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Domination in graphs
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Tarr, Jennifer
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Domination
Fair Domination
Edge-Critical Graphs
Vizing's Conjecture
Graph Theory
Dissertations, Academic -- Mathematics and Statistics -- Masters -- USF   ( lcsh )
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Abstract:
ABSTRACT: Vizing conjectured in 1963 that the domination number of the Cartesian product of two graphs is at least the product of their domination numbers; this remains one of the biggest open problems in the study of domination in graphs. Several partial results have been proven, but the conjecture has yet to be proven in general. The purpose of this thesis was to study Vizing's conjecture, related results, and open problems related to the conjecture. We give a survey of classes of graphs that are known to satisfy the conjecture, and of Vizing-like inequalities and conjectures for different types of domination and graph products. We also give an improvement of the Clark-Suen inequality. Some partial results about fair domination are presented, and we summarize some open problems related to Vizing's conjecture.
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Thesis (MA)--University of South Florida, 2010.
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by Jennifer Tarr.
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ABSTRACT: Vizing conjectured in 1963 that the domination number of the Cartesian product of two graphs is at least the product of their domination numbers; this remains one of the biggest open problems in the study of domination in graphs. Several partial results have been proven, but the conjecture has yet to be proven in general. The purpose of this thesis was to study Vizing's conjecture, related results, and open problems related to the conjecture. We give a survey of classes of graphs that are known to satisfy the conjecture, and of Vizing-like inequalities and conjectures for different types of domination and graph products. We also give an improvement of the Clark-Suen inequality. Some partial results about fair domination are presented, and we summarize some open problems related to Vizing's conjecture.
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Domination
Fair Domination
Edge-Critical Graphs
Vizing's Conjecture
Graph Theory
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DominationinGraphs by JenniferM.Tarr Athesissubmittedinpartialfulllment oftherequirementsforthedegreeof MasterofArts DepartmentofMathematics&Statistics CollegeofArtsandSciences UniversityofSouthFlorida MajorProfessor:StephenSuen,Ph.D. Nata saJonoska,Ph.D. BrendanNagle,Ph.D. DateofApproval: May19th,2010 Keywords:Domination,FairDomination,Edge-CriticalGra phs,Vizing'sConjecture,GraphTheory c r Copyright2010,JenniferM.Tarr

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Acknowledgments Firstandforemost,IamindebtedtoDr.StephenSuenforhisadvice,enc ouragementand supportduringthelastyear.Hedevotedcountlesshourstoassistingmewith thisthesisandI amtrulygratefulforthat.Iwouldalsoliketothankmyothercommitteemembers,Dr. Nata sa JonoskaandDr.BrendanNagle.Finally,Iappreciatethetremendousamo untofsupport receivedfromallmyfamilyandfriendsthroughoutthisprocess.

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TableofContents ListofTables......................................... .ii ListofFigures........................................ ..iii Abstract............................................ .iv Chapter1Introduction................................... .1 1.1History........................................11.2Graph-TheoreticDenitions............................. 3 1.3DominationinGraphs.................................6 Chapter2Vizing'sConjecture.............................. ..15 2.1ClassesofGraphsSatisfyingVizing'sConjecture.............. ....15 2.2Vizing-LikeConjecturesforOtherDominationTypes.............. ..22 2.3Clark-SuenInequalityandImprovement..................... ..30 Chapter3FairDomination.................................34 3.1DenitionandGeneralResults............................ 34 3.2EdgeCriticalGraphs.................................3 7 Chapter4Conclusion.................................... 43 References......................................... ...46 i

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ListofTables Table1Symbols.......................................45 ii

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ListofFigures Figure1Therstimagedepictsastandard 8 8 chessboard.Thesecondimagehasaqueen placedintheupperrightcorner.Ifwerepresenteverysquareonth eboardbyavertex inagraph,thenwewoulddrawanedgefromthequeentoeveryvertexre presenting oneoftheshadedsquares............................... ..4 Figure2Cycles C 4 and C 5 ..................................5 Figure3Atree T andthestar K 1 ; 4 .............................5 Figure4Completegraphs K 4 and K 5 ............................5 Figure5Graphs G and H ,andthecorona G H ......................9 Figure6Coronas K 1 K 1 and K 2 K 1 andcycle C 4 ....................10 Figure7Family A ......................................10 Figure8Family B ......................................10 Figure9Independentdominationinnon-claw-freeandclaw-freegrap hs..........12 Figure10Anexampleofequalityindominationandtotaldomination........... 13 Figure11Anexampleofequalityindomination,totaldomination,connecteddomina tion, andcliquedomination...................................14 Figure12Agraph G with r ( G )= 2 ( G ) andadecomposablegraph H formedbyadding edgesto G .........................................20 Figure13ExampleofaType X graphwithaspecialclique.................22 Figure14Thedaisywith3petals.............................. 27 Figure15Partitions i andthesets D v ;S v ,and C i ;andpartitions j andthesets D u ;S u ,and C j .....................................33 Figure16Exampleofagraphwith r ( G )= r F ( G )+1 ...................37 Figure17Examplesof3-edge-criticalgraphs.................. ......39 Figure18Fairdominationin3-edge-criticalgraphs:Ineachgraph,let S 1 =thevertices thatareblue, S 2 =setofgreenvertices,and S 3 =redvertices.Thesesetsformafair receptionofeachgraphofsize3........................... ...40 iii

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DominationinGraphs JenniferM.Tarr ABSTRACT Vizingconjecturedin1963thatthedominationnumberoftheCartesianproduc toftwographsis atleasttheproductoftheirdominationnumbers;thisremainsoneofthebigge stopenproblemsin thestudyofdominationingraphs.Severalpartialresultshavebeenprov en,buttheconjecturehas yettobeproveningeneral.ThepurposeofthisthesiswastostudyVizing' sconjecture,related results,andopenproblemsrelatedtotheconjecture.Wegiveasurveyof classesofgraphsthatare knowntosatisfytheconjecture,andofVizing-likeinequalitiesandconjectur esfordifferenttypes ofdominationandgraphproducts.WealsogiveanimprovementoftheClark -Sueninequality[17]. Somepartialresultsaboutfairdominationarepresented,andwesummarizes omeopenproblems relatedtoVizing'sconjecture. iv

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Chapter1 Introduction Mathematicalstudyofdominationingraphsbeganaround1960.Thefollowin gisabriefhistoryof dominationingraphs;inparticularwediscussresultsrelatedtoVizing'sconje cture.Wethenprovidesomebasicdenitionsaboutgraphtheoryingeneral,followedbyadis cussionofdomination ingraphs.1.1HistoryAlthoughmathematicalstudyofdominationingraphsbeganaround1960,ther earesomereferencestodomination-relatedproblemsabout100yearsprior.In1862,de Jaenisch[21]attemptedto determinetheminimumnumberofqueensrequiredtocoveran n n chessboard.In1892,W.W. RouseBall[42]reportedthreebasictypesofproblemsthatchessplaye rsstudiedduringthistime. Theseincludethefollowing: 1. Covering: Determinetheminimumnumberofchesspiecesofagiventypethatarenecess ary tocover(attack)everysquareofan n n chessboard. 2. IndependentCovering: Determinethesmallestnumberofmutuallynonattackingchesspieces ofagiventypethatarenecessarytodominateeverysquareofan n n board. 3. Independence: Determinethemaximumnumberofchesspiecesofagiventypethatcanbe placedonan n n chessboardsuchthatnotwopiecesattackeachother.Notethatiftheche ss piecebeingconsideredisthequeen,thistypeofproblemiscommonlyknowna stheN-queens Problem. Thestudyofdominationingraphswasfurtherdevelopedinthelate1950'sa nd1960's,beginning withClaudeBerge[5]in1958.Bergewroteabookongraphtheory,inwh ichheintroducedthe 1

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“coefcientofexternalstability,”whichisnowknownasthedominationnumb erofagraph.Oystein Ore[39]introducedtheterms“dominatingset”and“dominationnumber”inhisbo okongraph theorywhichwaspublishedin1962.Theproblemsdescribedabovewere studiedinmoredetail around1964bybrothersYaglomandYaglom[48].Theirstudiesresulte dinsolutionstosomeof theseproblemsforrooks,knights,kings,andbishops.Adecadelater, CockayneandHedetniemi [16]publishedasurveypaper,inwhichthenotation r ( G ) wasrstusedforthedominationnumber ofagraph G .Sincethispaperwaspublished,dominationingraphshasbeenstudiedexte nsively andseveraladditionalresearchpapershavebeenpublishedonthisto pic. Vizing'sconjectureisperhapsthebiggestopenproblemintheeldofdomin ationtheoryin graphs.Vizing[45]in1963rstposedaquestionaboutthedominationnu mberoftheCartesian productoftwographs,denedinsection1.2.Vizingstatedhisconjecture thatforanygraphs G and H r ( G H ) r ( G ) r ( H ) in1968[46]. Thisproblemdidnotreceivemuchimmediateattentionafterbeingconjectured; however,since thelate1970s,severalresultshavebeenpublished.Theseresultsesta blishthetruthofVizing's conjectureforcertainclassesofgraphs,andforgraphsthatmeetce rtaincriteria.Notethatwesay agraph G satisesVizing'sconjectureif,foranygraph H ,theconjecturedinequalityholds.The rstmajorresultrelatedtoVizing'sconjecturewasatheoremfromBarca lkinandGerman[4]in 1979.Theystudiedwhatisreferredtoasdecomposablegraphsandes tablishedaclassofgraphs knownasBG-graphsforwhichVizing'sconjectureholds.Acorollaryo fthisresultisthatVizing's conjectureholdsforallgraphswithdominationnumberequalto2,graphsw ithdominationnumber equalto2-packingnumber,andtrees.TheresultthatVizing'sconjectu reistruefortreeswasalso provedseparatelybyFaudree,SchelpandShreve[22],andChen, PiotrowskiandShreve[13]. HartnellandRall[27]in1995establishedVizing'sconjectureforalarge rclassofgraphs.They foundanewwayofpartitioningtheverticesofagraphthatisslightlydiffere ntfromtheway BarcalkinandGermanpartitionedtheverticesindecomposablegraphs.The Type X classofgraphs thatresultedfromHartnellandRall'sworkisanextensionoftheclassofB G-graphs. AnotherapproachtoVizing'sconjectureistondaconstant c> 0 suchthat r ( G H ) cr ( G ) r ( H ) .In2000,ClarkandSuen[17]wereabletoprovethisinequalityfor c =1 = 2 .Theyused whatiscommonlyreferredtoasthedoubleprojectionmethodintheirproof.As willbeproven,this resultcanbeimprovedto r ( G H ) 1 2 r ( G ) r ( H )+ 1 2 min f r ( G ) ;r ( H ) g 2

