerestrictedtocertainclassesofgraphs.HararyandHaynesdenedtheconditionaldominationnumber,denotedby(G:P)asthesmallestcardinalityofadominatingsetDV,suchthatthesubgraphhDiinducedbythesetDsatisesthepropertyP.Someofthepropertieshavebeenlistedhere.

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Noteverygraphsatisestheseproperties.AdominatingsetD,andhencethegraphGmustnothaveisolatedverticesinordertosatisfythepropertyP1.ItshowsthatthegraphGmustbeconnectedinordertohaveadominatingset,thatsatisespropertyP2.PropertyP1leadstoanewdominationcalledtotal(open)domination.AdominatingsetDisatotaldominatingset,ifV=N(D).Theminimumofthecardinalitiesofthetotaldominatingsetgivesthetotaldominationnumber,denotedast(G). SampathkumarandWalikar[20]denedthedominatingsetDtobeaconnecteddominatingset,iftheinducedsubgraphhDiisconnected(propertyP2).Since,aconnecteddominatingsetincludesatleastonevertexfromeachcomponentofG,thereexistsaconnecteddominatingsetifandonlyifGisconnected.Theminimumofthecardinalitiesoftheconnecteddominatingsetsistheconnecteddominationnumber,denotedbyc(G).Itisobviousthat(G)c(G).GareyandJohnson[9]showedthattheproblemofndingaminimumconnecteddominatingsetisNP-complete.Pfa,Laskar,andHedetniemi[18]showedthatCONNECTEDDOMINATINGSETisNP-completeforbipartitegraphs. AstheproblemisNP-complete,manyupperandlowerboundswerecomputedfortheconnecteddominationnumberforanarbitrarygraph.Forexample,manyboundswerecomputedbySampathkumarandWalikar[20].KleitmanandWest[15]studiedconnectedgraphsthathavespanningtreeswithmanyleaves.TheconnecteddominationnumberofaspanningtreeofagraphGisthenumberofnon-leafnodesinthetree.Hence,ndingtheminimumconnecteddominatingsetDisequivalenttondaspanningtreeofGwithmaximumnumberofleaves.TheresultsofKleitmanandWest[15]giveseveralboundsforc(G).In[5],Caro,WestandYustergaveanupperboundwhichisanimprovementoftheresultofKleitmanandWest,andisasymptoticallysharp. InChapter2,wepresentvariousknownboundsonthedominationnumberaswellastheconnecteddominationnumberofaundirectedsimplegraphG.Thegraphs3

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InChapter3,wepresentsomeearlierknownapproximationalgorithmstondtheconnecteddominatingsetDinanundirectedsimplegraphG.TheNP-completenessoftheDOMINATINGSETproblemisalsodiscussedinsection3.1.Wepresentaproofasstatedin[13]toshowthatthedominatingsetproblemisNP-complete.Insection3.3,somespecialgraphsforwhichaconnecteddominatingsetcanbefoundinpolynomialtimearediscussed. InChapter4,wepresentourresults.WepresentinSection4.1theBreadthFirstSearch(BFS)asaheuristicinndingtheconnecteddominatingsetofanundirectedsimplegraphG.TheconnecteddominatingsetofthegraphGisthesetofnon-leafverticesintheBFStree.InSection4.2,twolocaloptimizationproceduresaregiventhat,whenimplementedonatree(hereitisBFStreewithanarbitraryvertexastheroot),willimprovethenumberofverticesintheconnecteddominatingsetofthegraphG.Therstoptimizationprocedureisthe\Internal-opt",inwhichweaimtoturntheinternalverticesofthetreeTtoleavesinordertogainmoreleavesinthetreeT.Thesecondprocedureisthe\Leaf-opt"inwhichwepicktheleavesoftheBFStreeandturnitintoaninternalvertextogainmoreleavesinthetreeT.Finally,inSection4.3,wetabulatetheexperimentalresultsobtainedbyimplementingtheBreadthFirstSearchtogetherwiththelocaloptimizationproceduresonrandomgraphsandrandomregulargraphs.AcomparisonofourresultswiththeresultsoftheearlierknowalgorithmsgivenbyGuhaandKhuller[11]andbyDuckworthandMans[7]isalsopresentedinthissection.4

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1(G)n

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thereexistsavertexv2VDforwhichN(v)\D=fug. Proof:Sinceaminimaldominatingsetincludesatleastonevertexfromeverycom-ponentofG,itisclearthatthevertexuiseitheracomponentofhDi,thatis,uisanisolatedvertexinD.Or,sinceDisaminimaldominatingset,thereexistsavertexv62D,suchthatvisadjacenttoonlyuinD.Thus,N(v)\D=fug.Conversely,iftheabovetwoconditionsholdandassumethatDisnotaminimaldominatingset,thenthereexistsavertexu2DsuchthatDfugisadominatingset.Therefore,uisadjacenttoatleastonevertexinD,andhencecondition1oftheorem2.1.1doesnothold.Also,everyvertexinVDisadjacenttoatleastonevertexinDfug.Thatis,condition2oftheorem2.1.1doesnotholdforu,thuscontradictingtheas-sumptions.HenceDisaminimaldominatingset.Adirectconsequenceofcondition2oftheorem2.1.1isthefollowingtheoremgivenbyOre.Theorem2.1.2 Proof:ForaminimaldominatingsetD,weshallshowthatVDisalsoadominatingset.(1) Bycondition1oftheorem2.1.1,ifavertexu2DisnotadjacenttoanyvertexinDthen,umustbeadjacenttoavertexinVD,sinceGhasnoisolatedvertices.ThisimpliesthatuisdominatedbyavertexinVD.6

