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Towards Combinatorial Characterizations and Algorithms for Bar-And-Joint Independence and Rigidity in 3D and Higher Dimensions

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Towards Combinatorial Characterizations and Algorithms for Bar-And-Joint Independence and Rigidity in 3D and Higher Dimensions
Creator:
Cheng, Jialong
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[Gainesville, Fla.]
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University of Florida
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Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Computer Engineering
Computer and Information Science and Engineering
Committee Chair:
SITHARAM,MEERA
Committee Co-Chair:
UNGOR,ALPER
Committee Members:
THAI,MY TRA
DAVIS,TIMOTHY ALDEN
VINCE,ANDREW J
Graduation Date:
12/13/2013

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Algebra ( jstor )
Distance functions ( jstor )
Embeddings ( jstor )
Equations ( jstor )
Infinitesimals ( jstor )
Mathematics ( jstor )
Matroids ( jstor )
Nucleation ( jstor )
Roofs ( jstor )
Vertices ( jstor )
3d -- bound -- characterization -- combinatorial -- matroid -- rigidity
Computer and Information Science and Engineering -- Dissertations, Academic -- UF
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bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
Electronic Thesis or Dissertation
Computer Engineering thesis, Ph.D.

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Abstract:
Combinatorial characterization of generic bar-and-joint rigidity in 3D and higher dimensions is a long open problem. This translates to a combinatorial characterization of the exact rank of generic rigidity matroids of graphs whose vertices represent joints and edges represent bars. A further question is the algorithmic complexity of determining rank (and rigidity) combinatorially. Our program of research is to iteratively provide increasingly better upper bounds on the rank, and corresponding algorithms: we start with an approximate characterization (and algorithm) to upper bound the rank; construct families of graphs that illuminate the gap between the upper bound and the true rank; give better characterizations, upper bounds, and algorithms that overcome some of these obstacles and iterate this process. Along the way, we obtain the following results. (a) The best polynomial time rank upper bound currently known for general graphs: i.e, the size of any maximal subgraph satisfying a counting condition given by Maxwell in the 1800's  (b) Another rank upper bound obtained as a simple inclusion-exclusion formula applied to subgraphs in a specific type of graph cover; and the construction of a subgraph satisfying Maxwell's counting condition, whose size meets this bound (c) First proofs or shortened proofs of correctness of  existing algorithms for detecting rigidity in certain classes of graphs (d) Systematic constructions that answer a 20 year old open problem about so-called rigidity circuits, and illuminate the  limitations of the bounds in (a) and (b); and finally (e) A method to overcome these obstacles and  obtain a combinatorial characterization that potentially captures 3D rigidity  provided an existing conjecture, concerning a well-studied structure  called abstract rigidity matroid, holds in 3 dimensions. ( en )
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In the series University of Florida Digital Collections.
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Thesis (Ph.D.)--University of Florida, 2013.
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Adviser: SITHARAM,MEERA.
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Co-adviser: UNGOR,ALPER.
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RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2015-12-31
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by Jialong Cheng.

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TOWARDSCOMBINATORIALCHARACTERIZATIONSANDALGORITHMSFORBAR-AND-JOINTINDEPENDENCEANDRIGIDITYIN3DANDHIGHERDIMENSIONSByJIALONGCHENGADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2013

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c2013JialongCheng 2

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IdedicatethistoeveryonethathelpedinmydevelopmentasaPhDstudent. 3

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ACKNOWLEDGMENTS ThankstoallforthehelpIhavereceivedinmyPhDprogram.IwouldliketoexpressmyspecialappreciationandthankstomyadvisorDr.MeeraSitaram.Thepassionshehasforherresearchwasmotivationalforme,whichhelpedmepasstoughtimesinthePh.D.pursuit.Ialsowanttothankmyfamilyforalltheloveandsupport. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 10 CHAPTER 1INTRODUCTION ................................... 12 1.1BackgroundandMotivation .......................... 12 1.1.1CombinatorialRigidity ......................... 13 1.1.2OurStrategyandContribution ..................... 13 1.1.3ConnectionsbetweenCombinatorialRigidityandOtherMathematicalDomainandApplications ........................ 14 1.2GeometricConstraintSolvingandCombinatorialRigidityTheory ..... 15 1.3MetricSpaceEmbedding ........................... 17 1.4OtherApplicationsofCombinatorialRigidity ................. 20 2FORMALDEFINITIONSANDCONTRIBUTIONS ................. 22 2.1Denitions .................................... 22 2.1.1InnitesimalRigidity .......................... 22 2.1.2IndependentEdgeSetandRigidityMatroid ............. 24 2.1.3Rigidityin2D .............................. 25 2.1.4Rigidityin3D .............................. 26 2.2OrganizationandContribution ......................... 27 2.2.1MainResultofChapter 3 :RankUpperBoundsUsingMaximalMaxwell-IndependentSetsandInclusion-ExclusionCount ..... 27 2.2.2MainResultofChapter 4 :GeneralConstructionSchemeforNucleation-Free,IndependentGraphswithImpliedNon-Edges ............ 31 2.2.3MainResultofChapter 5 :aConcrete,PurelyCombinatorialDenitionofaClosureOperatorthatGivesanAbstractRigidityMatroid ... 33 2.2.4Miscellany ................................ 36 3MAXWELL-INDEPENDENCE ............................ 38 3.1MainResultandProof ............................. 38 3.2AlternativeUpperBoundsUsingIECounts ................. 48 3.2.1RelationtoKnownBoundsandConjecturesUsingIECounts ... 48 3.2.2AlternativeUpperBoundsforMaxwell-IndependentGraphs .... 50 3.2.3RemovingtheMaxwell-IndependenceCondition .......... 51 3.3OpenProblems ................................. 56 5