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OneofthemostrecentresultsrelatedtoVizing'sconjecturedealswiththene wconceptoffair reception,whichwasrstdenedbyBre sarandRall[11]in2009.Theydenedthefairdomination numberofagraph G ,denoted r F ( G ) ,andprovedthat r ( G H ) max f r ( G ) r F ( H ) ;r F ( G ) r ( H ) g Thus,foranygraph G having r ( G )= r F ( G ) ,Vizing'sconjectureholds.Bre sarandRallshowed thattheclassofsuchgraphsisanextensionoftheBG-graphsdistinctfr omType X graphs. 1.2Graph-TheoreticDenitionsThestudyofdominationingraphscameaboutpartiallyasaresultofthestudy ofgamesandrecreationalmathematics.Inparticular,mathematiciansstudiedhowchesspiecesofa particulartype couldbeplacedonachessboardinsuchawaythattheywouldattack,ord ominate,everysquare ontheboard.Withthisinmind,graphtheoreticaldenitionswillberelatedtotheg ameofchess whereapplicable. A graph G =( V;E ) consistsofaset V ofverticesandaset E ofedges.Weshallonlyconsider simplegraphs,whichcontainnoloopsandnorepeatededges.Thatis, E isasetofunordered pairs f u;v g ofdistinctelementsfrom V .The order of G is j V ( G ) j = n ,andthe size of G is j E ( G ) j = m .If e = f v i ;v j g2 E ( G ) ,then v i and v j are adjacent .Vertex v i andedge e aresaidto be incident Envisionastandard 8 8 chessboard,ascanbeseeninFigure1.Eachsquarecanberepres ented byavertexinagraph G .Considerplacingseveralqueensontheboard.Aqueenmaymoveany numberofspacesvertically,horizontally,ordiagonally.Anysquare(or vertex)towhichaqueen isabletomoveisadjacenttothesquarecontainingthequeen.Therefore,th ereisanedgebetween thosetwosquares,orverticesofthegraph G .Sincethechessboardis 8 8 ,witheachsquare reprentedbyavertexofthegraph G ,theorderof G is 64 .Thesizeof G dependsonthenumber, type,andplacementofchesspiecesontheboard. Wecallthesetofverticesadjacenttoavertex v inagraph G the openneighborhood N ( v ) of v .Theopenneighborhoodofasetofvertices S V ( G ) is N ( S )= S v 2 S N ( v ) .The closed neighborhood N [ v ] of v is N ( v ) [f v g ,andtheclosedneighborhoodofasetofvertices S V ( G ) is N [ S ]= N ( S ) [ S The degree ofavertex v ,denoted deg( v ) isthenumberofedgesincidentwith v .Alternatively, wecandene deg( v )= j N ( v ) j .Theminimumandmaximumdegreesofverticesin V ( G ) are 3

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Figure1. :Therstimagedepictsastandard 8 8 chessboard.Thesecondimagehasaqueen placedintheupperrightcorner.Ifwerepresenteverysquareonth eboardbyavertexinagraph, thenwewoulddrawanedgefromthequeentoeveryvertexrepresenting oneoftheshadedsquares. denotedby ( G ) and ( G ) ,respectively.If ( G )=( G )= r ,thenthegraph G is regular of degreer,or r-regular Consider,onceagain,placingseveralqueensonachessboard.As sumethespaceoccupiedbyone ofthequeensisdenotedbyvertex v .Thenthenumberofpossiblemovesforthequeenoccupying thatspace,includingthoseoccupiedbyotherqueens,isequalto deg( v ) .Ifwecountthenumber ofpossiblespacestowhichthequeeninFigure1canmove,weseethatitha s21possiblemoves. Thus,ifwerepresentthatchessboardbyagraphanddenotethespa cecontainingthequeenasvertex v ,wehave deg( v )=21 A walk oflength k isasequence w = v 0 ;v 1 ;v 2 ;:::;v k ofverticeswhere v i isadjacentto v i +1 for i =0 ; 1 ;:::;k 1 .Awalkconsistingof k +1 distinctvertices v 0 ;v 1 ;:::;v k isa path ,andif v o = v k thentheseverticesforma cycle .Agraph G is connected ifforeverypairofvertices v and x in V ( G ) ,thereisa v x path.Otherwise, G is disconnected .A component of G isaconnected subgraphof G whichisnotproperlycontainedinanyotherconnectedsubgraph. Ifthereisatleastone v x walkinthegraph G thenthe distance d ( v;x ) istheminumumlength ofa v x walk.Ifno v x walkexists,wesaythat d ( v;x )= 1 Wenowconsiderafewdifferenttypesofgraphs.The cycle C n oforder n 3 hassize m = n isconnectedand2-regular.SeeFigure2forthegraphs C 4 and C 5 .A treeT isaconnectedgraph 4

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Figure2. :Cycles C 4 and C 5 withnocycles.Everytree T with n verticeshas m = n 1 edges.The starK 1 ;n 1 hasonevertex ofdegree n 1 and n 1 verticesofdegree1.Observethatastarisatypeoftree.RefertoFigur e3 forexamplesofatreeandastar. Figure3. :Atree T andthestar K 1 ; 4 Inanygraphavertexofdegreeoneisan endvertex .Anedgeincidentwithanendvertexisa pendantedge .Wecanseethatthegraphs T and K 1 ; 4 inFigure3eachhavefourpendantedgesand fourendvertices.Specically,in T ,theendverticesare v 1 ;v 2 ;v 5 ,and v 6 ,andpendantedgesare f v 1 ;v 3 g ; f v 2 ;v 3 g ; f v 4 ;v 5 g ,and f v 4 ;v 6 g Figure4. :Completegraphs K 4 and K 5 The completegraph K n hasthemaximumpossibleedges n ( n 1) = 2 .SeeFigure4forthegraphs of K 4 and K 5 .The complement G ofagraph G has V ( G )= V ( G ) and f u;v g2 E ( G ) ifandonly if f u;v g = 2 E ( G ) .Thus,thecomplementofacompletegraphistheemptygraph. 5

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A bipartitegraph isonethatcanbepartitionedas V = V 1 [ V 2 withnotwoadjacentverticesin thesame V i .Wedenethe chromaticnumber ofagraph G tobetheminimum k suchthat V ( G ) canbepartitionedintosets S 1 ;S 2 ;:::;S k andeach S i isindependent.Thatis,foreach i ,notwo verticesin S i areadjacent.Denotethechromaticnumberof G by ( G ) .If ( G )= k ,then G is k -colorablewhichmeanswecancolortheverticesof G with k colorsinsuchawaythatnotwo adjacentverticesarethesamecolor.Observethatagraphis2-colorab leifandonlyifitisabipartite graph. Thegraph H isa subgraph of G if V ( H ) V ( G ) and E ( H ) E ( G ) .If H satisesthe propertythatforeverypairofvertices u and v in V ( H ) ,theedge f u;v g isin E ( H ) ifandonlyif f u;v g2 E ( G ) then H isan inducedsubgraph of G .Theinducedsubgraph H with S = V ( H ) is calledthe subgraphinducedbyS .Thisisdenotedby G [ S ] Thereareseveraldifferentproductsofgraphs G and H ;weshalldenetheCartesianproduct, strongdirectproduct,andcategoricalproduct.Allthreeofthesepro ductshavevertexset V ( G ) V ( H ) .The Cartesianproduct of G and H ,denotedby G H ,hasedgeset E ( G H )= ff ( u 1 ;v 1 ) ; ( u 2 ;v 2 ) gj u 1 = u 2 and f v 1 ;v 2 g2 E ( H ); or f u 1 ;u 2 g2 E ( G ) and v 1 = v 2 g : The strongdirectproduct of G and H hasedgeset E ( G H ) [ff ( u 1 ;v 1 ) ; ( u 2 ;v 2 ) gjf u 1 ;u 2 g2 E ( G ) and f v 1 ;v 2 g2 E ( H ) g andisdenotedby G H .The categoricalproduct ,denotedby G H ,hasedgeset E ( G H )= ff ( u 1 ;v 1 ) ; ( u 2 ;v 2 ) gjf u 1 ;u 2 g2 E ( G ) and f v 1 ;v 2 g2 E ( H ) g : 1.3DominationinGraphsWenowintroducetheconceptofdominatingsetsingraphs.Aset S V ofverticesinagraph G =( V;E ) isa dominatingset ifeveryvertex v 2 V isanelementof S oradjacenttoanelement of S .Alternatively,wecansaythat S V isadominatingsetof G if N [ S ]= V ( G ) .Adominating set S isa minimaldominatingset ifnopropersubset S 0 S isadominatingset.The domination number r ( G ) ofagraph G istheminimumcardinalityofadominatingsetof G .Wecallsuchaset a r -setof G 6

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Foragraph G =( V;E ) and S V avertex v 2 S isan enclave of S if N [ v ] S .For S V avertex v 2 S isan isolate of S if N ( v ) V S .Wesaythatasetis enclaveless ifitdoesnot containanyenclaves.Notethat S isadominatingsetofagraph G =( V;E ) ifandonlyif V S isenclaveless.Theorem1.1 [39]Adominatingset S ofagraph G isaminimaldominatingsetifandonlyiffor any u 2 S 1. u isanisolateof S ,or 2.Thereis v 2 V S forwhich N [ v ] \ S = f u g .Proof.[39]Let S bea r -setof G .Thenforeveryvertex u 2 S S f u g isnotadominatingsetof G .Thus,thereisavertex v 2 ( V S ) [f u g thatisnotdominatedbyanyvertexin S f u g .Now, either v = u ,whichimplies u isanisolateof S ;or v 2 V S ,inwhichcase v isnotdominatedby S f u g ,andisdominatedby S .Thisshowsthat N [ v ] \ S = f u g Inordertoprovetheconverse,weassume S isadominatingsetandforall u 2 S ,either u isan isolateof S orthereis v 2 V S forwhich N [ v ] \ S = f u g .Weassumetothecontrarythat S is nota r -setof G .Thus,thereisavertex u 2 S suchthat S f u g isadominatingsetof G .Hence, u isadjacenttoatleastonevertexin S f u g ,socondition(1)doesnothold.Also,if S f u g isa dominatingset,theneveryvertexin V S isadjacenttoatleastonevertexin S f u g ,socondition (2)doesnotholdfor u .Therefore,neither(1)nor(2)holds,contradictingourassumption. Theorem1.2 [39]Let G beagraphwithnoisolatedvertices.If D isa r -setof G ,then V ( G ) D isalsoadominatingset.Proof.[39]Let D bea r -setofthegraph G andassume V ( G ) D isnotadominatingsetof G Thismeansthatforsomevertex v 2 D ,thereisnoedgefrom v toanyvertexin V ( G ) D .But thentheset D v wouldbeadominatingset,contradictingtheminimalityof D .Weconcludethat V ( G ) D isadominatingsetof G Theorem1.3 [39]Ifagraph G hasnoisolatedvertices,then r ( G ) n 2 7