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IfuisavertexinDthatsatisescondition2oftheorem2.1.1thenitisobviousthatuisdominatedbyavertexinVD. Sinceeveryvertexu2Dmustsatisfy(1)and(2)(bytheorem2.1.1)foragraphGwithnoisolatedvertices,itisseenthattheverticesinDaredominatedbyverticesinVD.Hence,VDisalsoadominatingset.ThisshowsthateitherjDjorjVDjisatmostn=2.Hence,thedominationnumberofGisatmostn=2. TheupperboundisachievedifeverycomponentofGisa4-cycleorGisaspecialkindofcoronagraph.ThecoronaoftwographsG1andG2,asdenedbyFruchtandHarary[8],isthegraphG=G1G2formedfromonecopyofG1andjV(G1)jcopiesofG2wheretheithvertexofG1isadjacenttoeveryvertexintheithcopyofG2.Thefollowingtheoremillustratesthisresult.Theorem2.1.3 Theconnecteddominationnumberofacompletegraphisgivenbyc(Kn)=1.Sinceaconnecteddominatingsetisnecessarilyadominatingset,SampathkumarandWalikar[20]provedthefollowingresult.Theorem2.1.4 Proof:LetDbeadominatingset,andjDj=(G)bethedominationnumber.LetmbethenumberofcomponentsofthesubgraphhDiinducedbythedominatingsetD.Itisclearthat(G)m.Weshallshowthatthereexiststwocomponents(sayCiandCj,wherei6=j)ofhDisuchthatthelengthofashortestpathbetweenCiandCjisatmost3inG.Supposeforthepurposeofcontradictiontheshortestpath7

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Proof:EveryvertexinthegraphGcandominateatmost4(G)verticesanditself.Hence,(G)n Toprovetheupperbound,letvbeavertexwithmaximumdegree4(G)inG.FormaspanningtreeTofGsuchthateveryneighborofvinGisalsoaneighborofvinT.ThiswillresultinatreeTwithN(v)branchesinitandhencewithatleast4(G)leaves.Hence,theconnecteddominationnumberisatmostn4(G).2

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Proof:Letvbeavertexwithmaximumdegree4(T)inatreeT.IfTisaspiderwithvastheroot,thenweseethatthetreeThasexactly(T)branchesfromv(sinceverticesineachofthesebrancheshasadegreelessthan3andTisatree).ThusthenumberofleavesinthetreeTisexactly(T).Hencetheconnecteddominationnumberc(T)=n4(T).

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WestandKleitman[15]provedsomeboundsonc(G)whichwereproventobenearlyoptimalwhenkissmallwhere,(G)kforsomeintegerk.TheygaveanalgorithmicproofthatndsaspanningtreewithmanyleavesinaconnectedsimplegraphG.ForacyclicgraphCnwithnverticeswecanguaranteeonlytwoleavesforthespanningtreeT.WestandKleitman[15]consideredgraphsinwhicheveryvertexhavedegreeatleastk,thatis(G)k. LetGn;kdenotethecollectionofconnectedsimplegraphswithnverticesandaminimumdegreeatleastk.Letl(n;k)denotethemaximummsuchthateverygraphinGn;khasatreewithatleastmleaves.Noticethateverytreehasatleasttwoleaves,andhencel(n;k)2.SinceeveryspanningtreeofCn,acyclewithnvertices,hasexactly2leaves,Weseethat(Cn)=2andhencel(n;2)=2.Theorem2.2.1

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SinceanypairofedgesinZformanedgecut,atmostoneedgeofZisnotinT.Iffxj;yj+1g62T,forsomej,thenTmusthaveapathfromxitoyiinRi,foreveryi,inordertoremainconnected.ThisisbecausejRij3.HenceRi,foralli,musthaveanon-leafvertexotherthanxiandyi2Ri.Also,everyvertexoftheWmustbeanon-leafexceptperhapsxj,yj+1.Thus,thetreeThasatleast3m2internalvertices,andhenceatmostn3m+2leaves. IfTincludesalltheedgesinZ,thenthereisnoxiyi-pathinRiforonei(sayj)inT.Hence,theremustbeatleast3(m1)non-leavesinV(G)Rjandsincek2,eitherxioryjmustbeanonleaf.Therefore,connectedthedominationnumberc(G)isatmost3bn=(k+1)c2.Hence,themaximumnumberofleavesinthespanningtreeis,

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5n+8 5wasprovedinGriggsandWu[10]andinKleitmanandWest[15].GriggsandWu[10]alsoprovedthatl(n;5)3 6n+2.Thefollowingboundsinthissectionarenearlyoptimalwhen(G)islarge,andasn!1.TherstbounddiscussedhereisgivenbyAlonandSpencer[1].WealsopresenttheprobabilisticproofgivenbyAlonandSpencer[1].Theorem2.2.4 Proof:Letp=ln((G)+1)=((G)+1).ThedominatingsetDisconstructedasfollows.WeconstructasetSsuchthateveryvertexinSisselectedindependentlywiththeprobabilityp.Then,theexpectedvalueofthecardinalityofSisnp.LetBbethesetofverticesthatarenotdominatedbyanyoftheverticesinS,thatis,B=VN[S].AvertexvisinBifandonlyifvanditsneighborsarenotinS,thatis,ifandonlyifN[v]6D.Hencetheprobabilitythatv2Bis(1p)(1+deg(v)).Sinceep1p,anddeg(v)(G),itisclearthattheprobabilitythatvisinBisatmostep(1+(G)).ThereforetheexpectedvalueofjBjisatmostnep(1+(G)).ItisclearthatD=S[Bisadominatingset,andtheexpectedsizeofDisatmost SincetheaveragecardinalityofDisatmostn(1+ln((G)+1))=((G)+1),theremustbeaparticularsetSwithatmostthiscardinality.2

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(G)+1ln((G)+1)steps,therewillbeatmostn=((G)+1)uncoveredverticeswhichwhenaddedtothedominatingsetwillgivetheboundoftheorem2.2.4. Theprobabilisticargumentsusedintheproofoftheorem2.2.4arefurtherstudiedbyCaro,WestandYuster[5]forconnecteddominatingset.Theyobtainthefollowingresultforconnecteddominationnumber,whichisasymptoticallysharp.Theorem2.2.5