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3.3.1ExtendingRankboundtoHigherDimensions ............ 56 3.3.2StrongerVersionsofIndependence .................. 58 3.3.3BoundsforMaxwell-DependentGraphsUsing2-ThinCovers ... 59 3.3.4AlgorithmsforVariousMaximalMaxwell-IndependentSets ..... 60 4NUCLEATION-FREEGRAPHSWITHIMPLIEDNON-EDGES ......... 62 4.1Warm-UpExample:RingsofRoofs ...................... 62 4.1.1Flex-SignTechniqueforExistenceofImpliedNon-EdgesinRingsofRoofs ................................. 64 4.1.2Rank-SandwichTechniquefortheExistenceofImpliedNon-EdgesinRingofRoofs ............................ 66 4.1.2.1Independenceofringofroofs ................ 66 4.1.2.2Rankupperboundusing2-thincoverargument ...... 67 4.1.2.3Rankupperboundusingbody-hingeargument ...... 69 4.2NaturalExtensionsofWarm-UpExample .................. 69 4.3Roof-Addition:GeneralInductiveConstructionforNucleation-Free,IndependentGraphwithImpliedNon-Edges ........................ 75 4.3.1InductiveConstructionforIndependentGraphs ........... 75 4.3.2Roof-AdditionGivesNucleation-Free,IndependentGraphswithImpliedNon-Edges ........................... 84 4.3.3ExistenceandGenerationofStartingGraphsforTheorem 4.9 ... 89 4.4DependentGraphswithNoNucleusandFurtherConsequences ..... 93 4.5ConclusionsandOpenProblems ....................... 96 5GEM(GRADEDEXCHANGEMAXWELL)-MATROID ............... 98 5.1Motivation,ExamplesandDenitions ..................... 98 5.2Theorems .................................... 107 5.2.1GEM-ClosureIsComputable ..................... 107 5.2.2GEM-ClosureDenesaMatroid .................... 108 5.2.3GEM-ClosureDenesanAbstractRigidityMatroid ......... 109 5.2.4GEM-IndependenceV.S.Maxwell-Independence .......... 113 5.3OpenProblems ................................. 115 6OTHERPARTIALRESULTSANDFURTHEROPENPROBLEMS ....... 117 6.1ConvexCayleyCongurationSpaceandD-realizability .......... 117 6.2ResultsaboutGlobalRigidity ......................... 118 6.2.1Hendrickson'sConditionin1D ..................... 119 6.2.2OpenProblemsaboutGlobalRigidity ................. 120 6.2.3DetectingtheInsufciencyofHendrickson'sCondition ....... 120 7CONCLUSION .................................... 122 APPENDIX:ISSUEINTAY'SPAPER ........................... 123 REFERENCES ....................................... 125 6

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BIOGRAPHICALSKETCH ................................ 133 7

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LISTOFTABLES Table page 2-1AcomparisonofGEM-matroidwithotherrigidity-relatedmatroidconcepts. .. 37 4-1Constructionschemesfornucleation-free,independentgraphswithimpliednon-edges. ...................................... 96 5-1IllustrationoflevelsandgradesintheGEM-closure(GradedExchangeMaxwell-closure)ofanedgesetE. ................................... 103 8

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LISTOFFIGURES Figure page 1-1Agureshowingtherelationshipsbetweencombinatorialrigiditytheoryandgeometricconstraintsolvingandmetricspaceembedding. ........... 20 2-1Anon-genericrigidframeworkthatisinnitesimallyexible.~uisanon-trivialinnitesimalex. ................................... 23 2-2Bananaanddouble-banana ............................. 28 2-3Crapo'sgraphwithahingestructure:fa,bgisahinge,sinceitissharedbytworigidcomponents. ................................ 29 2-4AgureshowingmaximalMaxwell-independentsetscanhavedifferentsizes. 30 2-5AgureshowingthatnoteverymaximalMaxwell-independentsetofagivengraphGcontainsamaximalindependentsetofG. ................ 31 2-6Agureshowingminimum-sizedmaximalMaxwell-independentsetcanbeabadboundonrank. ................................. 34 3-1Thegureontheleftrepresentsthevertex-maximalcomponentsofagraph.Ontherightsideisits2-thincomponentgraph,wherecirclesrepresentcomponentnodesandsquaresrepresentedgenodes.Notethatthe2-thincomponentgraphmaynotbeconnected. ............................ 42 3-2Acoverofvertex-maximalcomponentsthatisnot2-thin.ThecirclesareK5'sandthetwolargerellipsesarevertex-maximal,Maxwell-rigidsubgraphsthatformthecover. .................................... 59 3-3AcounterexampletoshowthatIEfullcountofcoverXbyvertex-maximal,strongMaxwell-rigidsubgraphsturnsouttobesmallerthanthesizeofanymaximalMaxwell-independentset. .............................. 61 4-1Aroofanditsschematic. .............................. 63 4-2Connectingtworoofs.Ontheleftistherealgeometricconnection,whileontherightisaschematicshowinghowthetworoofsareconnectedviaahinge 64 4-3Aringof7roofs:connecting7roofsinthemannershowninFig. 4-2 andwecanseeeachroofcanbethusconnectedtoatmosttwoothers.Suchachainofsevenroofsisclosedbackintotheringshownhere.Ontheleftisthegeometricstructureoftheringandontherightistheschematicofthering. ........ 64 4-4Differenttypesofroofsusedinthewarm-upconstruction. ............ 65 4-5AmodiedoctahedralgraphobtainedfromtheHennebergextenderringschemefromaringofroofs.Thedashedlinesarethetwohingenon-edges. ...... 71 9

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4-6AschematicshowinghowtoapplygraphcuttingofGonfa,bg ......... 76 4-7Aschematicshowinghowtoinductivelyconstructindependentgraphsusingtheroof-additionscheme. .............................. 77 4-8Aschematicshowingtheframeworks,i.e.,includingthepositioningoftheverticesforthelasttworoofsfortheproofofTheorem 4.8 ................ 80 4-9Agureshowingthesymmetryoftheveedgesatc. .............. 80 4-10AgureshowingthefourofthesixtermsthatadduptozeroatBandtheirreectionthroughtheplanepassingcc0andAa2(thetwodashedline)tothefourtermsin( 4 ). .................................. 82 4-11Twotypesofbuildingblocksofringgraphs. .................... 90 4-12Atwo-body-sharingringswithicosahedra. ..................... 91 4-13AFour-body-sharingring. .............................. 92 4-14Adouble-ringof14roofsconsistingoftworingsof7roofs. ............ 94 4-15Abraceddouble-ringof14roofs:itconsistsoftworingsof7roofsandtwoextrabars. ....................................... 95 5-1AgureshowinghowtoobtainourclosureinR7. ................. 100 5-2Agureshowinghowtoobtainclosureforlevel9,grade1. ........... 104 5-3AgureshowingwhytheGEM-closureonanoctahedralgraphisacompletegraph. ......................................... 104 5-4AgureshowingwhytheGEM-closureonanicosahedralgraphisacompletegraph. ......................................... 105 5-5AgureshowingwhytheGEM-closureonaringof6roofsisacompletegraph. 106 A-1AcounterexampletoTay'sproof:therstsubgraphistheunionoftherestandweputtheminsuchapositionthatthechainissymmetricalonga1b1. ... 124 A-2AnothercounterexampletoTay'sproof. ...................... 124 10