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Proof.Let G beagraphwithnoisolatedverticesandlet D bea r -setof G .Assumetothecontrary that r ( G ) > n 2 .ByTheorem1.2, V ( G ) D isadominatingsetof G .But j V ( G ) D j
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Weconcludethat j D 0 ( i ) j =0 forall i =1 ; 2 ;:::;r ( G ) .Asanyset X with j D 0 ( X ) j =0 dominates G ,eachset i dominates G andso r ( G ) j i j .Therefore,wehave n = r ( G ) X i =1 j i j r ( G ) r ( G ) : Wedenethecorona G ofgraphs G 1 and G 2 asfollows.The corona G = G 1 G 2 isthegraph formedfromonecopyof G 1 and j V ( G 1 ) j copiesof G 2 wherethe i thvertexof G 1 isadjacentto everyvertexinthe i thcopyof G 2 .RefertoFigure5foranexampleofacoronaoftwographs.We taketheoriginalgraph G and,as j V ( G ) j =4 ,wehavefourcopiesof H .Bothverticesinthe i th copyof H areadjacenttothe i thvertexin G foreach i =1 ;:::; 4 Figure5. :Graphs G and H ,andthecorona G H Thefollowingtheorem,whichwasprovedindependentlybyPayanandXuo ngandbyFink,Jacobson,KinchandRoberts,tellsuswhichgraphshavedominationnumbere qualto n 2 .Thus,wecan usethisresulttondextremalexamplesofgraphswhichachievetheupper boundinTheorem1.3. Theorem1.5 [23][40]Foragraph G withevenorder n andnoisolatedvertices, r ( G )= n 2 if andonlyifthecomponentsof G arethecycle C 4 orthecorona H K 1 foranyconnectedgraph H .Proof.[40]Itcaneasilybeveriedthatifthecomponentsofagraph G are C 4 orthecorona H K 1 foraconnectedgraph H ,then r ( G )= n 2 Nowweassumethat r ( G )= n 2 .Wemayassumethat G isconnected.Let C = f S 1 ;S 2 ;:::;S p g beaminimalsetofstarswhichcoverallverticesof G .Since r ( G )= n 2 C mustbeamaximal matchingof p = n 2 edges.Foreach S i 2 C ,let S i = f x i ;y i g .Weconsidertwocases. If p 3 thenforevery i ,either x i or y i hasdegree1.Ifnot,thereis i suchthat deg( x i ) 2 and deg( y i ) 2 .Butthenwecanndadominatingsetof G withcardinalitylessthan n 2 .Thisimplies G isacorona H K 1 forsomeconnectedgraph H 9

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Figure6. :Coronas K 1 K 1 and K 2 K 1 andcycle C 4 If p 2 then G isisomorphictooneofthegraphsinFigure6.Notethatthersttwographsa re coronasandthethirdisthecycle C 4 Weconcludethat r ( G )= n 2 ifandonlyifthecomponentsof G arethecycle C 4 orthecorona H K 1 where H isaconnectedgraph. Figure7. :Family A Figure8. :Family B Wenowcharacterizeconnectedgraphswith r ( G )= b n 2 c bydeningthefollowingsixclassesof graphs.TheseresultswereprovedindependentlybyCockayne,Hay nesandHedetniemi[15]and byRanderathandVolkmann[41]. 1. G 1 = f C 4 g[f G j G = H K 1 where H isconnected g 2. G 2 = A[B where A and B arethefamiliesofgraphsdepictedinFigure7andFigure8. 3. G 3 = S H S ( H ) where S ( H ) denotesthesetofconnectedgraphs,eachofwhichcanbeformed from H K 1 byaddinganewvertex x andedgesjoining x toatleastonevertexin H 10

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4. G 4 = f ( G ) j G 2 G 3 g where y 2 V ( C 4 ) andfor G 2 G 3 ( G ) isobtainedbyjoining G to C 4 withthesingleedge f x;y g ,where x isthenewvertexaddedinforming G 5. G 5 = S H P ( H ) where u;v;w isavertexsequenceofapath P 3 .Foranygraph H P ( H ) isthe setofconnectedgraphswhichmaybeformedfrom H K 1 byjoiningeachof u and w toone ormoreverticesof H 6. G 6 = S H;X R ( H;X ) where H isagraph, X 2B ,and R ( H;X ) isthesetofconnectedgraphs obtainedfrom H K 1 byjoiningeachvertexof U V ( X ) tooneormoreverticesof H such thatnosetwithfewerthan r ( X ) verticesof X dominates V ( X ) U Theorem1.6 [15][41]Aconnectedgraph G satises r ( G )= b n 2 c ifandonlyif G 2G = 6 S i =1 G i AsaresultofTheorem1.5andTheorem1.6,wecancompletelyclassifygr aphswithdomination number r ( G )= b n 2 c Wenowdeneseveraladditionaltypesofdominationingraphs.Weshalls howVizing-like inequalitiesandconjecturesforthesetypesofdominationinSection2.2. Let f : V ( G ) [0 ; 1] beafunctiondenedontheverticesofagraph G ;thisisa fractionaldominatingfunction ifthesumofthevaluesof f overanyclosedneighborhoodin G isatleast1. The fractionaldominationnumber ofagraph G isdenoted r f ( G ) andistheminimumweightof afractional-dominatingfunction,wheretheweightofthefunctionisthesumo verallverticesof itsvalues.Asimilartypeofdominationisintegerdomination.Let k 1 andlet f : V ( G ) f 0 ; 1 ;:::;k g beafunctiondenedontheverticesofagraph G .Thisisa f k g -dominatingfunction ifthesumofthefunctionvaluesoveranyclosedneighborhoodof G isatleast k .Aswithfractional domination,theweightofa f k g -dominatingfunctionisthesumofitsfunctionvaluesoverallvertices.Wedenethe f k g -dominationnumber of G tobetheminimumweightofa f k g -dominating functionof G .Thisisdenotedby r f k g ( G ) Themaximumcardinalityofaminimaldominatingsetofagraph G iscalledthe upperdominationnumber andisdenotedby ( G ) .Wesaythataset S V ( G ) is independent ifforall u and v in S f u;v g = 2 E ( G ) .Themaximumcardinalityofamaximalindependentsetin G isthe independencenumber ( G ) ,andtheminimumcardinalityofamaximalindependentsetisthe lower independencenumber i ( G ) .Notethatthelowerindependencenumberisalsooftenreferredtoas the independentdominationnumber 11

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Figure9. :Independentdominationinnon-claw-freeandclaw-freegraphs Observethatclaw-freegraphs,orgraphsthatdonotcontainacopy of K 1 ; 3 asaninducedsubgraph,have r ( G )= i ( G ) .ThisresultwasprovedbyAllanandLaskarin1978[3].Referto Figure9.Itcaneasilybeveriedthatthegraphs G and H bothhavedominationnumberequalto2. Thegraph G isnotclaw-freeand i ( G )=3 ;anexampleofaminimalindependentdominatingsetof G isindicatedbythebluevertices.Thegraph H ,ontheotherhand,isclaw-freeandhas i ( H )=2 Wecanseethattheblueverticesin H formanindependentdominatingset. Aset S V ( G ) isa totaldominatingset of G if N ( S )= V .The totaldominationnumber r t ( G ) istheminimumcardinalityofatotaldominatingset.Notethatadominatingset S isatotal dominatingsetif G [ S ] ,thesubgraphinducedby S hasnoisolatedvertices.The uppertotaldominationnumber of G ,denotedby t ( G ) ,isthemaximumcardinalityofaminimaltotaldominating setofagraph G .Thefunction f : V ( G ) !f 0 ; 1 ;:::;k g isa total f k g -dominatingfunction if thesumofitsfunctionvaluesoveranyopenneighborhoodisatleast k .The total f k g -domination number r f k g t ofagraph G istheminimumweightofatotal f k g -dominatingfunctionof G Theabovedenedparametersofagraph G arerelatedbythefollowinglemma. Lemma1.1 [38]Foranygraph G r f ( G ) r ( G ) i ( G ) ( G ) ( G ) .If G hasnoisolated vertices,then r ( G ) r t ( G ) 2 r ( G ) Foranygraph G ,amatchingisasetofindependentedgesin G ,andaperfectmatchingof G isonewhichmatcheseveryvertexin G .Theset D V ( G ) isa paireddominatingset of G if D dominates G andtheinducedsubgraph G [ D ] hasaperfectmatching.Wedenotethe paired dominationnumber ,ortheminimumcardinalityofapaireddominatingset,by r pr ( G ) The independencedominationnumber ofagraph G ,denotedby r i ( G ) ,isthemaximum,overall independentsets I in G ,oftheminimumnumberofverticesrequiredtodominate I .Notethatthis isdifferentfromtheindependentdominationnumber. 12

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Thereareseveralothertypesofdomination,denedbelow,forwhichw ewillnotpresentfurther Vizing-likeresults. Let G =( V;E ) beabipartitegraph,withpartitesets V 1 and V 2 .Ifasetofvertices S V 1 dominates V 2 ,wesaythat S isa bipartitedominatingset of G A connecteddominatingset isadominatingsetthatinducesaconnectedsubgraphofthegraph G Wedenoteby r c ( G ) the connecteddominationnumber ,ortheminimumcardinalityofadominating set S suchthat G [ S ] isconnected.Clearly, r ( G ) r c ( G ) Observethatwhen r ( G )=1 r ( G )= r c ( G )= i ( G )=1 .Thisimpliesthatif G isacomplete graphorastar,thedominationnumber,connecteddominationnumber,andind ependentdomination numberallequal1.Also,sinceaconnecteddominatingsetof G isalsoatotaldominatingsetof G ,wehave r ( G ) r t ( G ) r c ( G ) .Anexampleofthesharpnessofthisboundcanbeseenin thecompletebipartitegraph K r;s ,inwhich r ( K r;s )= r t ( K r;s )= r c ( K r;s )=2 .SeeFigure10, whichdepictsthegraph K 2 ; 3 .Theblueverticesformbothaminimaldominatingsetandatotal dominatingset. Figure10. :Anexampleofequalityindominationandtotaldomination If D isadominatingsetof G and G [ D ] iscomplete,thenwecall D a dominatingclique .The minimumcardinalityofadominatingcliqueisthe cliquedominationnumber ,denoted r cl ( G ) .Not everygraphhasadominatingclique;forexample,anycycle C n where n 5 doesnotcontaina dominatingclique.Clearly,if r ( G )=1 ,then r ( G )= r c ( G )= r cl ( G )=1 .If G hasadominating cliqueand r ( G ) 2 then r ( G ) r t ( G ) r c ( G ) r cl ( G ) .Anexampleofthesharpnessofthese boundscanbeseeninthecorona K p K 1 ,whichhas r ( K p K 1 )= r t ( K p K 1 )= r c ( K p K 1 )= r cl ( K p K 1 )= p .Theblueverticesinthegraphofthecorona K 3 K 1 inFigure11formaminimal dominatingsetwhichisalsoatotaldominatingset,connecteddominatingset,anda dominating clique. A cycledominatingset isadominatingsetof G whoseverticesformacycle. 13

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Figure11. :Anexampleofequalityindomination,totaldomination,connecteddomination,and cliquedomination 14