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Thesimplestprocedurewouldenumeratethe2nsubsetsofV(G)anddeterminewhethertheenumeratedsubsetDVisaconnecteddominatingsetandifso,outputthecardinalityofDas(G)andhalt.SuchanalgorithmiseasytoconstructbutrequiresO(2n)stepsintheworstcase.Thatis,ithasanexponentialtimecomplexityintheorderofthegraphG. Weareinterestedtoknowifthereexistanalgorithmthatndsthevalueof(G)foranarbitrarygraphGandrunsinpolynomialtime.Todate,noonehasconstructedadominationalgorithmthathasbetterthanexponentialtimecomplexityforarbitrarygraphs.Furthermore,theNP-completenessofthedominatingsetsuggeststhatitis14

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GareyandJohnson[9]mentionedthatthedominationproblemisNP-completeforarbitrarygraphs.Thebasiccomplexityquestionsconcerningthedecisionproblemforthedominationnumbertakesthefollowingform:DOMINATINGSETINSTANCE:AgraphG=(V;E)andapositiveintegerk.QUESTION:DoesGhaveadominatingsetofsizek?.DavidJohnsonshowedthatthedominatingsetisNP-complete.Hisproofisasfollows.(Alsosee[13]fortheproof.)Theorem3.1.1 Proof:Therearetwosteps.TherststepistoprovethattheDOMINATINGSETproblemresidesintheclassofNP.Thisinvolvesaneasyvericationofa\yes"instanceofDOMINATINGSETinpolynomialtime,thatis,foragraphG=(V;E),apositiveintegerkandanarbitrarysetDVwithjSjk,itiseasytoverifyinpolynomialtimewhetherDisadominatingsetornot.ThesecondstepofanNP-completenessproofistoselectaknownNP-completeproblem,anddeneatransformationfromthisproblemtotheDOMINATINGSETproblem.Weusethewellknown3-SATproblemhere.Thisproblemcanbestatedinthefollowingform.3-SATINSTANCE:AsetU=fu1;u2;:::;ungofvariables,andasetC=fC1;C2;:::;Cmgof3-elementsets,calledclauses,whereeachclauseCicontainsthreedistinctoccur-rencesofeitheravariableuioritscomplimentu0i.Forexample,theclauseC1inFigure3.1isC1=fu1;u2;u03g.QUESTION:DoesthesetChaveasatisfyingtruthassignment,thatis,anassign-mentofTrueandFalsetothevariablesinUsuchthatatleastonevariableineachclauseinCisassignedthevalueTrue?GivenaninstanceCof3-SAT,wecanconstructaninstanceG(C)ofDOMINAT-15

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WeneedtoshowthatCisa\yes"instanceofthe3-SATproblemifandonlyifG(C)isa\yes"instanceoftheDOMINATINGSET,fork=n.InotherwordsChasasatisfyingtruthassignmentifandonlyifthegraphG(C)hasadominatingsetofcardinalityatmostn. LetChaveasatisfyingtruthassignment.Forexample,inFigure3.1,weseethatu1=False,u2=True,u3=True,u4=False,u5=False.ThesetDofverticesinG(C)iscreatedsuchthatalltheverticesinthesetDhasatruthassignment.Forexample,ifui=True,thenui2D,andifui=False,thenu0i2D.ThesetDisthedominatingsetofG(C)since, Therefore,thesetDisadominatingofG(C)withcardinalityn.16

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WemustalsoshowthattheconstructionexplainedbytheFigure3.1,forcreatinganinstanceofDOMINATINGSETfromaninstanceof3-SAT,canbecarriedoutinpolynomialtime.Thelengthofaninstanceof3-SATisgivenbymsetseachofsizethreeplusnvariables,thatis,O(3m+n).ThegraphG(C)has3n+mverticesand3n+3medges.Hence,thecardinalityofG(C)isatmostaconstanttimesthecardi-nalityofC,andthusG(C)canbeconstructedinpolynomialtimefromaninstanceof3-SATproblem.2

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TwoapproximationalgorithmsforndingtheconnecteddominatingsetofagraphGwasproposedbyGuhaandKhuller[11].Therstalgorithmisamodiedgreedyal-gorithmthatachievesaratiojCDSj=jDSOPTjwhichisapproximately2(1+H(4(G))),whereHistheharmonicfunction.Thealgorithmisexplainedasfollows. Initially,allverticesaremarkedaswhite.Astherststep,weselectavertexwiththemaximumnumberofwhiteneighborsasadominatingvertex,andmarkitsneighborsasgrey.Iteratively,thegreyverticesarescanned.Theprocessofscanninginvolvesaddingthegreyvertextotheconnecteddominatingsetandcoloringallitsneighborstogrey(dominated).Ateachstepofthescanningprocess,weeitherselectthegreyvertexorthegreyvertexanditswhiteneighbor,whicheverdominatesmorewhiteneighbors.Thisiscalledthelookaheadprocedure.Attheend,wegetaspanningtreeT,andtheconnecteddominatingsetincludestheverticesinthetreeTthatarenotleaves.Theorem3.2.1

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Forad-regulargraphonnvertices,dnpointsaretakeninnbucketslabeled1:::nwithdpointsineachbucket.Thenadisjointpairingofthesednpointsischosenuniformlyatrandom.Thebucketsaretheverticesoftherandomlygeneratedgraphandeachpairrepresentsanedgewhoseend-pointsaregivenbythebucketsofthepointsinthepair.Thisprocessiscalledthepairingprocess.Thealgorithmiscombinedwiththepairingprocessthatuniformlyatrandomgeneratesthegraphasdescribedabove. Thegraphbeinggeneratediscalledtheevolvinggraph,andavertexissaidtobesaturatedifithasadegreedintheevolvinggraph.LetVi=Vi(t)bethesetofverticesofdegreeioftheevolvinggraph(graphbeinggenerated)attimet,andYi=Yi(t)denotejVij.19