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyTOWARDSCOMBINATORIALCHARACTERIZATIONSANDALGORITHMSFORBAR-AND-JOINTINDEPENDENCEANDRIGIDITYIN3DANDHIGHERDIMENSIONSByJialongChengDecember2013Chair:MeeraSitharamMajor:ComputerEngineeringCombinatorialcharacterizationofgenericbar-and-jointrigidityin3Dandhigherdimensionsisalongopenproblem.Thistranslatestoacombinatorialcharacterizationoftheexactrankofgenericrigiditymatroidsofgraphswhoseverticesrepresentjointsandedgesrepresentbars.Afurtherquestionisthealgorithmiccomplexityofdeterminingrank(andrigidity)combinatorially.Ourprogramofresearchistoiterativelyprovideincreasinglybetterupperboundsontherank,andcorrespondingalgorithms:westartwithanapproximatecharacterization(andalgorithm)toupperboundtherank;constructfamiliesofgraphsthatilluminatethegapbetweentheupperboundandthetruerank;givebettercharacterizations,upperbounds,andalgorithmsthatovercomesomeoftheseobstaclesanditeratethisprocess.Alongtheway,weobtainthefollowingresults.(a)Thebestpolynomialtimerankupperboundcurrentlyknownforgeneralgraphs:i.e,thesizeofanymaximalsubgraphsatisfyingacountingconditiongivenbyMaxwellinthe1800's(b)Anotherrankupperboundobtainedasasimpleinclusion-exclusionformulaappliedtosubgraphsinaspecictypeofgraphcover;andtheconstructionofasubgraphsatisfyingMaxwell'scountingcondition,whosesizemeetsthisbound(c)Firstproofsorshortenedproofsofcorrectnessofexistingalgorithmsfordetectingrigidityincertainclassesofgraphs(d)Systematicconstructionsthatanswera20yearoldopenproblemaboutso-called 11

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rigiditycircuits,andilluminatethelimitationsoftheboundsin(a)and(b);andnally(e)Amethodtoovercometheseobstaclesandobtainacombinatorialcharacterizationthatpotentiallycaptures3Drigidityprovidedanexistingconjecture,concerningawell-studiedstructurecalledabstractrigiditymatroid,holdsin3dimensions. 12

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CHAPTER1INTRODUCTION 1.1BackgroundandMotivationRigidityofbar-and-jointstructures,orlinkages,hasbeenimplicitlystudiedsincethetimeofJamesMaxwell.Questionsaboutrigidityarespecialcasesofalargerclassofproblemsaboutthepropertiesoflinkagerealizationspaces.Thelatterhasbeenstudiedforevenlonger,sinceAlbrechtDurerinthe1500'sandAlfredKempeinthe1800's.Mathematiciansandcomputerscientistshaveonlybeguntostudytheseareasinthelast3decades.Theareasarearichsourceofbasicunansweredquestionsattheconuenceofgeometry,algebra,combinatorics,andanalysis.Solutionstothesequestionsarelikelytohavewiderangingapplicationsinmanydomainsincluding:computer-aideddesign,molecularmodellingandmachinelearning,andmoregenerally,inanydomainattheintersectionofgeometricconstraintsolvingand(partial)metricspaceembedding(moreinSections 1.2 and 1.3 ).Thebasicproblemstudyingrigidityofbar-and-jointstructuresinvolvesasystemofpolynomialequations,whereeachequationrepresentsadistancerelation,i.e.,abar,betweenapairofjointsandthusthesystemcanberegardedasagraphofrelationships,andndingthesolutions(i.e.,realizations)ofthesysteminRd.However,somepropertiesoftherealizationspacecanbedeterminedgenericallybythegraphitself,i.e.,independentoftheactualnumericcoordinatevaluesofthejoints.I.e.,genericityplaysanimportantroleinrigiditytheory.Whenagenericsystemhasnitelymanysolutions,wecallthecorrespondinggraphrigid.Anotherwaytostudythesystemistoregarditasadistancemap,i.e.,afunctionthatmapsarealizationofagraphGinRd,i.e.,thevertexcoordinatevalues,tothecorrespondingsetofedge-distances,orbar-lengths.TheJacobianofthismapiscalledtherigiditymatrixR(G)ofG,withrowsindexedbyedgesandcolumnsbythevertex 13

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coordinates.Whentherigiditymatrixofarealizationhasmaximalrank,weknowthereareonlynitelymanysolutionstothesystem,andhencetherealizationisrigid. 1.1.1CombinatorialRigidityStartingwitharigiditymatrixofagenericrealizationofG,itsindependentsetsofrows(edges)deneamatroid(see[ 72 ]andSection 2.1 ),whichiscalledthegenericrigiditymatroid([ 35 ])ofG.Matroidsarecombinatorialwaysofcapturinglinear(andalgebraic)dependences,andthusrigidityofagenericrealizationisequivalenttowhethertherankofthegenericrigiditymatroidofGinddimensions,denotedbyrankd(G),ismaximal.WhentherankofthegenericrigiditymatroidofGhasmaximalrank,everygenericrealizationofGhasnitelymanysolutionsandhencethegraphGisrigid.In1864,JamesMaxwellshowedthatasimplecountingcondition,whichinvolvesonlythevertexsetandtheedgesetofthegraphG,isnecessaryforrankd(G)tobemaximal.In1970,LamanshowedthatMaxwell'scountingconditionisalsosufcientforrank2(G)tobemaximal,givingacombinatorialcharacterizationofgenericrigidityin2dimensions.However,in3andhigherdimensions,thereisnoknowncombinatorialcharacterizationofgenericrigidity,andthisremainsanopenquestion.Thisdissertationdealsdirectlywiththisquestion. 1.1.2OurStrategyandContributionWeapproachtheprobleminaniterativemannerandfocusongivingupperboundsonrank3(G).Westartwithnewnon-trivialupperboundontherank;constructfamiliesofgraphsthatilluminatethegapbetweentheupperboundandthetruerank;thenweanalyzeexamplesforwhichtheupperboundisnottight,isolateanobstacleposedbytheseexamples,andremovethisobstacle,thusgivingbettercharacterizations,upperbounds,andcleaneranalysisofalgorithmsthatovercomesomeoftheseobstacles. 14