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Chapter2 Vizing'sConjecture SinceVizing'sconjecturewasrststatedinthe1960s,severalresultsh avebeenpublishedwhich establishthetruthoftheconjectureforclassesofgraphssatisfyingcerta incriteria.Astheproblem hasnotyetbeensolvedingeneral,researchershavealsostudiedsimila rproblemsfordifferent typesofgraphproductsandforothertypesofdomination.Someoftheses imilarproblemsalso remainconjectures,whileothershavebeenproven.Here,wedescribe theclassesofgraphswhich areknowntosatisfyVizing'sconjectureandprovideabriefdiscussiono fthesimilarVizing-like conjectureswhichhavealsobeenstudied.Anothercommonapproachtoso lvingtheconjectureis tondaconstant c suchthatforanygraphs G and H r ( G H ) cr ( G ) r ( H ) .AsClarkandSuen [17]provedin2000,thisistruefor c = 1 2 .Weprovideaslightimprovementofthislowerboundby tighteningtheirarguments.2.1ClassesofGraphsSatisfyingVizing'sConjectureVizing'sconjectureisthatforanytwographs,thedominationnumberoftheC artesianproduct graphof G and H isgreaterthanorequaltotheproductofthedominationnumbersof G and H Theconjectureisstatedasfollows:Conjecture2.1 [46]Foranygraphs G and H r ( G H ) r ( G ) r ( H ) RecallthattheCartesianproductofgraphs G and H hasvertexset V ( G H )= V ( G ) V ( H )= f ( x;y ) j x 2 V ( G ) and y 2 V ( H ) g andithasedgeset E ( G H )= ff ( x 1 ;y 1 ) ; ( x 2 ;y 2 ) gj x 1 = x 2 and f y 1 ;y 2 g2 E ( H ); 15

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or f x 1 ;x 2 g2 E ( G ) and y 1 = y 2 g : Denea 2-packing of G asaset X V ( G ) ofverticessuchthat N [ x ] \ N [ y ]= ; foreach pairofdistinctvertices x;y 2 X .Alternatively,wecandenea2-packingasaset X ofvertices in G suchthatforanypairofvertices x and y in X d ( x;y ) > 2 .Themaximumcardinalityofa 2-packingof G iscalledthe 2-packingnumberof G andisdenotedby 2 ( G ) Observethatforanygraph G 2 ( G ) r ( G ) .Let S beamaximal2-packingof G .Then,as d ( u;v ) > 2 foreverypairofvertices u and v in S ,weneedatleastonevertexin V ( G ) todominate eachvertexin S .Hence,thecardinalityofaminimaldominatingsetisgreaterthanorequaltothe cardinalityofamaximal2-packing. Notethatwesayagraph G satisesVizing'sconjectureif,foranygraph H ,theconjectured inequalityholds.SeveralresultsestablishthetruthofVizing'sconjecturef orgraphssatisfying certaincriteria.Thecasewhere r ( G )=1 istrivial.AcorollaryofBarcalkinandGerman's[4] proofthatVizing'sconjectureholdsfordecomposablegraphsisthatViz ing'sconjectureistruefor anygraph G with r ( G ) 2 .In2004,Sun[44]veriedVizing'sconjectureholdsforanygraph G with r ( G ) 3 WenowconsiderclassesofgraphsthatareproventosatisfyVizing'sco njecture. Lemma2.1 [26]If G satisesVizing'sconjectureand K isaspanningsubgraphof G suchthat r ( G )= r ( K ) ,then K satisesVizing'sconjecture.Proof.Let K beaspanningsubgraphof G obtainedbyanitesequenceofedgeremovalswhich doesnotchangethedominationnumber.Since K isasubgraphof G K H isasubgraphof G H Thuswehave r ( K H ) r ( G H ) r ( G ) r ( H ) byassumptionon G .Byassumptionon K ,we have r ( G ) r ( H )= r ( K ) r ( H ) .Weconcludethat K satisesVizing'sconjecture. Theorem2.1 [28]Let G beagraphandlet x 2 V ( G ) suchthat r ( G x )
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j A j + j B j
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Theorem2.3 [45]Foranygraphs G and H r ( G H ) min f r ( G ) j V ( H ) j ; j V ( G ) j r ( H ) g .Proof.Let A bea r -setof G .Nowlet D = f A f v gj v 2 V ( H ) g .Then D isadominating setof G H ofcardinality r ( G ) j V ( H ) j .Similarly,wecanlet B bea r -setof H anddene D = ff u g B j u 2 V ( G ) g .Thus,wehave r ( G H ) min f r ( G ) j V ( H ) j ; j V ( G ) j r ( H ) g Theorem2.4 [35]Foranygraphs G and H r ( G H ) max f r ( G ) 2 ( H ) ; 2 ( G ) r ( H ) g : NoticethatthisresultfromJacobsonandKinchcanbeimprovedbythefollo wingtheoremfrom Chen,PiotrowskiandShreve.Theorem2.5 [13]Foranygraphs G and H r ( G H ) r ( G ) 2 ( H )+ 2 ( G )( r ( H ) 2 ( H )) : TheearliestsignicantresultrelatedtothedominationnumberofaCartesian productwasproducedbyBarcalkinandGerman[4]in1979.BarcalkinandGermanstudied graphs G whichhave dominationnumberequaltothechromaticnumberof G .Recallthatthechromaticnumber ( G ) of agraph G isthesmallestnumberofcolorsneededtocolortheverticesof G insuchawaythatno twoadjacentverticesarethesamecolor.Observethatanypropercolor ingof G isapartitionofthe verticesof G intocliques,orcompletesubgraphsof G .Asinglevertexmaybechosenfromeach cliquetoformadominatingsetof G and,therefore,itisalwaystruethat r ( G ) ( G ) BarcalkinandGermandened decomposablegraphs asfollows.Let G beagraphwith r ( G )= k ,andassume V ( G ) canbepartitionedinto k sets C 1 ;C 2 ;:::;C k suchthateachinducedsubgraph G [ C i ] isacompletesubgraphof G .If G satisestheseconditions,thenitisadecomposable graph.Theyalsodenethe A-class ,whichconsistsofallgraphs G 0 thatarespanningsubgraphs ofadecomposablegraph G ,where r ( G 0 )= r ( G ) .TheresultofBarcalkinandGerman's1979 paperestablishedVizing'sconjectureforanygraphwhichbelongstothe A-class.Notethatwenow commonlyrefertothisclassofgraphsasBG-graphs.Theorem2.6 [4]Let G beadecomposablegraphandlet K beaspanningsubgraphof G with r ( G )= r ( K ) .Then K satisesVizing'sconjecture. 18

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Proof.[28]Weassumethat G isadecomposablegraphwith r ( G )= k .Let f C i j G [ C i ] is acompletesubgraphof G; 1 i k g beapartitionof V ( G ) .Wenowconsiderthepartition f C i V ( H ) j i =1 ;:::;k g of V ( G H ) for H anarbitrarygraph.Let D bea r -setof G H Denoteby D j thesetofverticesin D thatarealsoin C j V ( H ) .Thatis, D j = D \ ( C j V ( H )) for j =1 ;:::;k: Let u j 2 C j anddenoteby P j theprojectionofverticesin C j V ( H ) onto f u j g V ( H ) Let L j bethesetofallvertices v suchthat ( u j ;v ) isnotdominatedby P j ( D j ) .Thatis, L j = f v j ( u j ;v ) = 2 N [ P j ( D j )] g : Weobservethatif v 2 L i ,thenthevertices C j f v g aredominated“horizontally”.Obviously,if P j ( D j ) dominates u j V ( H ) j L j j =0 .However,if j D j j = r ( H ) m thenwehave j D j j + j L j jj P j ( D j ) j + j L j j r ( H ) : Thisimpliesthat j L j j m Wenowconsider v 2 V ( H ) suchthat v 2 L i foratleastone i =1 ;:::;k .Denethesets D v S v ,and A v asfollows.Welet S v = f C i j v 2 L i and i =1 ;:::;k g .Dene A v tobethe setofcliques C j suchthatthereisatleastoneedgefromavertexin C j toamemberof S v and D \ ( C j f v g ) 6 = ; .Finally,welet D v = f u 2 V ( G ) j ( u;v ) 2 D and u 2 C j 2 A v g Weobservethat j D v jj S v j + j A v j ,forotherwisewewouldhave ^ D v = D v [f ( u j ;v ) j C j = 2 S v [ A v g isadominatingsetof V ( G ) f v g ofcardinalitylessthan k Alsoobservethatforeach i =1 ;:::;k either j D i j r ( H ) ,inwhichcasesummingover i givesthedesiredinequality;or j D i j = r ( H ) m .Inthelattercase,wehaveshownthat j D v j j S v j + j A v j .Fromthis,wehave j S v j X u 2 D v ( j D \ ( C j f u g ) j 1) : (2.1) Thus,wehavesufcientextraverticesin D inneighboringcliquessothatwestillhaveanaverage of r ( H ) foreach j D j j .Weconcludethat r ( G H )= j D j r ( G ) r ( H ) If K isaspanningsubgraphofadecomposablegraph G satisfying r ( G )= r ( K ) ,thenweapply Lemma2.1toprovethat K alsosatisesVizing'sconjecture. 19

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Corollary2.1 [4]Let G beagraphsatisfying r ( G )=2 or 2 ( G )= r ( G ) .Then G satises Vizing'sconjecture. Thiscorollaryfollowsfromtheprevioustheorem.Anygraph G with r ( G )=2 isasubgraph ofadecomposablegraph.Toestablishthesecondpartofthecorollary, weassume G isagraph satisfying r ( G )= 2 ( G ) .Let S = f v 1 ;v 2 ;:::;v k g bea2-packingof G .Thenwecanaddedgesto G tomake N [ v 1 ] ;N [ v 2 ] ;:::;N [ v k 1 ] and V ( G ) ( N [ v 1 ] [ N [ v 2 ] [ ::: [ N [ v k 1 ]) intocliques. Theresultinggraphisdecomposableandstillhas k pairwisedisjointclosedneighborhoods.Hence, itfollowsfromTheorem2.6thatanygraphwith r ( G )= 2 ( G ) satisesVizing'sconjecture.An exampleofthiscanbeseeninFigure12.Thelabeledvertices v 1 ;v 2 ; and v 3 in G forma2-packing ofthegraph.Wecanaddedgesasdescribedabovetogetthedecompos ablegraph H Figure12. :Agraph G with r ( G )= 2 ( G ) andadecomposablegraph H formedbyaddingedges to G ObservethatthiscorollaryimpliesVizing'sconjectureistrueforanytree.W ealsohavethe followingresultfromHartnellandRallasacorollaryofTheorem2.6and Corollary2.1. Corollary2.2 [28]Let G beagraphsuchthat G is3-colorable.Then G satisesVizing'sconjecture.Proof.Weconsiderthreecasesbasedonthechromaticnumberof G Case1: ( G )=1 .Then G isacompletegraphandtheresultholds. Case2: ( G )=2 .Then G belongstotheA-classandVizing'sconjectureholds. 20