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Therestofthealgorithmisdividedintotwostages.Intherststage,weselectavertex,v,uniformlyatrandomfromV1andthenexposeoneoftheedgesincidentwithvtoavertexw(say).Ifv2V1,thenweaddvtoD(sincevdominatesw)andexposetheremainingedgesofv.Ifw62V1,thenthealgorithmproceedswithoutaddingvtoD. Inthesecondstageofthealgorithm,wedenotektobethecurrentminimumdegreeofalltheverticesthathavenon-zerodegree.Ifk=1,thenweselectavertexu(say),uniformlyatrandomfromV1andexposeitsremainingedges.Ifk6=1,weselectavertexufromVkuniformlyatrandomandexposeanedgeincidentwithutov.Ifv2V1,thenuisaddedtoD,andalltheremainingedgesincidentwithuareexposed.Otherwise,theoperationterminateswithoutincreasingthesizeofD.Thereasonbehindexposingk1edgesincidentwithuistoincreasetheminimumdegreeoftheverticesthathavenon-zerodegree. AvertexuisaddedtoDifandonlyifoneormoreneighborsofu,alongtheexposededgeshavedegree1.thisensuresthatDisadominatingsetofG.Also,sinceeachvertex,u,chosenforpossibleadditiontoD,isselecteduniformlyatrandomfromthoseverticesofaparticularnon-zerodegree.ThisensuresthatDisconnected.20

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Westartfromthevertexvadjacenttooneoftheendverticesinthepath,andincludeitinthedominatingsetD.Ateachstep,weaddthevertexwhichisatadistanceof3fromthevertexaddedpreviouslytothedominatingset,untilalltheverticesaredominatedinthepathorinthedominatingsetD. Theconnecteddominationnumberofacycleissimplytwoverticeslessthanthenumberofverticesinthecycle.Sincethespanningtreeofacycleisapath,ndingaminimumdominatingsetforanycycleissimilartondingadominatingsetofapath.Hence,ifGisapathoflengthnthen(G)=dn

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Wecanseethatthecyclegraphsarenotsubgraphsoftheintervalgraphs.Itwasshownin[4]thattheproblemofdeterminingwhetheragraphisanintervalgraphcanbedoneino(n)time.Also,thereexistsalgorithmsthatruninpolynomialtimetondthetotaldominationnumbert(G),andtheconnecteddominationnumberc(G).WepresenthereanalgorithmasexplainedbyHaynes,Hedetniemi,andSlater[13]tondaminimumconnecteddominatingset. ThecollectionofintervalsofanintervalgraphG=(V;E)iscalledtheintervalmodelofGandisdenotedbyI.Thecoordinatesoftheintervalsareobtainedbylabelingtheendpointsfromlefttorightby1;2;:::;2n,wherenisthenumberofvertices.Theintervalsarethuslabeledfrom1toninincreasingorderoftheirrightendpoints. Theintervalgraphsaresaidtohaveaboxicityof1.Agraphhasboxicity1,ifithasanintersectionmodelconsistingofboxesin1-dimensionalspace.ForintervalgraphstheDOMINATINGSETproblemcanbesolvedinpolynomialtime.Agraphhasboxicity2ifithasanintersectionmodelconsistingofboxesin2-dimensionalspace.Theproblemofndingadominatingsetforgraphswithboxicity2isNP-complete. Theintervalgraphisconstructedasfollows.LetVi=f1;2;:::;igandGi=hViibethesubgraphofG,inducedbytheverticeslabeledfrom1;2;:::;i.GiisobtainedbyaddingthevertexitoGi1andjoiningittotheverticesinGi1whoseintervalintersectwiththeintervalofi.Thisgivesusthe\leftdegree"ofthevertexi.Hence22

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LetCD(i)denotetheconnecteddominatingsetofGi,whichincludesvertexi,andletLowNbr(i)denotetheleastvertextowhichiisadjacentto.LetMinCD(i)denotetheCD(i)withminimumcardinality.IfLowNbr(i)=1,thenvertexidominatesallofthevertices1;2;:::;i1inGiandhenceMinCD(i)=fig.IfLowNbr(i)>1,thentheremustbeanothervertexotherthaniinCD(i),whichisadjacenttoiinGi.LetjbethemaximumvertexinCD(i)figsuchthatLowNbr(j)
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Initially,allverticesaremarkedunvisited.2. Choosetherootasthestartingvertex.3. Marktherootasvisited.4. Addtherootattheendofthequeue,andaddittothetreeT.5. Chooseavertexvfromthefrontofthequeue,andvisitallunvisitedneighborsofv.6. Foreachunvisitedneighboruofv, markuasvisited.25

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addthevertexu,andtheedgefv;ugtothetreeT.7. Repeatsteps5and6untilthequeueisnotempty.8. ThetreeTthusobtainedistheBFStree. ThePseudocodeisgiveninappendixC. Whybreadthrstsearchisgoodtotry?.TheBFSexpandsthetreebetweenthevisitedandtheunvisitedverticesuniformlyacrossthebreadthofthetree.Thatis,thealgorithmvisitsallverticesatadistancekfromthe\root"beforevisitinganyvertexatadistanceofk+1.Therandomgraph,denotedbyGn;pwithnverticesandanedgeprobabilityp,isgeneratedsuchthatthereexistsanedgebetweentwoverticesindependentlyofotherpossibleedges,withprobabilityp.Thegraphisgeneratedasfollows.Foreverypairofvertices,arandomnumberbetween0and1isgenerated.Thereexitsanedgebetweenapairofverticesiftherandomnumbergeneratedforthatpairisatmosttheedgeprobabilityp.Theconnecteddominationnumberob-tainedbytheBFSwithanarbitraryvertexastherootisdenotedasBFS(G).ForanyconnectedgraphGn;p,thefollowingconjectureclaimsthatwhennissucientlylarge,thenBFS(G)nln(d) where,d=dn=np=npn,andpistheedgeprobabilityofthegraphGn;p.AheuristicargumentfortheConjecture26