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Alongtheway,weobtainthefollowingresults.(a)Thebestpolynomialtimerankupperboundcurrentlyknownforgeneralgraphs:i.e,thesizeofanymaximalsubgraphsatisfyingacountingconditiongivenbyMaxwellinthe1800's(b)Anotherrankupperboundobtainedasasimpleinclusion-exclusionformulaappliedtosubgraphsinaspecictypeofgraphcover;andtheconstructionofasubgraphsatisfyingMaxwell'scountingcondition,whosesizemeetsthisbound(c)Firstproofsorshortenedproofsofcorrectnessofexistingalgorithmsfordetectingrigidityincertainclassesofgraphs(d)Systematicconstructionsthatanswera20yearoldopenproblemaboutso-called3Drigiditycircuits,i.e.,minimaldependentsetsofedges,andilluminatethelimitationsoftheboundsin(a)and(b);andnally(e)Amethodtoovercometheseobstaclesandobtainacombinatorialcharacterizationthatpotentiallycaptures3Drigidityprovidedanexistingconjecture,concerningawell-studiedstructurecalledabstractrigiditymatroid,holdsin3dimensions.WegiveformaldenitionsandcontributionsinChapter 2 .Theremainderofthischapterisdevotedtoexplainingtheconnectionbetweencombinatorialrigiditytheoryandgeometricconstraintsolvingandmetricspaceembedding. 1.1.3ConnectionsbetweenCombinatorialRigidityandOtherMathematicalDomainandApplicationsCombinatorialrigiditytheoryhasstronginterplaywithgeometricconstraintsolvingandmetricspaceembedding,inthatcombinatorialrigidityborrowstechniquesfromtheseareasandviceversa.Forexample,combinatorialrigidityborrowsalgebraictechniquesfromgeometricconstraintsolvingandanalytictechniquesfrommetricspaceembedding,whileitcontributescombinatorialtechniquestothosedomains,especiallyforgenericresultsaboutEuclideandistanceconstraintsolvingandpartialmetricspaceembedding. 15

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1.2GeometricConstraintSolvingandCombinatorialRigidityTheoryTheproblemofgeometricconstraintsolvingconsistsofagivensetofgeometricelementsandadescriptionofgeometricconstraintsbetweentheelements.Thegoalistondallplacementsofthegeometricentitieswhichsatisfythegivenconstraints.Geometricconstraintsolvinghasalonghistory([ 10 30 38 40 58 ])andapplicationsinComputer-Aided-Design[ 39 41 42 62 82 ],geometry-basedteachingsoftware[ 53 63 106 ],molecularmodelling([ 2 25 52 87 ])androbotics[ 73 81 83 ].AspecialcaseofgeometricconstraintsolvingrestrictstheconstraintstobeEucldieandistances.GivengraphG=(V,E),ifweassignaEuclideandistancevaluetoeveryedge,weobtainaEuclideandistanceconstraintsystem(EDCS),oralinkage.I.e.,alinkageisagraphG=(V,E)withadistancemap:E7!R1.Forsomesetsofdistances,onecanndaconguration/realizationamappingofverticestopointsind-dimensionalEuclideanspacesuchthatalledgedistancesaresatisedoftheEDCSinaspecieddimension.ThoseEDCSarecalledrealizableinRd.In1979,itwasprovedthatlinkagerealizabilityisaveryhardproblem[ 76 ]:foragivensetof(integer)edgelengths,determiningifthereisarealizationofagraphinRdisstronglyNP-hard.However,forspecialclassesofthelinkageembeddingproblem,manyresultsareknown.Forexample,whenacompletesetofpair-wisedistanceconstraintsisgiven,Schoneberg'stheorem(Theorem 1.1 )belowgivestheminimumdimensionforwhichthereisarealizationthatsatisestheconstraints.WhentheEuclideandistanceconstraintsystemisnotcomplete,i.e.,somepairsdonothaveanexplicitdistanceconstraint,theproblemofrealizingexactdistancesremainsopen.Foraspecialclassofsuchlinkages,i.e.,treedecomposablelinkages,theproblemhasbeensolvedefcientlybymany[ 10 88 ].Ifweallowdistortionsingenerallinkages,i.e.,forsomeconstraints,insteadofimposingexactvalues,asmallintervalissometimespermitted,thenvariousresults 16

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havebeengiven,showingthetrade-offbetweenminimumrealizabledimensionandamountofdistortionallowed[ 74 75 107 ].Besideexistenceofanembeddingandndingembeddingalgorithms,linkageembeddingproblemsalsoconcerncharacterizingalgebraicsystemswithuniquesolutionorwithanitezeroset(inaxeddimension).Thuscombinatorialrigiditytheoryisaspecialcaseoflinkageembeddingproblems,sincecombinatorialrigiditytheoryingeneralconcernscharacterizingthezerosetofanyalgebraicsystemwhichcanberepresentedbyalinkage.Morespecically,rigiditytheoryinvestigatesalgebraicsystems(e.g.,linkages)withgenericconstraintsthathavearealizationinacertaindimension(realizable),havenitelymanyrealizationsinacertaindimension(rigid),orhaveauniquerealization(globallyrigid)inacertaindimension.Withgenericconstraints,linkageembeddingproblemsaresometimescalledgraphembeddingproblems([ 4 69 ]).Agraphissaidtoberealizable(resp.rigid,globalrigid)ifithasagenericlinkagethatisrealizable(resp.rigid,globalrigid).Here,genericmeansthesetofdistancesdoesnotsatisfyanypolynomialsovertherationalnumbers.Note.Inthescopeofthispaper,whenwetalkaboutthenumberofrealizationsofalinkage,oraframework(tobeintroducedlater),wemeannumberofrealizationsmoduloEuclideantransformations.I.e.,wetreatEuclideancongruenceasthesamerealization.Othershaveworkedonafnecongruenceandprojectivecongruence.Also,incombinatorialrigiditytheory,peoplehaveworkedonothertypesofalgebraicsystems,see[ 9 17 60 ].Fortherealizabilityofagraph,Barvinok[ 7 ]gaveatheoremthatassumingagraphG=(V,E)isrealizableinRd,thenthereexistsaEuclideanembeddingofGinadimensionxedbydandjEj.Fortherigidityofagraph,ithasbeenshownthatforanygraphG,eitherallgenericlinkagesofGarerigid,ornoneofitsgenericlinkagesisrigid.Hence,wecallrigidityagenericproperty.ThuswhenagraphGinRdisrigid,we 17