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Case3: ( G )=3 .If r ( G )=3 then G isdecomposableandresultholdsbyTheorem2.6. Otherwise r ( G ) 2 andresultholdsbyCorollary2.1. WenowdeneType X graphs,asintroducedbyHartnellandRall[27]in1995.Thisclassof graphscontainstheBG-graphsasapropersubsetand,hence,isan improvementofBarcalkinand German's[4]1979result.HartnellandRall,indeningType X graphs,tookanapproachsimilar tothatofBarcalkinandGermaninthattheyconsideredaparticularwayofp artitioningagraph G ThedifferenceisthatnoteverysetinthepartitionofaType X graphinducesacompletesubgraph. Type X graphsaredenedasfollows.Let k;t;r benonnegativeintegers,notallzero.Let G be agraphwith r ( G )= k + t + r +1 whoseverticescanbepartitionedas S [ SC [ BC [ C ,where S;SC;BC ,and C satisfythefollowing. Let BC = B 1 [ B 2 [ ::: [ B t .Each B i for i =1 ;:::;t isreferredtoasa bufferclique Let C = C 1 [ C 2 [ ::: [ C r Eachof SC;B 1 ;:::;B k ;C 1 ;:::;C r inducesaclique. Every v 2 SC hasatleastoneneighboroutsideof SC .Theset SC iscalleda specialclique Each B i ,for i =1 :::;k hasatleastonevertexwhichhasnoneighborsoutsideof B i Let S = S 1 [ S 2 [ ::: [ S k whereeach S i isstar-like.Thatis,each S i hasavertex v i whichis adjacenttoall v 2 S i v i .Thevertex v i hasnoneighborsotherthanthosein S i .Notethat S i doesnotinduceaclique,andnoedgesmaybeaddedto S i withoutdecreasingthedomination numberof G Therearenoedgesbetweenverticesin S andverticesin C ObservethatnoteverygraphthatisType X hasaspecialclique.Wecanalsohave t;r; or k equal tozero.TheexampleinFigure13,isaType X graphwithaspecialclique.Inthisgraph,theblue verticesrepresenttheset S ,theredverticesrepresentthebufferclique B ,andthegreenvertices representthespecialclique SC .OnecaneasilyverifythatthisgraphsatisesthedenitionofType X graphsabove. 21

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Figure13. :ExampleofaType X graphwithaspecialclique Theorem2.7 [27]Let G beaType X graph.Thenforanygraph H r ( G H ) r ( G ) r ( H ) TheproofofHartnellandRall'stheoremissimilartotheproofthatVizing'sc onjectureistrue forBG-graphs.Wepartitiontheverticesof G asindicatedbythedenitionofaType X graphand consideranydominatingset D of G H .HartnellandRallusedtheideathatsomeverticesinthe productgraphmustbedominated“horizontally”andfound r ( G ) disjointsetsin D ,eachofwhich havecardinalityatleast r ( H ) ,thusimplyingthatVizing'sconjectureholdsforanyType X graph. Theorem2.8 [27]Let G beaType X graphandlet K beaspanningsubgraphof G suchthat r ( G )= r ( K ) .ThenVizing'sconjectureistruefor K Thistheoremcanbeprovedinthesamewayweshowedthatanyspannings ubgraph K ofa decomposablegraph G with r ( G )= r ( K ) satisesVizing'sconjecture. HartnellandRallwerealsoabletoshowthatanygraphwithdominationnumber onemorethan its2-packingnumberisaType X graphand,hence,wehavethefollowingresult. Corollary2.3 [27]Let G beagraphsatisfying r ( G )= 2 ( G )+1 .ThenVizing'sconjectureis truefor G Bre sarandRall[11]recentlydiscoveredanewclassofgraphswhichsatis fyVizing'sconjecture. Theydenedfairdominationandprovedthatanygraphwithfairdominationn umberequaltoits dominationnumbersatisestheconjecture.Furthermore,theyprovedthatth isclassofgraphsisan extensionoftheBG-graphsdistinctfromType X graphs.TheirresultsarepresentedinChapter3. 2.2Vizing-LikeConjecturesforOtherDominationTypesAsVizing'sconjecturehasnotyetbeenproveningeneral,research erssuchasFisher,Ryan,Domke andMajumdar[25];NowakowskiandRall[38];Bre sar[7];andDorbec,HenningandRall[19] 22

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havestudiedvariationsoftheoriginalproblem.Thesesimilarproblemsdealw ithothertypesof graphproductsanddifferentgraphparameters.Aswewillsee,seve ralofthesevariationsremain openconjectures,whileothershavebeenproven. FractionalDomination OneoftherstVizing-likeresultswasprovedforthefractionaldomination number.Recallthat thefractionaldominationnumberofagraph G istheminimumweightofafractional-dominating function,wheretheweightofthefunctionisthesumoverallverticesofitsv alues.Wenotethatfor anygraph G r f ( G ) r ( G ) .Fisher,Ryan,Domke,andMajumdarprovedthefollowingresultin their1994paper.Theorem2.9 [25]Foranypairofgraphs G and H r f ( G H ) r f ( G ) r f ( H ) Thistheoremcanbeprovedbyrstshowingthat r f ( G H )= r f ( G ) r f ( H ) .Recallthat G H denotesthestrongdirectproductof G and H ,whichhasvertexset V ( G ) V ( H ) andedgeset E ( G H ) [ff ( u 1 ;v 1 ) ; ( u 2 ;v 2 ) gjf u 1 ;u 2 g2 E ( G ) and f v 1 ;v 2 g2 E ( H ) g .Since G H isa subgraphof G H ,wehave r f ( G H ) r f ( G H ) Fisher[24]alsoprovedthefollowingsimilartheoremin1994;animprovedpr oofwasgivenby Bre sar[6]in2001. Theorem2.10 [24]Foranypairofgraphs G and H r ( G H ) r f ( G ) r ( H ) AnobviouscorollaryofthistheoremisthatVizing'sconjectureistruefora nygraphwithfractionaldominationnumberequaltodominationnumber. IntegerDomination Arelatedconcepttofractionaldominationisintegerdomination,whichwasstud iedrstby Domke,Hedetniemi,Laskar,andFricke[18].Werecallthattheweightof a f k g -dominatingfunctionisthesumofitsfunctionvaluesoverallvertices,andthe f k g -dominationnumberof G r f k g ( G ) istheminimumweightofa f k g -dominatingfunctionof G .Domke,et.al.provedthefollowing theoremrelatingfractionaldominationtointegerdomination.Theorem2.11 [18]Foranygraph G r f ( G )=min k 2 N r f k g ( G ) k 23

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ThefollowingVizing-likeconjectureforintegerdominationisfromHouandLu [33]. Conjecture2.2 [33]Foranypairofgraphs G and H andanyinteger k 1 r f k g ( G H ) 1 k r f k g ( G ) r f k g ( H ) Thisconjectureremainsopen,butBre sar,HenningandKlav zar[9]proveseveralrelatedresults intheir2006paper.Notethatifthisconjectureistrueforall k ,inparticular k =1 ,thenVizing's conjectureistrue. UpperDomination NowakowskiandRall's[38]1996papergivesresultsandconjecture sonseveralassociativegraph products,twoofwhicharetheCartesianproductandthecategoricalpr oduct,aspreviouslydened inSection1.2. Recallthattheupperdominationnumber ( G ) ofagraph G isthemaximumcardinalityofa minimaldominatingsetof G .Alsorecallthattheminimumcardinalityofamaximalindependent setistheindependentdominationnumber i ( G ) NowakowskiandRall[38]madethefollowingconjecturesintheir1996pap er. i ( G H ) i ( G ) i ( H ) ( G H ) ( G )( H ) ( G H ) ( G )( H ) ThelastoftheseconjectureswasprovedbyBre sar[7]in2005.Infact,heprovidedaslight improvementoftheconjecturedlowerbound.Theorem2.12 [7]Foranynontrivialgraphs G and H ( G H ) ( G )( H )+1 : TheproofBre sarprovidedforthistheoremisconstructiveinnature.Hebeginswitharb itrary graphs G and H andcreatesaminimaldominatingset D oftheproductgraph G H whichcontains atleast ( G )( H )+1 vertices. 24

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TotalDomination HenningandRall's[30]2005paperwasthersttointroduceresultsonto taldominationin Cartesianproductsofgraphs.Recallthataset D V ( G ) isatotaldominatingsetif N ( D )= V ( G ) .Thetotaldominationnumberistheminimumcardinalityofatotaldominatingsetof G andisdenotedby r t ( G ) .HenningandRallconjecturedthat 2 r t ( G H ) r t ( G ) r t ( H ) andthey provedthisinequalityholdsforcertainclassesofgraphs G withnoisolatedverticesandanygraph H withoutisolatedvertices.Thisconjecturewasprovedforgraphswithout isolatedverticesbyHo. Theorem2.13 [32]Let G and H begraphswithoutisolatedvertices.Then 2 r t ( G H ) r t ( G ) r t ( H ) : Recallthatthetotal f k g -dominationnumber r f k g t ( G ) isdenedastheminimumcardinalityofa total k -dominatingset D ofagraph.In2008,LiandHou[37]provedthatforanygraphs G and H withoutisolatedvertices, r f k g t ( G ) r f k g t ( H ) k ( k +1) r f k g t ( G H ) .NotethatTheorem2.13is easilyprovedusingthisinequality. UpperTotalDomination Recallthatwedenetheuppertotaldominationnumberof G ,denotedby t ( G ) ,tobethe maximumcardinalityofaminimaltotaldominatingsetofagraph G .Dorbec,HenningandRall [19]publishedresultsin2008onaVizing-likeinequalityfortheuppertotald ominationnumber. Theyachievedthefollowingtworesults.Theorem2.14 [19]If G and H areconnectedgraphsoforderatleast3and t ( G ) t ( H ) then 2 t ( G H ) t ( G )( t ( H )+1) andthisboundissharp. Inordertoprovethistheoremwemustrstdenethesets epn ( S;v ) ipn ( v;S ) ,and pn ( v;S ) .Let S V ( G ) andlet v 2 S .Theset epn ( v;S ) ofexternalprivateneighborsof v is epn ( v;S )= f u 2 V ( G ) S j N ( u ) \ S = f v gg .Thesetofinternalprivateneighborsof v 2 S is ipn ( v;S )= f u 2 S j N ( u ) \ S = f v gg .Wedenotethesetofallprivateneighborsof v 2 S by pn ( v;S ) .Thisisthe unionofallexternalandinternalprivateneighborsof v .Thatis, pn ( v;S )= epn ( v;S ) [ ipn ( v;S ) Cockayne,et.al.makethefollowingobservation. 25

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Observation2.1 [14]Let S beatotaldominatingsetinagraph G withnoisolatedvertices.Then S isaminimaltotaldominatingsetifandonlyifforall v 2 S 1. epn ( v;S ) 6 = ; ,or 2. pn ( v;S )= ipn ( v;S ) 6 = ; Wewillalsoneedthefollowinglemma. Lemma2.3 [19]Let G beagraph.Every t ( G ) -setcontainsasasubseta r -set D suchthat j D j 1 2 t ( G ) andforall v 2 D j epn ( v;D ) j 1 WewillnowproveTheorem2.14.Proof.[19]Weassume G and H areconnectedgraphswithorderatleast3,where t ( G ) t ( H ) Bytheabovelemma,thereisa r -set S of G with j S j 1 2 t ( G ) andforeach v 2 S j epn ( v;S ) j 1 Foreach u 2 V ( G ) ,denote H u = f u g V ( H ) .Similarly,for w 2 V ( H ) ,let G w = V ( G ) f w g Now,let D = S V ( H ) ,andobservethat D dominates G H since S dominatesV(G).Also, foreach u 2 S ,thevertices V ( H u ) aretotallydominated“vertically”;thus, D isatotaldominating setof G H .Weclaimthat D isaminimaltotaldominatingsetof G H Let ( u;w ) 2 D andconsider ( u 0 ;w ) ,where u 0 2 epn ( u;S ) in G .Then ( u 0 ;w ) 2 epn (( u;w ) ;D ) in G H .Thus,forall ( u;w ) 2 D j epn (( u;w ) ;D ) j 1 .Then,byObservation2.1, D isaminimal totaldominatingsetof G H andso t ( G H ) j D j .Notethatsince H isaconnectedgraphwith orderatleast3, j V ( H ) j t ( H )+1 .Therefore, t ( G H ) j D j = j S jj V ( H ) j 1 2 t ( G )( t ( H )+1) : Equalityholdswhenboth G and H are daisies with k 2 petals .Thatis,webeginwith k copies of K 3 andidentifyonevertexfromeachcopytoformasinglevertex.Theresu ltinggraphisadaisy. Figure14showsthedaisywith3petals. ThefollowingtheoremiseasilyprovedusingTheorem2.14andthefacttha tforagraph G with noisolatedvertices, t ( G ) t ( K 2 ) 2 t ( G K 2 ) .Equalityholdsifandonlyif G isadisjoint unionofcopiesof K 2 .Let u 2 V ( K 2 ) .Then V ( G ) f u g isaminimaltotaldominatingsetof G K 2 ,givingthat t ( G K 2 ) j V ( G ) j t ( G )= 1 2 t ( G ) t ( K 2 ) : 26