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Ateachstepoftheiteration,weaddanexpectednumberof(ny)pedgestotheBFStree.Hence,thechangeinthenumberofverticesintheBFStreeisgivenby,4y=(ny)p.Since,thechangeinyissmall,wehavethat,dy dx=(ny)pSolvingthedierentialequation,weget,Zdy ny=Zpdx;whichgivesthat,ln(ny)=px+c givingthat,px=lnn ny:(4.1.2) Let(ny)p=A,forsomelargenumberA.Then,y=nAn d:27

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n.Usingthisin(4.1.2),wegetpx=lnn An=d=ln(d=A):Therefore,x=1 A=n d(ln(d A))(4.1.3) NoticethatthenumberofinternalverticesintheBFStreeisatmostx.Thusfromequation(4.1.3),whenx=n dln(d=A),thenumberofverticesintheBFStreeisy=nAn d,andthenumberofinternalverticesisatmost dln(d=A). SincethereareAn=dverticesnotintheBFStree(sofar),andeachoftheseverticeswillcreateatmostonenewinternalvertexwhenBFSiscompleted,thenumberofinternalverticesinthenalBFStreeisatmostx+An d=n d(ln(d)ln(A))+An d=n d(ln(d)ln(A)+A): dln(d).RunningTimeLetGbeagraphwithnverticesandmedges.ItiswellknownthatBFStraversalofGtakesarunningtimeofO(n+m).Ateachstepofthealgorithm,wepicktheun-visitedverticesthatareadjacenttoavertexwhichisvisitedbefore.Theseunvisitedverticesarenowvisited,andareaddedtothequeue.Hence,everyvertexisaddedtothequeueexactlyonce,andtakesarunningtimeofO(1).Therefore,thetotaltimeittakestoenqueueanddequeuealltheverticesisO(n).Sinceavertexisdequeuedfromthequeue,itsadjacencylistisscannedatmostonce.Scanningtheadjacency28

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13whichcanbeverybadwhendisverybig. Inviewofthisbadexample,itisdesirabletodevisesomelocaloptimizationprocedurestotheBFStreesothattheconnecteddominationnumberofthegraphGcanbeimproved.Inthefollowingsection,wediscusstheseoptimizationprocedures.TheselocaloptimizationprocedureshoweverdonotguaranteetoincreasethenumberofleavesintheBFStreeforallthegraphsG.29

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Initially,aninternalnodevismadethe\root"ofthetreeT.ThisgivesatmostdegT(v)branchesinthetree,thatis,atmostdegT(v)subtrees,where,degT(v)isthedegreeofthevertexvinthetreeT.Initially,therearenofailuresforthevertexv.2. Allthesesubtreesarejoinedtogetherbyajoiningprocess.Thejoiningprocessinvolvesaddinganon-treeedgebetweenu2N(v)oroneofthedescendantsofuwithavertexinanothersubtreeofthetreeTwiththerootv.Thejoiningprocesshasthefollowingrestrictionsinit.Therestrictionsaredesignedsothatnonewinternalverticesarecreatedinthejoiningprocess.(a) Twointernalnodesfromdierentsubtreescanbejoinedwithanon-treeedge.30

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Ifexactlyoneofthenodesbeingjoinedisaleaf,thentheleafmustbeaneighbortothevertexvinthetree.(c) Ifboththeverticestobejoinedareleaves,thendonothing.3. WhentwoverticesintwodierentsubtreesofthetreeTarejoinedwegetacycleinthetree,andhenceweremoveanedgebetweenvandoneofitsneighborsinthesetwosubtrees. Therearetwocasesofremovinganedgebetweenthevertexv,anditsneighbor.(a) Iftwointernalverticesfromdierentsubtreesarejoinedduringthejoiningprocess,theedgebetweenthevertexv,andanyoneofitstwoospringsinthosetwosubtreesisremoved.(b) Inthecaseof2bofthejoiningprocess,theedgebetweenthevertexvanditsneighborwhichisaleafisremoved.Thejoiningprocessforthecurrentbranchiscontinued.4. Ifthereisabranchthatcannotbejoinedtoanyoftheotherbranches,thenthereisafailure.5. Repeatsteps2,3,and4untildegT(v)=1,andthereisnotmorethanonefailure.Iftwoormoresubtreescannotbejoinedtogetherthen,thevertexvcannotbeturnedintoaleaf.HenceitremainsasaninternalvertexinthetreeT,andthechangesmadetothetreearediscarded.6. Thewholeprocedureisrepeateduntilalltheinternalverticesinthetreehavebeenconsidered. Case2bandcase2cofthejoiningprocessensuresthatnoleafinthetreeisturnedintoaninternalvertexduringInternal-optprocedure.Thejoiningandremovingprocessesensuresthatnoleafisturnedintoaninter-nalvertexduringtheoptimizationofaninternalvertexinthetreeT,andhencewe31

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Proof:IfthereisafailureinoptimizinganinternalvertexvinthetreeT,thenitisclearthatatleasttwooftheospringsofthevertexvcannotberemoved.SincenoleafisturnedintoaninternalvertexintheprocessofjoiningandremovinganedgefromthetreeT,wegainnointernalvertexwiththeoptimizationprocedure.Therefore,ifaninternalvertexcannotbeturnedintoaleaf,thenitremainsasaninternalvertexinthetreeT.Hence,itsucestooptimizeeachoftheinternalverticesofthetreeTexactlyonce.2RunningtimeTheworstcaserunningtimeoftheInternal-OptprocedureisO(n242).Ateachiter-ationoftheoptimizationprocedure,aninternalvertexvinTispicked.ThisincuratmostO(n)iterations.EdgesbetweenvanditsospringsinTcanberemoved,andthisrequiresatmostO(4)iterations.Foreachsuchospring,eithertheospringoroneofitsdescendantsisjoinedwithavertexinanotherbranch.Tondthatneighbor,weperformabreadthrstsearchwiththeospringasthe\root",andeachsuchBFSrequiresO(n)steps.TheprocessofjoiningtwobranchesrequiresBFStogothroughtheverticesofonebranch,andforeachsuchvertex,lookforaneighborinanotherbranch.Hence,ittakesatmostO(n4)steps.ThejoiningprocessrequiresatotalofO(n+n4)iterations.Therefore,thetotalrunningtimerequiredisatmostO(n242). Thefollowingexampleshowsthattheinternaloptimizationdoesnotworksowellforsomegraphs.32