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knowforgenericlinkages,thereareonlynitelymanyrealizationsmoduloEuclideantransformations.WhenagraphisgloballyrigidRd,weknowforgenericlinkages,thereisauniquerealizationinRd.Hendrickson[ 36 ]rstgaveanecessaryconditionforagraphtobegloballyrigid.Healsoshowedthathisconditionwassufcientin2D.For3Dorhigherdimensions,hisconditionwasshowntobeinsufcientbyConnelly[ 19 ].AsufcientconditionforagenericrealizationinanydimensiontobegloballyrigidwasgivenbyConnelly[ 20 ].Gortleretal[ 33 ]showedthatConnelly'sconditionwasnecessaryforgenericglobalrigidity,thusprovingthatglobalrigidityisagenericproperty.Hencewecansayagraphisgloballyrigidifanyofitsgenericrealizationisgloballyrigid.Inthefollowing,weturntoarelatedresearchareaofgeometricconstraintsolving.Whenageometricdistanceconstraintsystemhasinnitelymanyrealizations(exible,notrigid),thestudyoftherealizationorcongurationspaceisofspecialimportance.Onewaytodescribethecongurationspaceofalinkageistodrawthelinkage.Linkagedrawinghasalonghistoryandoriginatesandprospersinmechanicengineering[ 3 80 97 ].2DlinkagedrawinghasbeenshowntobeequivalenttothesetofallalgebraiccurvesbyKempe[ 57 ]andalgorithmstodrawKempecurveshavealsobeengiven[ 31 ].LinkagecongurationspacealsohasawideapplicationinCAD[ 39 41 42 62 82 ],geometry-basedteachingsoftware[ 53 63 106 ],molecularmodelling([ 2 25 52 87 ])androbotics[ 73 81 83 ].Thestudyofgenericpropertiesofcongurationspaces,i.e.,thosepropertiesthatonlydependonthegraphs,hasrecentlyattractedattention([ 92 93 ])andfoundapplicationsinCADandmolecularmodelling([ 90 ]). 18

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1.3MetricSpaceEmbeddingInthissection,weintroducemetricspaceembedding,whichhaswideapplicationsinapproximationalgorithmstoNP-hardproblems[ 5 6 12 ],anddatacompression[ 11 28 54 55 67 ],andrelateittocombinatorialrigiditytheoryvialinkageembedding.GivenasetofpointsX,adistancemaponXisamap:XX7!R+thatissymmetricand(xi,xi)=0forallxi2X.Thedistanceissaidtobeametric[ 29 ]ifinadditionitsatisesthetriangleinequality,i.e,8xi,xj,xk2X(xi,xj)+(xi,xk)(xj,xk).Ametricspace(X,)isasetofpointsXtogetherwithadistancemapthatisametric.ThemostcommonlyusedmetricistheLpnorm,i.e.,8p2N,8x2Rd,Lp(x):=(Pdijxijp)1=p,wherex1,x2,...,xdarethedcoordinatesofxinRd.AspecialcaseistheL1normspeciedbyL1(x)=maxdifjxijg.Sometimes,wedenotebyjjxjjptheLpnormofx.OnecanimmediatelyndoutthattheEuclideanspaceisa2-normspace.OtherexamplesforametricareHammingdistanceofavectortozeroandF-norminF-spaces.Givenmetricspaces(X,)and(X0,0)withanymetric,amapf:X7!X0isanembedding,where(X,)and(X0,0)arecalledembeddedspaceandembeddingspacerespectively.Adistance-preservingembedding,i.e.,8xi,xj,(xi,xj)=0(xi,xj),iscalledisometric.Metricspaceembedding,especiallyisometricembedding,hasplayedanimportantroleinmanyeldsofcomputersciencesuchasalgorithmsdesign,computervision,computationalbiologyandmachinelearning,tonameafew.Thedimensionoftheembeddingnormedspaceisanimportantmeasureforembeddings.Forexample,aclassicaltheoremofSchoenberg[ 77 79 ]characterizeswhenanisometricembeddingofametricintoEuclideanspacecanbefound,andtheminimumdimensionofsuchanembedding. Theorem1.1. GivenasetofpointsX,letij=jjxi)]TJ /F3 11.955 Tf 12.41 0 Td[(xjjjbesomemetricandbeannnmatrix=(ij)nn.Thenthereisanembeddingof(X,)intoEuclideanspaceif 19

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andonlyifisnegativesemidenite,andtheminimumdimensionforwhich(X,)hasanembeddingisrank())]TJ /F4 11.955 Tf 11.95 0 Td[(1.However,itisnotalwayseasytondisometricembeddingsbetweentwoarbitrarymetricspacesofinterest,evenwhenthedimensionofembeddingspaceisnotspeciedtobexed.Sooftenweallowsomedistortionsbetweentheembeddedspace(X,)andembeddingspace(X0,0).Therearedifferenttypesofdistortions.Forexample,inmachinelearning,oneoftenndsthedatapointsareoftoomanydimensionsandforsomeapplications,itisnotalwaysnecessarytokeepallthedistancevalues.Principalcomponentanalysis(PCA)[ 55 ]ndsthemost-signicantcomponents(i.e,projectionsoforiginalpoints)whosedistancevaluesarenotthatfarfromtheoriginalmetric.Withthosecomponents,theembeddingspacecanhaveafairlylowdimensionwhilethedistortionofdistances(absolutedifference)isnottoolarge.AnotherexampleisthefamousJohnson-Lindenstrausslemma[ 54 ]thatmapsasetofpointsofhigh-dimensionalrealspacetoalow-dimensionalrealspacesuchthatforevery,theratiobetweenthe2-normofanypairofpointsintheembeddedspaceandthe2-normofthesamepairofpointsintheembeddingspaceiswithin.I.e.,8xi,xj2X,(1)]TJ /F5 11.955 Tf 11.96 0 Td[()2(xi,xj)(0)2(xi,xj)(1+)2(xi,xj).Dvoretzky'stheorem[ 28 ]inthe1960sgaveanotherexampleofembeddingwithlow-distortionandthedistortionheremeansthedifferencebetweenthenormintheembeddedspaceandthenormintheembeddingspaceissmall,i.e.,jxjjjxjj(1+)jxj,wherejxjdenotesthenormintheembeddingspaceandjjxjjdenotesthenormintheembeddedspace.BougainandMatousek[ 11 67 ]gaveatheoremthatguaranteestondalow-distortionembeddingofanymetricspaceintoanLpspace(aspaceequippedwithLpnorm).Here,distortionmeanstheproductofmaximumratiobetweenthedistancesintheembeddedspaceandembeddingspaceandthemaximumratiobetweenthedistancesintheembeddingspaceandembeddedspace,i.e.,maxi,j(xi,xj) 0(xi,xj)maxi,j0(xi,xj) (xi,xj). 20