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Figure14. :Thedaisywith3petals Inorderforequalitytohold,wemusthave t ( G )= j V ( G ) j ,andso G mustbeadisjointunionof copiesof K 2 Theorem2.15 [19]If G and H havenoisolatedvertices,then 2 t ( G H ) t ( G ) t ( H ) withequalityifandonlyifboth G and H aredisjointunionsofcopiesof K 2 PairedDomination Bre sar,HenningandRall[10]publishedresultsin2007aboutVizing-likeine qualitiesforpaired domination.Recallthataset D V ( G ) isapaireddominatingsetof G if D dominates G andthe inducedsubgraph G [ D ] hasaperfectmatching.Notethatineverygraphwithoutisolatedvertices, amaximalmatchingformsapaireddominatingset.Thepaireddominationnumber r pr ( G ) isthe minimumcardinalityofapaireddominatingset. TheinequalitiesestablishedbyBre sar,HenningandRallrelatethepaireddominationnumberof theCartesianproductof G and H tothe 3 -packingnumberof G .Recallthata2-packingofagraph G isasetofvertices S V ( G ) suchthatforanyvertices u and v in S d ( u;v ) > 2 .Wedenea 3-packingsimilarly.Thatis,a3-packingofthegraph G isaset S ofverticessuchthatthedistance betweenanypairofverticesin S isgreaterthan3.The3-packingnumberof G ,denoted 3 ( G ) ,is themaximumcardinalityofa3-packingin G Theorem2.16 [10]If G and H aregraphswithoutisolatedvertices,then r pr ( G H ) max f r pr ( G ) 3 ( H ) ;r pr ( H ) 3 ( G ) g : Bre sar,HenningandRallwerealsoabletoshowthat r pr ( T )=2 3 ( T ) inanynontrivialtree T Thus,thefollowingresultfollowsfromTheorem2.16. 27

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Theorem2.17 [10]Let T beanontrivialtree.Thenforanygraph H withoutisolatedvertices, r pr ( T H ) 1 2 r pr ( T ) r pr ( H ) ,andthisboundissharp. ThenalmajorresultfromBre sar,HenningandRallin2007isthefollowingtheoremrelating paireddominationintheCartesianproductof G and H tothe3-packingnumbersof G and H Theorem2.18 [10]If G and H havenoisolatedvertices,then r pr ( G H ) 2 3 ( G ) 3 ( H ) IndependenceDomination AharoniandSzab o[2]in2009providedaVizing-likeresultfortheindependencedomination number.Recallthatthisisdifferentfromtheindependentdominationnumbe r;welettheindependencedominationnumber r i ( G ) denotethemaximum,overallindependentsets I in G ,ofthe minimumnumberofverticesrequiredtodominate I .ItwasprovenbyAharoni,BergerandZiv[1] that r ( G )= r i ( G ) foranychordalgraph G ,whereagraphischordalifanycycleofmorethanfour verticescontainsatleastonechord,oredgeconnectingverticesthata renotadjacentinthecycle. AharoniandSzab oprovedthefollowingtheorem. Theorem2.19 [2]Forarbitrarygraphs G and H r ( G H ) r i ( G ) r ( H ) .Proof.[2]Let G and H begraphs.Wemayassumethat G hasnoisolatedvertices,forifitdid haveanisolatedvertex v thenthevalidityofthetheoremfor G v impliesthevalidityfor G Assume I V ( G ) isanindependentsetwhichrequiresatleast r i ( G ) verticestodominateit. Wewillshowthat r ( I H ) r i ( G ) r ( H ) byshowingthat j D j r i ( G ) r ( H ) ,where D isaset thatdominates I V ( H ) Let f v 1 ;v 2 ;:::;v r ( H ) g bea r -setof H .Usetheseverticestopartition V ( H ) intosets f i j v i 2 i and v 2 i ifandonlyif v = v i or f v;v i g2 E ( H ) g .Notethat,forevery J f 1 ; 2 ;:::;r ( H ) g wehave r ( [ j 2 J j ) j J j (2.2) Let S u = f i jf u g i isdominatedverticallybysomevertices ( u;v ) 2 D g ,andlet S i = f u 2 I jf u g i isdominatedverticallybysomevertices ( u;v ) 2 D g .Summing S u and S i ,wehave S = X u 2 I S u = r ( H ) X i =1 S i 28

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By(2.2),foreach u 2 I wehave j D \ ( f u g V ( H )) jj S u j : Sumover v 2 I toget j D \ ( I V ( H )) jjSj : (2.3) Nowconsider k r ( H ) ;eachsetofvertices f u g k whichisnotin S containsatleastone vertex ( u;v ) whichisnotdominatedbyanyvertexin f u g V ( H ) .Thus, ( u;v ) isdominated “horizontally”bysomevertex ( w;v ) where w = w ( v ) .Notethat w= 2 I since I isindependentand sotheset f w ( v ) jf v g k = 2Sg dominates j I jj S j j verticesin I andhascardinalityatleast r i ( G ) j S j j .Sumover k toget j D \ (( V ( G ) I ) V ( H )) j r i ( G ) r ( H ) jSj : (2.4) Combineequations(2.3)and(2.4)toget r ( G H ) r i ( G ) r ( H ) : CombiningthisresultwiththatofAharoni,BergerandZiv[1],anobvious corollaryisthat Vizing'sconjectureholdsforchordalgraphs. IndependentDomination Bre sar,et.al.[8]provideafewopenconjecturesintheirsurveypaper,in cludingthefollowing. Conjecture2.3 [8]Foranygraphs G and H r ( G H ) min f i ( G ) r ( H ) ;r ( G ) i ( H ) g ThetruthofthisconjecturewouldimmediatelyimplyVizing'sconjectureholdsfora nypairof graphs G and H ,as r ( G ) i ( G ) byLemma1.1.Wealsohavethefollowingconjecture,whichis impliedbyVizing'sconjecture.Bre sar,et.al.suggestthatperhapsthiscouldbeestablishedwithout rstprovingVizing'sconjecture.Conjecture2.4 [8]Foranygraphs G and H i ( G H ) r ( G ) r ( H ) Inaddition,thesurveypapermakesthefollowingpartitionconjecture,which wouldalsoimply thetruthofVizing'sconjecture. 29

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Conjecture2.5 [8]Let G and H bearbitrarygraphs.Thereisapartitionof V ( G ) into r ( G ) sets 1 ;:::; r ( G ) suchthatthereisaminimaldominatingset D of G H suchthattheprojectionof D \ ( i V ( H )) onto H dominates H forall i =1 ;:::;r ( G ) 2.3Clark-SuenInequalityandImprovementWehavegivenseveralresultsestablishingthetruthofVizing'sconjectur eforclassesofgraphs satisfyingcertainproperties.AnotherapproachtoprovingVizing'scon jectureistondaconstant c suchthatforanygraphs G and H r ( G H ) cr ( G ) r ( H ) .ClarkandSuen[17]in2000proved thatthisinequalityistruefor c = 1 2 .Here,wepresentanimprovementofthisresult. Theorem2.20 Foranygraphs G and H r ( G H ) 1 2 r ( G ) r ( H )+ 1 2 min f r ( G ) ;r ( H ) g .Proof.Let G and H bearbitrarygraphs,andlet D bea r -setoftheCartesianproduct G H Let f u 1 ;u 2 ;:::;u r ( G ) g bea r -setof G .Partition V ( G ) into r ( G ) sets 1 ; 2 ;:::; r ( G ) ,where u i 2 i forall i =1 ; 2 ;:::;r ( G ) andif u 2 i then u = u i or f u;u i g2 E ( G ) Let P i denotetheprojectionof ( i V ( H )) \ D onto H .Thatis, P i = f v 2 V ( H ) j ( u;v ) 2 D forsome u 2 i g : Dene C i = V ( H ) N H [ P i ] asthecomplementof N H [ P i ] ,where N H [ X ] isthesetofclosed neighborsof X ingraph H .As P i [ C i isadominatingsetof H ,wehave j P i j + j C i j r ( H ) ;i =1 ; 2 ;:::;r ( G ) : (2.5) For v 2 V ( H ) ,let D v = f u j ( u;v ) 2 D g and S v = f i j v 2 C i g : Observethatif i 2 S v thentheverticesin i f v g aredominated“horizontally”byverticesin D v f v g .Let S H bethenumberofpairs ( i;v ) where i =1 ; 2 ;:::;r ( G ) and v 2 C i .Then obviously S H = X v 2 V ( H ) j S v j = r ( G ) X i =1 j C i j : Since D v [f u i j i= 2 S v g isadominatingsetof G ,wehave j D v j +( r ( G ) j S v j ) r ( G ) ; 30

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givingthat j S v jj D v j : (2.6) Summingover v 2 V ( H ) ,wehave S H j D j : (2.7) Wenowconsidertwocasesbasedon(2.5). Case1 Assume j P i j + j C i j >r ( H ) forall i =1 ;:::;r ( G ) .Thenas j ( i V ( H )) \ D jj P i j wehave r ( G ) X i =1 ( j C i j + j ( i V ( H )) \ D j ) r ( G ) X i =1 ( r ( H )+1) ; whichimpliesthat S H + j D j r ( G ) r ( H )+ r ( G ) : (2.8) Combining (2.7) and (2.8) givesthat r ( G H )= j D j 1 2 r ( G ) r ( H )+ 1 2 r ( G ) : (2.9) Case2 Assume j P i j + j C i j = r ( H ) forsome i =1 ;:::;r ( G ) .Notethat P i [ C i isa r -setof H Wenowusethis r -setof H topartition V ( H ) inthesamewayas V ( G ) ispartitionedabove.That is,labeltheverticesin P i [ C i as v 1 ;v 2 ;:::;v r ( H ) ,andlet f j j 1 j r ( H ) g beapartitionof H suchthatforall j =1 ;:::;r ( H ) v j 2 j andif v 2 j ,either v = v j or f v;v j g2 E ( H ) .We nextdenethesets P j ;C j ;S u and D u inthesameway P i ;C i ;S v and D v aredenedabove.To bespecic,for 1 j r ( H ) ,let P j = f u 2 V ( G ) j ( u;v ) 2 D forsome v 2 j g ; and C j = V ( G ) N G [ P j ] ; andfor u 2 V ( G ) ,let D u = f v j ( u;v ) 2 D g and S u = f j j u 2 C j g : Similarly,wehave S G = X u 2 V ( G ) j S u j = r ( H ) X j =1 C j : For u 2 V ( G ) ,let ^ D u = f v j j ( u;v j ) 2 D u ; 1 j r ( H ) g .Weclaimthat j S u jj D u jj ^ D u j : (2.10) 31