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LetTdenotethecurrentspanningtreeofG,withnverticesandlleaves.AleafxinTisexpandedifanedgeisaddedtoT,fromxtoanyoneofitsneighborsnotinT.Whenaleafisexpanded,thetreeThasacycleinit,andhencewede-cycleit.Theprocessofde-cyclinginvolvesremovinganedgeinthecyclebetweentwovertices,whereatleastoneoftheverticeshasadegreetwointhetreeT.Theprocedureisexplainedasfollows. Ateachstepoftheprocedure,wepickaleafx2T,andexpandit.NowthereisacycleinthetreeT,andhencewede-cyclethetreeT.Intheprocessofde-cycling,33

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SincethetreeThasatleasttwoleavesinit,theleafoptimizationprocedurecanrunindenitely.Toavoidsuchasituation,weimplementthefollowingstoppingruleonthetreeT.Theleaf-optisterminatedifitcomesacrossa(current)treeforwhichtheexpansionofallitsleavesresultinnogaininthenumberofleaves. Itcanbeeasilycheckedthattheleaf-optperformedonthegraphsinFigure4.2andFigure4.1producesaminimumconnecteddominatingset.Theorem4.2.2 Proof:Bydenitionofthestoppingrule,thetreeTisupdatedwiththecurrenttree,ifandonlyifthereisanoverallgainofleaves,thatis,ifandonlyifthenumberofleavesinthetreeTincreases.Sincethegraphisnite,andthenumberofleavesinatreeisboundedfromabove,theoptimizationproceduremusteventuallystop.RunningtimeAsimpleimplementationappearstogiveaworstcaserunningtimeofO(n34).Ateachiteration,wechoosealeafforoptimization.ItisclearthatwemayhaveO(n)iterations,sincethereareO(n)leavesinthetree.Ineachiteration,everynon-treeedgeoftheleafisaddedtothetreeT.HenceateachiterationatmostO(4(G))edgesisaddedtothetreeT.Foreachedgeadded,wendacycleandanedgetoberemovedfromthecycle.AtmostO(n)stepsareusedinndingtheedgetoberemovedfromthecycle,sincethereareO(n)edgesinthetree.Theoptimizationprocedureisrepeateduntilthereisnogainofleaves.ThetreeTisupdatedatmost34

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RegularGraphgenerationTheregulargraphisgenerateduniformlyatrandombythefollowingprocedure. Forad-regulargraphonnvertices,rsttakednpointsinanarrayofsizedn.Thearrayisarrangedinascendingorderofthednpoints,andtherstdpointscorrespondtotherstvertex,andsoon.Usethisarraytogeneratearandompermutationofthepoints.Placeanedgebetweenthepointsintheithandthe(i+1)starray,wherei=1;3;5;:::;dn1.Groupthepointsandtheedgescorrespondingtoeveryvertextoformadregulargraph.Weremovethemultipleedgesandloopsinthegraph,whichgivesusa\near"regulargraph. ABFSisperformedontherandomregulargraph,toensurethatthegraphisconnected.Ifthegraphisnotconnected,thenthewholeprocessisrepeateduntilaconnectedrandomregulargraphisobtained.RandomGraphGeneration

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Intherststageofthealgorithm,apairofvertices,uandv,ischosenfromthevertexsetV(G)ofthegraphG.Foreachpairofverticesfu;vg,wherev2V(G)fugarandomnumberbetween0and1isgenerated.ThereexistsanedgebetweenuandvinthegraphifandonlyiftherandomnumbergeneratedforthepairuandvisatmosttheedgeprobabilitypofthegraphG.Thisoperationisrepeateduntilallpairsofverticeshavebeenconsidered. Inthesecondstageofthealgorithm,abreadthrstsearchisperformedontheran-domgeneratedgraphtoensurethatthegraphisconnected.Ifthegraphisconnected,thenweretaintheadjacencylistofthegraph.Theabovealgorithmisrepeateduntilaconnectedsimplegraphisobtained.Breadth-FirstSearchandLocalOptimization1. GivenarandomlygeneratedgraphGasinput,wearbitrarilychooseavertexastherootandperformtheBFSontheinputgraph.TheBFStreethusobtainedisretainedasaninputtothelocaloptimizationprocedure,andthenumberofinternalverticesintheBFStreegivestheBFS(G).2. TheBFS(G)canbefurtherimprovedbyapplyingthelocaloptimizationpro-cedurestotheBFStree.Aquestionarisesaboutwhichproceduretobeim-37

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TheInternal-optisperformedontheBFStreebychoosinganinternalvertexfromtheBFStree.Thetreeisupdatedonlyifthereisagainofleavesinthetree.4. TheLeaf-optproceduretakesthetreeobtainedfromtheInternal-optasitsinput.Theoutputoftheleaf-optprocedureisaspanningtreewithpossiblymorenumberofleavesthanthespanningtreeobtainedfromtheInternal-optprocedure.5. ABFSisperformedonthespanningtreeobtainedfromtheoptimizationpro-cedures.Thisensuresthatthespanningtreeproducedbytheoptimizationproceduresareconnectedandhasthesamenumberofleavesasgivenbytheleaf-optprocedure.ImplementationResults Experimental DuckworthandMans bound 3 0.542592n 0.5854n 0.3466n 4 0.413667n 0.4565n 0.3219n 5 0.342275n 0.3860n 0.2986n 10 0.212892n 0.2397n 0.2180n 20 0.132000n 0.1493n 0.1450n 30 0.097333n 0.1121n 0.1108n 40 0.078308n 0.0910n 0.0906n 50 0.066183n 0.0771n 0.0771n Table4.1:TheBoundsonc(G)forarandomdregulargraphG.38