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Figure1-1. Agureshowingtherelationshipsbetweencombinatorialrigiditytheoryandgeometricconstraintsolvingandmetricspaceembedding.Here,Grepresentsgeometricconstraintsolving,Mrepresentsmetricspaceembedding,andLrepresentslinkageembedding,whichistheintersectionofgeometricconstraintsolvingandmetricspaceembedding.EArepresentsembeddingalgorithms,CSrepresentscongurationspacesandCRrepresentscombinatorialrigiditytheory. Insteadofndingisometricembeddingofthecompletemetricintoarbitraryembeddingspaces,sometimeswearejustinterestedinembeddingapartialmetricspace[ 68 ],i.e.,xingasubsetofthepair-wisedistances,andembeddingthosedistancesintoEuclideanspaces.Thisversionofthepartialmetricspaceembeddingproblemisexactlythelinkageembeddingproblem,whichprimarilyinvestigatesthecaseofmetricspaceembeddingwhereweareonlyinterestedinembeddinginto2-normspaces,i.e.,Euclideanspaces,andthedimensionoftheembeddingspaceisoftenxed.Moreover,linkageembeddingisanisometricembeddingontheedges,butthenon-edgescanobtainvaluesfreely(theyhavetoattainthevaluesthatensuretheedgedistancesaresatised).TherelationshipbetweentheresearchdomainscoveredsofarisdrawninFig. 1-1 21

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1.4OtherApplicationsofCombinatorialRigidityBesidesestablishedapplicationsinCAD,teachingsoftware,molecularmodellingandrobotics,combinatorialrigiditytheoryhasfoundnewapplications.Oneexampleisindictionarylearning,abranchofmachinelearningthattriestondasparserepresentationofthedataandatransformation(dictionary)betweentheoriginaldataanditsrepresentation.Oneversionofthedictionarylearningproblemisrelatedtotherigidityoftheso-calledpinnedline-incidenceframework.Dictionarylearningholdsadualrelationshipwithdimensionalityreduction,whichfacilitatesaniteratedorhierarchicalformofdimensionalityreduction.InterestedreaderscanndinChapters3and4of[ 98 ]formoredetails. 22

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CHAPTER2FORMALDEFINITIONSANDCONTRIBUTIONSInthischapter,wegiveformaldenitionsandstateourcontributions.Alongtheway,wealsorelateourworktoexistingresults. 2.1DenitionsInthissection,wegiveformaldenitionsandterminologiesinrigiditytheory.Thesearestandarddenitionsfromcombinatorialrigiditytheory([ 34 35 111 112 ]).AframeworkG(p)inRdisagraphG=(V,E),whereVisthevertexsetandEistheedgeset,togetherwithacongurationp:V7!RdwhichmapsthevertexsettoRd.ForagivengraphG,congurationqisequivalenttocongurationp,denotedasG(q)G(p)ifforall(Vi,Vj)2E,jp(Vi))]TJ /F9 11.955 Tf 12.3 0 Td[(p(Vj)j2=jq(Vi))]TJ /F9 11.955 Tf 12.31 0 Td[(q(Vj)j2.Inaddition,qissaidtobecongruenttop,denotedasqpifjp(Vi))]TJ /F9 11.955 Tf 12.23 0 Td[(p(Vj)j2=jq(Vi))]TJ /F9 11.955 Tf 12.22 0 Td[(q(Vj)j2forall(Vi,Vj)2KV,whereKVisacompletegraphonV.AframeworkG(p)isrigidinRdifforanyqinaneighborhoodofp,G(q)G(p)impliesqp.AframeworkG(p)isgenericinRdifthereexistsaneighbhoodN(p)ofG(p),s.t.foranyqinN(p),G(q)isrigidifandonlyifG(p)isrigid.AgraphGisrigidinRdifthereisagenericframeworkofGthatisrigidinRd.TheminimumnumberofedgesneededtobeaddedtoGtomakeitrigidiscalledthedegree-of-freedom(dof)ofG 2.1.1InnitesimalRigidityForframeworkG(p)inRd,therigiditymap:RjVjd7!RjEjisdenedby(p)=(...,jp(Vi))]TJ /F9 11.955 Tf 12.68 0 Td[(p(Vj)j2,...).TheJacobianofthemap,whichhasdjVjcolumnsandjEjrows,iscalledtherigiditymatrixofG(p).Eachrowoftherigiditymatrixcorrespondstoanedgeinthegraphandeachcolumncorrespondstoacoordinateofavertex.Aninnitesimalmotion([ 34 ])u:V7!Rdsatises(u(Vi))]TJ /F9 11.955 Tf 11.96 0 Td[(u(Vj))(p(Vi))]TJ /F9 11.955 Tf 11.96 0 Td[(p(Vj))=0,forall(Vi,Vj)2E 23

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Figure2-1. Anon-genericrigidframeworkthatisinnitesimallyexible.~uisanon-trivialinnitesimalex. ItiseasytoseethatthesetofallinnitesimalmotionsofG(p)formsthekernelofrigiditymatrixofG(p).NotethatallEuclideanmotionskeepstheedgedistancesxedinRjEj,i.e.,theyareinnitesimalmotions(theyarealsocalledtrivialinnitesimalmotions).SothekernelofrigidmatrixisofdimensionatleastthedimensionofEuclideanmotions,i.e.,)]TJ /F14 7.97 Tf 5.47 -4.38 Td[(d+12.HencethemaximumrankofanyrigiditymatrixisdjVj)]TJ /F15 11.955 Tf 18.57 9.69 Td[()]TJ /F14 7.97 Tf 5.48 -4.38 Td[(d+12.IftherankisequaltodjVj)]TJ /F15 11.955 Tf 17.93 9.68 Td[()]TJ /F14 7.97 Tf 5.48 -4.38 Td[(d+12,wesayG(p)isinnitesimallyrigid.Afundamentalresultconnectingrigidityandinnitesimalrigidityisthefollowing: Theorem2.1. IfG(p)isinnitesimallyrigidinRd,thenG(p)isrigidinRd.Aproofcanbefoundin[ 4 21 32 ].TheproofideaisthatassumingaframeworkG(p)isnotrigid,thenforanyneighborhoodofG(p),thereexistanotherframeworkequivalentbutnotcongruenttoG(p).OnecanchooseasequenceofneighborhoodsandeventuallyapproachG(p)andobtainanon-trivialinnitesimalmotion.However,theconversedirectiondoesnotholdgenerally.Forexample,Figure 2-1 showsaframeworkthatisnon-genericallyrigidbutisinnitesimallyexible.Nevertheless,theconversedoesholdiftheframeworkisgeneric.Thefollowingisaformalstatementoftheresult. 24