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Thisisbecause D u [f v j j j= 2 S u g isadominatingsetof H ,with D u \f v j j j= 2 S u g = ^ D u ; andtheargumentforproving (2.10) followsinthesamewayas (2.6) isproved.Tomakeuseof theclaim,wenotethatwhenwepartitiontheverticesof H ,wehaveatleast r ( H ) verticesin D thatareoftheform ( u;v k ) .Indeed,foreach k =1 ; 2 ;:::;r ( H ) ,either v k 2 P i ,whichimplies ( u;v k ) 2 D forsome u 2 i ,or v k 2 C i ,whichimpliesthattheverticesin i f v k g are dominated“horizontally”bysomevertices ( u 0 ;v k ) 2 D .Itthereforefollowsthat X u 2 V ( G ) j ^ D u j r ( H ) ; andhencesummmingbothsidesof (2.10) X u 2 V ( G ) j S u j X u 2 V ( G ) ( j D u jj ^ D u j ) givesthat S G j D j r ( H ) : (2.11) Tocompletetheproof,wenotethatsimilarto (2.5) ,wehave j P j j + j C j j r ( G ) ;j =1 ; 2 ;:::;r ( H ) ; andsummingover j givesthat j D j + S G r ( G ) r ( H ) : (2.12) Combining (2.11) and (2.12) ,weobtain r ( G H ) 1 2 r ( G ) r ( H )+ 1 2 r ( H ) : (2.13) Aseither(2.9)or(2.13)holds,itfollowsthat r ( G H ) 1 2 r ( G ) r ( H )+ 1 2 min f r ( G ) ;r ( H ) g : Thisapproachmayalsobeusedtoproveasimilarinequalityinvolvingtheindepe ndencenumber ofagraph,where G isaclaw-freegraph.Recallthattheindependencenumberofagraph G isthe maximumcardinalityofamaximalindependentsetin G ,andisdenotedby ( G ) .Alsorecallthat agraphisclaw-freeifitdoesnotcontainacopyof K 1 ; 3 asaninducedsubgraph.Bre sar,et.al.[8] provedthefollowing. 32

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Figure15. :Partitions i andthesets D v ;S v ,and C i ;andpartitions j andthesets D u ;S u and C j Theorem2.21 [8]Let G beaclaw-freegraphandlet H beagraphwithoutisolatedvertices. Then r ( G H ) 1 2 ( G )( r ( H )+1) : Observethat r ( G ) ( G ) foreverygraph G ,sowehavethefollowingcorollary. Corollary2.4 [8]Let G beaclaw-freegraphandlet H beagraphwithoutisolatedvertices. Then r ( G H ) 1 2 r ( G )( r ( H )+1) : Fromthiscorollarywecanconcludethatanyclaw-freegraphsatisfying ( G )=2 r ( G ) satises Vizing'sconjecture. 33

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Chapter3 FairDomination ArecentdevelopmentinattemptstoproveVizing'sconjectureisBre sarandRall's[11]ideaoffair domination.Their2009paperdenesthisconceptandestablishesthetruth ofVizing'sconjecture forgraphswithfairdominationnumberequaltodominationnumber.Furthermor e,theyverifythat theclassofsuchgraphscontainstheBG-graphsandisdistinctfromtheT ype X graphsdenedby HartnellandRall.Wewilldenefairreceptionandfairdomination,providea proofthatVizing's conjectureholdsfortheclassofgraphswithfairdominationnumberequalto dominationnumber, examinefairdominationinedge-criticalgraphs,andsummarizesomeopenque stionsrelatedtofair domination.3.1DenitionandGeneralResultsArecentpaperbyBre sarandRall[11]publishedin2009introducestheconceptoffairdomina tion ofagraph.Bre sarandRallwereabletoverifythatVizing'sconjectureholdsforanygr aph G with afairreceptionofsize r ( G ) Inordertodenefairdomination,wemustrstdeneexternaldomination. Wesaythataset X V ( G ) externallydominates set U V ( G ) if U \ X = ; andforeach u 2 U thereis x 2 X suchthat f u;x g2 E ( G ) LetGbeagraphandlet S 1 ;:::;S k bepair-wisedisjointsetsofverticesofG.Let S = S 1 [ S 2 [ ::: [ S k andlet Z = V ( G ) S .Thesets S 1 ;:::;S k forma fairreceptionofsizek ifforeach l 2 Z 1 l k ,andanychoiceof l sets S i 1 ;:::;S i l thefollowingholds:if D externallydominates S i 1 [ ::: [ S i l then j D \ Z j + X j;S j \ D 6 = ; ( j S j \ D j 1) l: Noticethatontheleft-handsideoftheaboveinequality,wecountallthever ticesof D thatarenot in S .Forverticesof D thatareinsome S j ,wecountallbutonefrom D \ S j .Thelargest k such 34

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thatthereexistsafairreceptionofsize k ingraph G iscalledthe fairdominationnumber of G and isdenotedby r F ( G ) Proposition3.1 [11]Foranygraph G 2 ( G ) r F ( G ) r ( G ) .Proof.Let T bea2-packingofG.Leteach S i consistofexactlyonevertex v 2 T .Thisgives usafairreceptionofsize j T j .Thus, 2 ( G ) r F ( G ) .Nowassumethereexistsagraph G with r = r ( G )
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Nowwedene T i = f u 2 V ( G ) j u= 2 N [ P i ] g .Observethat T i [ P i isadominatingsetof G and,thus, j T i j + j P i j r ( G ) (3.2) Let T u = f i j u 2 T i ;i =1 ; 2 ;:::;k g .Bydenitionof T i and T u ,thefollowingholds k X i =1 j T i j = X u 2 V ( G ) j T u j (3.3) Observethatif i 2 T u thevertices f u g S i arenotdominatedby D i ,andso P u externally dominates S i forall i 2 T u .Therefore,bydenitionoffairreception,wehave j D uZ j + k X i =1 d ui j T u j (3.4) Now,wehave: j D j = k X i =1 j D i j + j D Z j = k X i =1 ( j P i j +( j D i jj P i j ))+ j D Z j = k X i =1 j P i j + k X i =1 X u 2 V ( G ) d ui + j D Z j = k X i =1 j P i j + X u 2 V ( G ) ( k X i =1 d ui + j D uZ j ) k X i =1 j P i j + X u 2 V ( G ) j T u j (3.5) = k X i =1 ( j P i j + j T i j ) (3.6) kr ( G )= r ( G ) r F ( H ) : (3.7) Note,(3.5)holdsby(3.4),(3.6)holdsby(3.3),and(3.7)holdsby(3.2) Similarly,wedeneafairreceptionofGandrepeattheproofwiththeroleso fGandHreversedtoconcludethat r ( G H ) r F ( G ) r ( H ) .Therefore,weconcludethat r ( G H ) max f r F ( G ) r ( H ) ;r ( G ) r F ( H ) g Corollary3.1 Let G beagraphwith r ( G )= r F ( G ) .Then G satisesVizing'sconjecture. Therearesomeknownexamplesofgraphs G forwhich r F ( G ) 6 = r ( G ) .Onesuchexamplecan beseeninFigure16.Itcanbeeasilyveriedthatthisgraph G has r ( G )=3 .Bre sar,et.al.[8] veriedbycomputerthat r F ( G )=2 36

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Figure16. :Exampleofagraphwith r ( G )= r F ( G )+1 Bre sarandRallobservethattheclassofgraphssatisfying r ( G )= r F ( G ) isanextensionof theclassofBG-graphswhichisdistinctfromType X graphs.Anopenquestionregardingfair dominationiswhetheralowerboundmaybefoundfor r F ( G ) intermsof r ( G ) .If,forexample, onecouldndaconstant c> 1 2 suchthat r F ( G ) cr ( G ) ,thatwouldimprovetheClark-Suen inequality.3.2EdgeCriticalGraphsTherearetwoclassesofgraphsthatarecriticalwithrespecttothedomina tionnumber: edge-critical graphs and vertex-criticalgraphs .Inanedge-criticalgraph,thedominationnumberdecreasesifan edgeisadded;invertex-criticalgraphs,thedominationnumberdecrease sifavertexisdeleted. Here,weconcentrateontheclassofedge-criticalgraphs. Agraph G is k-edge-domination-critical ,orsimply k -edge-criticalif r ( G )= k andforevery pairofnonadjacentvertices u;v 2 V ( G ) r ( G + f u;v g )= k 1 .Inotherwords,thedomination numberdecreasesifanymissingedgeisaddedtothegraph G Notethatagraph G is1-edge-criticalifandonlyif G isacompletegraph.Itisalsostraightforwardtocharacterize2-edge-criticalgraphs,usingthefollowingtheore m. Theorem3.2 [43]Agraph G is2-edge-criticalifandonlyif G = t S i =1 K 1 ;p i forsome t 1 Inotherwords,theonly2-edge-criticalgraphsarecomplementsofunion sofstars.Although Vizing'sConjecturehasalreadybeenestablishedforgraphs G with r ( G )=2 ,wecanprovide adifferentmethodofprooffor2-edge-criticalgraphs.Wewillshowth atthedominationnumber equalsthefairdominationnumberina2-edge-criticalgraphand,therefor e,wecanapplyBre sar andRall's[11]resulttoshowthatVizing'sconjectureholds. 37