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(G)+1inthefourthcolumnofthetable4.1.Weusedfor(G).TheboundsgivenbyDuckworthandMans[7]wereproventobeasymptoticallyalmostsureupperboundsontheconnecteddominationnumberofrandomregulargraphs. Comparingourboundswiththeboundsin[7]and[1],weseethattheourboundsaresimilartotheboundsin[7]and[1].ThisshowsthatBFStogetherwiththelocaloptimizationproceduresworkwellinndingaconnecteddominatingset. Thefollowingtableliststheexperimentalboundsandtheearlierknownboundsontheconnecteddominationnumberc(G)ofarandomgraphG. Probabilityp Bound Bound 500 0.178000n 0.271012n 0.248400n 1000 0.170400n 0.261181n 0.239000n 0.169733n 0.255790n 0.239200n 2000 0.162200n 0.252113n 0.232800n 500 0.077600n 0.118102n 0.103600n 1000 0.067600n 0.100292n 0.089400n 0.063067n 0.091733n 0.082133n 2000 0.058400n 0.086326n 0.078000n 500 0.015600n 0.026906n 0.019200n 1000 0.013000n 0.020570n 0.017000n 0.011867n 0.017778n 0.015067n 2000 0.010600n 0.016102n 0.013100n Table4.2:Boundsonthec(G)ofarandomgraphG.

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BFS(G)IOPT(G)LOPT(G) 455327326 383 154115114 140 272121 27 Table4.3:Theconnecteddominationnumbersoftheoptimizationprocedures.

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WehavepresentedtheBreadthFirstSearch(BFS)asaheuristicinndingtheconnecteddominatingsetDofasimpleundirectedconnectedgraphG.WehavemadeaheuristicargumentshowingthatBFSgivesaconnecteddominatingsetwhosecardinalityissimilartotheAlon-Spencerbound.ItremainstoprovetheconjectureandhenceshowthattheBFSgivesaboundsimilartotheboundgivenbyAlonandSpencer[1]. TherearegraphsforwhichtheBFSgivesabadperformanceratioinndingamin-imumconnecteddominatingset(seeFigure4.1).TwolocaloptimizationproceduresthataimtoincreasethenumberofleavesintheBFStree,andhencetheconnecteddominationnumber,havebeenproposedinthiswork.ProvidingatheoreticalprooftoshowthattheselocaloptimizationprocedureswillsignicantlyimprovetheresultofBFSneedstobeinvestigated. TheexperimentalresultssuggestthattheBFSandthelocaloptimizationproce-durescanworkwellatleastforrandomgraphsGn;pandrandomregulargraphs.ItwasalsoshownthattherearegraphsforwhichtheInternal-optproceduredoesnotgiveanyimprovementinthenumberofleavesintheBFStree.AlthoughtheBFSandthelocaloptimizationproceduresarepolynomialtimealgorithms,theydonotguaranteeanoptimalsolutionfortheboundonc(G).Furtherinvestigationisneededtoseehowwellthelocaloptimizationproceduresperformintheworstcase.41

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(N(v)andN[v])Theopenneighborhoodofv2V(G)isthesetofverticesadjacenttov,N(v)=fwjfv;wg2E(G)g,andtheclosedneighborhoodofvisN[v]=N(v)[fvg.DenitionA.0.2 (N(S)andN[S])ForasetSV(G),N(S)=[s2SN(s)andN[S]=[s2SN[s].DenitionA.0.3 ((G)and4(G))TheminimumdegreeofGis(G)=minfdeg(v):v2V(G)gandthemaximumdegreeofGis4(G)=maxfdeg(v):v2V(G)g.DenitionA.0.5 (InducedSubgraph)Aninducedsubgraphisasubset,S,oftheverticesofagraphGtogetherwithanyedgesofG,whoseendpointsarebothinthissubset.Thatis,42

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(Diameterofagraph)ThediameterofaconnectedgraphGistheleastintegerDsuchthatforallverticesuandvinGwehaved(u;v)D,whered(u;v)denotesthedistancefromutovinG,thatis,thelengthoftheshortestpathbetweenuandv.DenitionA.0.7 (EdgeCut)AnedgecutfortwoverticesuandvisthesetofedgeswhoseremovalfromthegraphGdisconnectsuandv.AnedgecutforthewholegraphGisthesetofedgeswhoseremovalrendersthegraphdisconnected.DenitionA.0.8 (Dominatingset)AsetofverticesDisaDominatingsetofagraphG=(V;E),iseveryvertexinVDisadjacenttoatleastonevertexinD.Thedominationnumber(G)ofagraphGistheminimumofthecardinalitiesofthedominatingsetsofthegraphG.DenitionA.0.9 (TotalDominatingset)AsetDisatotaldominatingset,alsocalledanopendominatingset,ifforeveryvertexu2Vthereexistsavertexv2D,suchthatuisadjacenttov.Thetotal(open)dominationnumberofagraphGist(G)=minfjDj:DV(G)andV=N(D)g.DenitionA.0.10 (ConnectedDominatingset)AsetDVofverticesinanyconnectedgraphGiscalledtheConnectedDominatingsetofthegraphGifthesubgraphhDiinducedbythesetofverticesDisconnected.TheminimumofthecardinalitiesoftheconnecteddominatingsetsofGiscalledtheconnecteddominationnumberc(G)ofthegraphG. SinceadominatingsetmustcontainatleastonevertexfromeverycomponentofG,thereexistsaconnecteddominatingsetforagraphGifandonlyifGisconnected.43

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n AverageBFSL(G) 274270271270269 0.5416 1000 539547538551541 0.5432 3 1500 817809817821815 0.5438 2000 10811084108210841086 0.5417 500 212205206210208 0.4164 1000 410416414403416 0.4118 4 1500 621621621617627 0.4142 2000 820822827818835 0.4122 500 174171165178169 0.3436 1000 340344345340343 0.3424 5 1500 505515509517516 0.3416 2000 687690681685680 0.3423 TableB.1:TheconnecteddominationnumberBFSL(G)foradregulargraphG.44