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Theorem2.2. IfagenericframeworkG(p)isrigidinRd,thenG(p)isinnitesimallyrigidinRd.Theproofusesimplicitfunctiontheorem.Adetailedproofcanagainbefoundin[ 4 ].WithTheorems 2.1 and 2.2 ,onecandeterminewhetheragivengenericframeworkisrigidornotinRdbylookingattherankoftherigiditymatrix.However,foranarbitraryframework,itisnotalwayseasytoproveitsgenericity,althoughmostoftheframeworksaregenericforanygivengraph.Anotherreasonwhycombinatorialcharacterizationisdesiredisthateventhoughforaframeworkwecandetermineitsindependencebylookingattherankofitsrigiditymatrix,thereisnoeasywaytonditsmaximalrigid/independentsub-framework. 2.1.2IndependentEdgeSetandRigidityMatroidAsetofedgesisindependentinRdiftheircorrespondingrowsintherigiditymatrixofanygenericframeworkisindependentinRd.IndependentsetsofedgesinRdformthegenericrigiditymatroidinRd.Thefollowingistheformaldenitionofamatroid([ 35 ]): Denition2.1. GivenanitesetE,thepowersetofE,denotedbyP(E),isthecollec-tionofallsubsetsofE.AmatroidMisanitesetEtogetherwithanoperator<>mappingP(E)intoP(E)suchthatthefollowingfourconditionsaresatisedforeachsubsetTofE. C1 T; C2 IfRT,then; C3 <>=. C4 Ifs,t2(E)]TJ /F5 11.955 Tf 12.62 0 Td[(),thens2ifandonlyift2TherankofthegenericrigiditymatroidofanygraphGisequaltothesizeofanymaximalindependentset(setthatcannotbeextendedtoalargerindependentset)ofG,andinturnequaltotherankofrigiditymatrixofanygenericframework.Hencein 25

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ordertodeterminetherigidityofagraph,onecantrytondamaximalindependentsetandcheckwhetherithastherequiredsize(djVj)]TJ /F15 11.955 Tf 17.94 9.68 Td[()]TJ /F14 7.97 Tf 5.47 -4.38 Td[(d+12inRd.)Determiningindependenceofanedgesetalsoinvolvesgenericityofthegraph.Soacombinatorialcharacterizationofindependence/rigidityismoredesirable. 2.1.3Rigidityin2DAsmentionedbefore,JamesClerkMaxwell[ 70 ]rstgaveanecessaryconditionforagraphtobeindependentinRd:Maxwell'scondition.AgraphG=(V,E)satisesMaxwell'sconditioninRdifforanysubgraph(V0,E0)s.t.jV0jd,thenjE0jdjV0j)]TJ /F15 11.955 Tf 17.93 9.68 Td[()]TJ /F14 7.97 Tf 5.48 -4.38 Td[(d+12.IfanedgesetEsatisesMaxwell'scondition,thenitisMaxwell-independent.AmaximalMaxwell-independentsubsetE0ofEisaMaxwell-independentsetsuchthatforanyedgee2EnE0,e[E0isnotMaxwell-independent.ForagraphG=(V,E),ifthereisasubgraph(V0,E0),s.t.E0isMaxwell-independentandjE0j=djV0j)]TJ /F15 11.955 Tf 18.38 9.68 Td[()]TJ /F14 7.97 Tf 5.48 -4.38 Td[(d+12,thenwesayGisMaxwell-rigid.IthasbeenprovedbyLaman[ 59 ]thatMaxwell'sconditionisalsoasufcientconditionin2D,thusgivingacombinatorialcharacterizationofindependenceandrigidityin2D. Theorem2.3. In2D,ifagraphGsatisesMaxwell'scondition,thenGisindependent.AlgorithmsbasedonLaman'scharacterizationhavebeenproposedandareofpolynomialtimecomplexity[ 51 ].AsanasideofLaman'stheorem,onecanobtainthatmaximalMaxwell-independentsetsofagivengraphin2Dhavethesamesize.ThusMaxwell-independentsetsalsodeneamatroidin2D.Thismatroidhappenstobeequivalenttotherigiditymatroidin2D.Anotherwaytoapproachindependenceistocombinatoriallyconstructindependentgraphsfromsmallerindependentgraphs.Thereareseveralmethods[ 105 ].ThemostsignicantconstructionistheHennebergconstructions.AHennebergkconstructiononagraphGinRdistorstchooseasetvertexsetXofd+k)]TJ /F4 11.955 Tf 12.17 0 Td[(1verticesofGsuchthat 26

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thenumberofedgesinducedbyXisnolessthank)]TJ /F4 11.955 Tf 12.21 0 Td[(1,deletingk)]TJ /F4 11.955 Tf 12.21 0 Td[(1edgesbetweentheverticesofX,andthenaddinganewvertexvandedgesfromvtoallverticesinX.Ithasbeenshownin[ 105 ]thatHennebergIandIIconstructionsin2Dpreserveindependence.Ithasalsobeenshown[ 37 105 ]thateveryindependentgraphcanbeobtainedusingHennerbergIandHennebergIIconstructionsfromK2.However,in3D,thisisnotthecase. 2.1.4Rigidityin3DAsmentionedearlier,thereisnoknowncombinatorialcharacterizationfor3Dbar-and-jointrigidity.Allexistingcharacterizationsareapplicabletoothertypesofframeworks.Abody-barframeworkisasetofrigidbodies(sub-frameworks)connectedviaxed-lengthbars[ 100 ].Ad-dimensionalbody-hingeframeworkisastructureconsistingofrigidbodiesconnectedbyhingesind-dimensionalspace.Therearemanyresultsrelatingbody-barframeworkandbody-hingeframeworkwithbar-and-jointframeworks.[ 101 ]and[ 111 ]showthathingescanberepresentedwith5bars.Tay[ 100 ]givesacharacterizationofminimallyrigidbody-and-barframeworksinarbitrarydimensiond:abody-barframeworkinRdisrigidifandonlyifitcontains)]TJ /F14 7.97 Tf 5.48 -4.38 Td[(d+12disjointspanningtrees.Body-barandbody-hingeframeworkscanbeusefulinthecombinatorialcharacterizationofgenericrigidityforaspecialclassofgraphs:in[ 49 ],itwasshownthatforgraphsG2thataresquaressomegraphG,Maxwell-independenceisequivalenttotrueindependent.HereasquareG2ofagraphG=(V,E)isagraphobtainedfromGbyaddinganewedge(u,v)foreachpairfu,vg2Vsuchthatbetweenuandv,thereisapathofdistancetwoinG.Theresultusesthemolecularconjectureforbody-hingeframeworks(posedin[ 104 ]andprovedin[ 56 ]).Butforgeneralgraphs,wewillseeinSection 2.2.1 thatevenasimpleexamplecanrevealthegapbetweenMaxwell-independenceandtrueindependence. 27