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Theorem3.3 Forany2-edge-criticalgraph G r ( G )= r F ( G ) .Proof.Let G bea2-edge-criticalgraph.ByTheorem3.2,every2-edge-criticalg raphisthe complementofaunionofstars.Consider H = K 1 ;n 1 1 [ K 1 ;n 2 1 [ ::: [ K 1 ;n t 1 ,where j V ( K 1 ;n i 1 ) j = n i and t 1 .Let v i bethevertexofmaximumdegreein K 1 ;n i 1 for i = 1 ; 2 ;:::;t .Nowlet G = H .Let S 1 = f v i j i =1 ; 2 ;:::;t g andlet S 2 = V ( G ) S 1 Weneedtoshowthat S 1 and S 2 formafairreceptionof G .Considertheset S 1 .Inorderto externallydominatethisset,weneedatleast2verticesfrom S 2 .Take v 2 K 1 ;n i 1 where v 6 = v i Then v externallydominates v j forall j 6 = i .Thuswemustchooseatleastonemorevertexfrom S 2 toexternallydominate v i .Thisimplies j D \ S 2 j 1 1 forallsetsofvertices D thatexternally dominate S 1 Nowconsider S 2 .Choose v i 2 S 1 .Then v i externallydominatesallverticesof S 2 exceptthose thatwereinthestar K 1 ;n i 1 in H .Thus,wemustchooseanadditionalvertex v j 6 = v i toexternally dominatethosevertices.Wehave j D \ S 1 j 1 1 Therefore,forany2-edge-criticalgraph G r F ( G ) 2 .But r F ( G ) r ( G ) byProposition3.1 andsince r ( G )=2 ,wehave r F ( G )= r ( G )=2 SinceweknowthatVizing'sconjectureholdsforanygraph G thathas r ( G )= r F ( G ) ,thisresult impliesVizing'sconjectureholdsforall2-edge-criticalgraphs. Unfortunately,3-edge-criticalgraphsarenoteasilycharacterizeda s1-and2-edge-criticalgraphs are.Weprovideafewexamplesof3-edge-criticalgraphs. Figure17providessevenexamplesof3-edge-criticalgraphs.Obser vethatwecanndafair receptionofsize3inveofthesegraphs,asshowninFigure18;howev er,itisdifculttotellif thereisafairreceptionofsize3intheremainingtwographsinFigure17.Wed ohavethefollowing resultwhichmayhelpinndingfairreceptionofsize r ( G ) inanedge-criticalgraph G Theorem3.4 Let G be k -edge-criticalwith r ( G )= r F ( G )= k .Thenif S 1 ;S 2 ;:::;S k forma fairreceptionof G ,each S i for i =1 ; 2 ;:::;k isacompletesubgraphof G .Proof.Assume G is k -edge-criticalandthat r ( G )= r F ( G )= k .Let S 1 ;S 2 ;:::;S k formafair receptionof G .Withoutlossofgenerality,assume S 1 doesnotformacompletesubgraphof G .For u;v 2 S 1 suchthat f u;v g = 2 E ( G ) ,drawtheedge f u;v g .Thenwestillhaveafairreceptionofsize 38

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Figure17. :Examplesof3-edge-criticalgraphs 39

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Figure18. :Fairdominationin3-edge-criticalgraphs:Ineachgraph,let S 1 =theverticesthatare blue, S 2 =setofgreenvertices,and S 3 =redvertices.Thesesetsformafairreceptionofeachgraph ofsize3. 40

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k .Butadding f u;v g decreases r ( G ) ,sonowwehave r F ( G ) >r ( G ) ,acontradiction.Therefore, foreach i =1 ; 2 ;:::;k S i formsacompletesubgraphof G Wedenethe linegraph ofthecompletegraphon [ k ] asfollows:let [ k ] denotethe k -set f 1 ; 2 ;:::;k g andconsiderthesetof2-subsetsof [ k ] .Letthese n 2 2-subsetsbethevertices v 1 ;v 2 ;:::;v ( k 2 ) ofthelinegraph G k .Thereisanedge f v 1 ;v 2 g betweenvertices v 1 ;v 2 2 V ( G ) ifandonlyif v 1 \ v 2 6 = ; .Forthelinegraph G k r ( G k )= d k 1 2 e .If k iseventhena r -set of G k is f 1 ; 2 g ; f 3 ; 4 g ;:::; f k 1 ;k g ,and r ( G )= d k 1 2 e .If k isoddthena r -setof G k is f 1 ; 2 g ; f 3 ; 4 g ;:::; f k 2 ;k 1 g ,and r ( G )= k 1 2 Lemma3.1 If k iseven,thenthelinegraph G k isedge-critical.Proof.Let D beadominatingsetfor G k ,where k iseven.Withoutlossofgenerality,let D = ff 1 ; 2 g ; f 3 ; 4 g ;:::; f k 1 ;k gg .Nowaddanedgebetweentwoverticesin D ,say ff 1 ; 2 g ; f 3 ; 4 gg toformthegraph G 0k .Then D 0 = ff 1 ; 2 g ; f 5 ; 6 g ;:::; f k 1 ;k gg isadominatingsetof G 0k and j D 0 j = j D j 1 .Hence, G k isedge-criticalwhen k iseven. Consequently,ifthereisafairreceptionof G k ofsize d k 1 2 e ,theneachset S i i =1 ; 2 ;:::; k 1 2 isacompletesubgraphof G k Notethatforany k ,wecanndafairreceptionof G k ofsize b k 3 c .Considerpartitioningthe set [ k ] into3-subsets;withoutlossofgenerality,saywehave f 1 ; 2 ; 3 g ; f 4 ; 5 ; 6 g ,andsoon.Then theverticesgeneratedbyeachsetformthesets S 1 ;S 2 ;:::;S b k 3 c .Sowehave,forexample, S 1 = ff 1 ; 2 g ; f 1 ; 3 g ; f 2 ; 3 gg .Byformingoursets S i inthisway,weensurethatnovertexin S j dominates avertexin S i for i 6 = j .Wealsorequireatleasttwoverticesfrom V ( G k ) S todominate each S i andsothesesetssatisfythecriteriatobeafairreception.As r ( G k )= d k 1 2 e wehave r F ( G k ) b k 3 c 2 3 r ( G k ) .Now,observethatwehavealowerboundon r F ( G k +6 ) intermsof r F ( G k ) Lemma3.2 Forany k r F ( G k +6 ) r F ( G k )+2 .Proof.Let S 1 ;S 2 ;:::;S r F ( G k ) formafairreceptionof r ( G k ) .Nowaddthe6points f 1 ; 2 ; 3 ; 4 ; 5 ; 6 g to [ k ] andconsider G k +6 .Wecanformafairreceptionofthisgraphbyadding S r F ( G k )+1 = ff 1 ; 2 g ; f 1 ; 3 g ; f 2 ; 3 gg and S r F ( G k )+2 = ff 4 ; 5 g ; f 4 ; 6 g ; f 5 ; 6 gg to S 1 ;S 2 ;:::;S r F ( G k ) .Thus, r F ( G k +6 ) r F ( G k )+2 41

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Notethatndingagoodupperboundfor r F ( G k ) ismuchmoredifcult.Weknowthat r F ( G k ) r ( G k ) .Itremainsanopenproblemwhetherwecanimprovethisupperbound. Weobservethatthelinegraph G k isclaw-free,andsowecanapplyCorollary2.4,whichstates thatforaclaw-freegraphandanygraph H withoutisolatedvertices, r ( G k H ) 1 2 r ( G k )( r ( H )+1) : Note,also,thatwecanapplyTheorem3.1toget r ( G k H ) r F ( G k ) r ( H ) 2 3 r ( G k ) r ( H ) 42

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Chapter4 Conclusion Vizing'sconjecture,asstatedin1963,isthatthedominationnumberoftheCar tesianproductof twographsisatleasttheproductoftheirdominationnumbers.Therstmajo rresultrelatedtothe conjecturewasfromBarcalkinandGerman[4]in1979whentheydened decomposablegraphs andprovedVizing'sconjectureholdsfortheso-calledA-class,nowco mmonlycalledBG-graphs. HartnellandRall's[27]1995breakthroughestablishedthetruthofVizing 'sconjectureforwhat theycalledType X graphs;thisclassofgraphsisanextensionoftheBG-graphs.Bre sarandRall [11]in2009denedfairreceptionandfairdomination.Theyprovedthat Vizing'sconjectureholds forgraphswithdominationnumberequaltofairdominationnumber.Theclasso fsuchgraphsis anextensionoftheBG-graphswhichisdistinctfromType X graphs.WealsoknowthatVizing's conjectureistrueforanygraphwithdominationnumberlessthan4;thiswaspr ovedin2004by Sun[44]. AnotherapproachtoprovingVizing'sconjectureistondaconstant c> 0 sothat r ( G H ) cr ( G ) r ( H ) ,withthehopethateventuallythisconstantwillimproveto1.ClarkandSuen[1 7]were abletodothisin2000for c = 1 2 ,andwewereabletotightentheirargumentstoproveaslightly improvedinequality. AsVizing'sconjectureisnotyetprovedforallgraphs,severalres earchershavestudiedVizinglikeconjecturesforothergraphproductsandothertypesofdomination.W eprovidedasummary ofsomeVizing-likeresultsforfractionaldomination,integerdomination,uppe rdomination,upper totaldomination,paireddomination,andindependencedomination.Inaddition, westatedafew conjectureswhichremainopenproblemsandwouldcontributetoeffortstopr oveVizing'sconjecture.Twooftheseconjecturesinvolveindependentdomination,andoneis knownastheprojection conjecture(Conjecture2.5).Aproofofanyofthesethreeconjectures wouldimplythetruthof Vizing'sconjecture. 43

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Wealsodenedfairreceptionandfairdomination,asintroducedbyBre sarandRall,andincluded aproofoftheirVizing-likeinequalityrelatingthedominationnumberoftheCartes ianproductof graphs G and H tothefairdominationnumbersof G and H .Itremainsanopenquestionwhether wecanndaconstant c> 1 2 sothat r F ( G ) cr ( G ) foranygraph G .Wedoknowthatthereare graphsforwhich r F ( G )= r ( G ) 1 ,andwebelievethelinegraph G k couldhavefairdomination numbermuchsmallerthandominationnumber;howeveritremainsdifculttonda lowerbound onthefairdominationnumberofagraphintermsofthedominationnumber. Finally,weconsideredfairdominationinedgecriticalgraphs.Wefoundtha tafairreception ofanedge-criticalgraph G ofsize r ( G ) musthaveeachset S i induceacompletesubgraphof G WealsoprovidedaproofthatVizing'sconjectureistruefor2-edge-cr iticalgraphs.Thisresult,of course,wasalreadyknownsinceweknowVizing'sconjectureholdsfo ranygraphwithdomination numberlessthan4;however,itisanexampleofhowwemightusetheideaoff airdominationto provethatVizing'sconjectureistrueforcertaingraphs. NotethatacommonmethodofproofinmostoftheVizing-likeresultsistopartitionad ominatingset D of G H andprojecttheverticesof D onto G or H .Itisunclearwhetherthisparticular methodwillbeusefultoproveVizing'sconjecture.AslongasVizing'scon jectureremainsunresolved,possiblenextstepsinattempttoproveitaretocontinuestudyingVizin g-likeconjectures, particularlythoserelatingdominationandindependentdomination.Onemightals ostudyfairdominationfurther,withhopesofndingalowerboundonthefairdominationnumb erofagraph.We alsonotethattheBG-graphs,Type X graphs,andgraphswithfairdominationnumberequalto dominationnumberarealldenedbyapartitionofthevertexsetofagraph. Itcouldbeusefulto ndanewwayofpartitioningtheverticesofagraphinsuchawaythatweca nestablishthetruth ofVizing'sconjectureforanevenlargerclassofgraphs. 44

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Table1 :Symbols SymbolDescription r ( G ) Dominationnumber r F ( G ) Fairdominationnumber ( G ) Minimumvertexdegree ( G ) Maximumvertexdegree ( G ) Chromaticnumber i ( G ) Independentdominationnumber r t ( G ) Totaldominationnumber r c ( G ) Connecteddominationnumber r cl ( G ) Cliquedominationnumber 2 ( G ) 2-packingnumber r f ( G ) Fractionaldominationnumber r f k g ( G ) f k g -dominationnumber ( G ) Independencenumber ( G ) Upperdominationnumber r f k g t ( G ) Total f k g -dominationnumber t ( G ) Uppertotaldominationnumber r pr ( G ) Paireddominationnumber r i ( G ) Independencedominationnumber 45

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