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n AverageBFSL(G) 108106105111105 0.2140 1000 211213211211207 0.2106 10 1500 322321326321314 0.2138 2000 435419425428424 0.2131 500 6768686666 0.1340 1000 138131135130132 0.1332 20 1500 192193198193193 0.1292 2000 266260259266265 0.1316 500 4949484851 0.0980 1000 97981009392 0.0960 30 1500 143145150143149 0.0973 2000 191198199195197 0.0980 500 3839404240 0.0796 1000 7381757580 0.0768 40 1500 121120114120117 0.0789 2000 159158156152154 0.0779 500 3232343336 0.0668 1000 6767646666 0.066 50 1500 989996100100 0.0657 2000 132131136128135 0.0662 TableB.2:TheconnecteddominationnumberBFSL(G)foradregulargraphG.45

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BFSL(G) AverageBFSL(G) 500 8992879186 0.17800 1000 173168168173170 0.17040 252251254259257 0.16973 2000 328326329318321 0.16220 500 4039383740 0.07760 1000 6471676868 0.06760 9993889499 0.06306 2000 116118120116114 0.05840 500 78879 0.01600 1000 1313141312 0.01300 1717181918 0.01186 2000 2221222021 0.01060 TableB.3:TheconnecteddominationnumberBFSL(G)forarandomgraphGn;p

PAGE 55

1. parent[u]=u markrootas\visited"3. Queue=frootg4. parent[v]=u endwhile5. returnfparent[]g47

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1. Ifdegree(v2)then OPT(root) OPT(root) 1. Descendants(u) Leaf(u) return; Leaf(u) 48

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Iffu;wg2E(G)thenAddfu;wgtoT Descendants(u) 1. Markuasadescendant2. AddutotheQueue3. Ifyisnotadescendantthen return; 1. AddutoQueue2.

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Iffx;yg2E(G)andyisnotadescendantthenAddtheedgefx;ygtoT

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1. forv2Tdo OPT(v) updatethetreestructure LEAF OPT(v) 1. Iffu;vg2Gthen

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N.AlonandJ.Spencer,TheProbabilisticMethod,WileyInterscience,NewYork,1992.[2] C.Berge,TheTheoryofGraphsanditsapplications,MethuenandCo,London,1962.[3] Bollabas,B.:RandomGraphs.AcademicPress,1985.[4] K.BoothandG.Lueker,TestingfortheConsecutiveOnesProperty,IntervalGraphs,andGraphPlanarityUsingPQ-TreeAlgorithms,J.ofComputerandSystemSciences,vol.13,pp.335-379,1976.[5] Y.Caro,D.B.West,andR.Yuster,ConnectedDominationandSpanningTreeswithManyLeaves,SIAMJ.Discr.Math.13,2000,202-211.[6] E.J.CockayneandS.T.Hedetniemi,Towardsatheoryofdominationingraphs,JohnWileyandSons,NY,7,1977,247-261.[7] W.DuckworthandBernardMans,OntheConnectedDominationNumberofRandomRegularGraphs,Springer-Verlag,BerlinHeidelberg,2002.[8] FruchtandHarary,Onthecoronaoftwographs,AequationesMath.,4,1970,322-324.[9] M.R.GareyandD.S.Johnson.ComputersandIntractability:AGuidetotheTheoryofNP-Completeness.Freeman,NewYork,1979.[10] J.R.Griggs,andM.Wu,Spanningtreesingraphswithminimumdegree4or5,Disc.Math.104,1992,167-183.[11] S.GuhaandS.Khuller,Approximationalgorithmsforconnecteddominatingsets,Algorithmica,20(4),Page374-387,Apr.1998.103

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T.W.Haynes,S.T.Hedetniemi,andPeterJ.Slater,DominationinGraphs,Advancedtopics,MarcelDekkerInc.,NewYork,1998.[13] TeresaW.Haynes,S.T.Hedetniemi,PeterJ.Slater,FundamentalsofDomina-tioninGraphs,MarcelDekker,NewYork,1998.[14] S.T.HedetniemiandRenuLaskar,ConnecteddominationinGraphs,InB.Bollobas,editor,GraphTheoryandCombinatorics,AcademicPress,London,1984,209-218.[15] D.J.KleitmanandD.B.West,Spanningtreeswithmanyleaves,SIAMJ.Disc.Math.4(1991),99-106.[16] S.L.Mitchell,E.J.Cockayne,andS.T.Hedetniemi,Linearalgorithmsonrecursiverepresentationsoftrees.J.Comput.SystemSci.,18(1),76-85,1979.[17] O.Ore,TheoryofGraphs,Amer.Math.Soc.Colloq.Pub.,Providence,RI381962.[18] J.Pfa,R.Laskar,andS.T.Hedetniemi.NP-completenessoftotalandconnecteddomination,andirredundanceforbipartitegraphs.TechnicalReport428,Dept.MathematicalSciences,ClemensonUniv.,1983[19] G.RamalingamandC.PanduRangan.Auniedapproachtodominationprob-lemsinintervalgraphs.inform.Process.Lett.,27:271-274,1988.[20] E.SampathkumarandH.B.Walikar,Theconnecteddominationofagraph.Math.Phys.Sci.13(1979),607-613.[21] Wormald,N.C.:TheDierentialEquationMethodforRandomGraphProcessesandGreedyAlgorithms.In:LecturesonApproximationandRandomizedAlgo-rithms,PWN(1999),73-155.104


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Connected domination in graphs
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ABSTRACT: A connected dominating set D is a set of vertices of a graph G=(V,E) such that every vertex in V-D is adjacent to at least one vertex in D and the subgraph induced by the set D is connected. The connected domination number is the minimum of the cardinalities of the connected dominating sets of G. The problem of finding a minimum connected dominating set D is known to be NP-hard. Many polynomial time algorithms that achieve some approximation factors have been provided earlier in finding a minimum connected dominating set. In this work, we present a survey on known properties of graph domination as well as some approximation algorithms. We implemented some of these algorithms and tested them with random graphs and compared their performance in finding a minimum connected dominating set D.
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Connected dominating set.
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Local optimization.
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