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Itisnothardtoseethatifwecanndacombinatorialcharacterizationofrigidityforbar-and-jointframeworks,thenwehaveacombinatorialcharacterizationofrigidityforbody-barorbody-hingeframeworks.Ontheotherhand,thestudyofbody-barandbody-hingeframeworkscanbeappliedtobar-and-jointframeworks.Forexample,wecanidentifysomerigidcomponentsofabar-and-jointframework,convertittobody-barorbody-hingestructureanduseresultsonthoseframeworks.Wewilluseapropertyofbody-hingeframeworkinChapter 4 toshowexistenceofimpliednon-edgeinaspecialclassofbar-and-jointframeworks.Note:Intheremainderofthisdissertation,whenwerefertoframeworks,wemeanbar-and-jointframeworksunlessotherwiseindicated. 2.2OrganizationandContributionInthissection,wemotivateandstatethemainresultsofthisdissertation.InSection 2.2.1 ,welistresultsinvolvingrankupperboundsusingmaximalMaxwell-independentsetsshowninChapter 3 .InSection 2.2.2 ,welistresultsofChapter 4 ,i.e.,generalconstructionschemesfornucleation-freedependentgraphs.InSection 2.2.3 ,welistresultsforChapter 5 :aconcrete,purelycombinatorialdenitionofaclosureoperatorthatgivesanabstractrigiditymatroid. 2.2.1MainResultofChapter 3 :RankUpperBoundsUsingMaximalMaxwell-IndependentSetsandInclusion-ExclusionCountIn3Dandhigherdimensions,Maxwell-independenceisnolongerequaltoindependence.AclassicalexampleillustratinginsufciencyofMaxwell'scountingconditionin3Distheso-calleddouble-bananagraphinFig. 2-2 .ItsatisesMaxwell'scountingcondition,butthegraphisclearlyexible,anddependent.Thereasonisthatsinceeachbanana(aK5with1edgemissing)isrigidasaninducedsubgraph,thedistancealongthenon-edgefa,bgsharedbythe2bananasisdeterminedbyeachbanana.Whenthedistancealonganon-edgefa,bgisdeterminedbyagraphG,i.e.,linearlydependentontherowsofG'sgenericrigiditymatrix,thenfa,bgiscalledan 28

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impliednon-edgeinG.Inthiscase,sincefa,bgisimpliedbybothbananas,thegraphsGisdependent. Figure2-2. Bananaanddouble-banana:ontheleftisthebananagraph,whichisaK5(completegraphonvevertices)withoneedgemissing;ontherightisadouble-banana,whichconsistsoftwobananasgluingtogetheralongtheirrespectivenon-edgeandthenon-edgeisimplied. Remark:Theconceptofimpliednon-edgeissimilartobutweakerthanthenotionofgloballylinkedpairfrom[ 50 ],whichreferstoapairofverticeswhosedistanceisgenericallyxedbythegraph.Thedistanceassociatedwithanimpliednon-edgeisincontrastgenericallyrestrictedtonitelymanyvalues.Analternativenameforanimpliednon-edgecouldbealinkedpair.AnotherexampleofadependentgraphillustratinginsufciencyofMaxwell'scountingcondition,i.e.,beingdependentwhilesatisfyingMaxwell'scountingcondition,isduetoCrapo(Fig. 2-3 ).Thegraphcontainsaso-calledhinge,i.e.,apairofverticescommontoatleasttworigidcomponents,wherearigidcomponentisamaximalsubsetSofverticesofagraphGsuchthatthenon-edgesinSareimpliedbytheedgesinG,possiblyoutsidethegraphinducedbyS.Forexample,thefa,bgpairinFig. 2-3 isa 29

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hinge.Again,herehingesareimpliednon-edgesthatarethecausesoftheinsufciencyofMaxwell'scondition. Figure2-3. Crapo'sgraphwithahingestructure:fa,bgisahinge,sinceitissharedbytworigidcomponents. AmoreimportantfactillustratingthegapbetweenMaxwell-independenceandgenericindependenceisthatMaxwell-independentedgesetsdonotdeneamatroidford3.I.e.,noteverymaximalMaxwell-independentsethasthesamesize.OneexampleisthegiveninFigure 2-4 .Clearly,anymaximalindependentsubgraphofGisitselfMaxwell-independent,sotherankofthegenericrigiditymatroidofagraphisatmostthesizeofsomemaximalMaxwell-independentsetandthisgeneralizestoanydimension.Butthisonlyyieldsthetrivialupperbound,i.e.,numberofedges,forMaxwell-independentgraphs.Thisleadstothefollowingnaturalquestionconcerningtherankofthe3-dimensionalgenericrigiditymatroid.ThequestionwasposedbyTiborJordanduringthe2008BIRSrigidityworkshop[ 1 ]. Problem1. Doeseverymaximal,Maxwell-independentsubgraph(subsetsofedges)ofagraphGhavesizeatleasttherankofthe3-dimensionalgenericrigiditymatroidofG? 30

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Figure2-4. AgureshowingmaximalMaxwell-independentsetscanhavedifferentsizes.Thegraphontheleftisadouble-bananawiththemiddlenon-edgeaddedandconsistsoftwoK5'sintersectingonanedge.ThegraphsonthemiddleandtherightaretwomaximalMaxwell-independentsetsofdifferentsizesforthegraphontheleft(themiddleisofsize18andtherightisofsize17). NotethattheanswertoProblem 1 wouldbeobviousifeverymaximalMaxwell-independentsetofagivengraphGcontainsamaximalindependentsetofG.However,thisisnotthecase.SeeFigure 2-5 .InChapter 3 ,wegiveanafrmativeanswertoProblem 1 inTheorem 3.1 .BillJackson[ 44 ]hasextendedthisresultuptod=5.Hisproofisbycontradictionandishencenonconstructive.Ourproofisconstructive:forMaxwell-independentgraphs,wegivecombinatorialformulaebasedoninclusion-exclusion(IE)countsupperboundingtherank;andweconstructsubgraphs(independenceassignments)whosesizesmeetthisbound,andmoreovercontainamaximaltrueindependentset(Theorems 3.2 and 3.3 );thisconstructionisofalgorithmicinterest.TheconstructionleadstoalternativeupperboundsonrankrelatedtoDress'formula([ 26 ],Section 3.2.1 )forcertainclassesofnon-Maxwell-independentgraphsthatadmitcertaintypesofcoversinSection 3.2.3 (Theorems 3.4 and 3.5 ).However,algorithmsforcomputingthesecoversarebeyondthescopeofthisdissertation.Severalalgorithmsexistforcombinatoriallyrecognizingcertaintypesofdependencesinford=3([ 64 85 94 ]).Thesimplestofthesealgorithmsisaminormodication([ 64 ]) 31

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Figure2-5. AgureshowingthatnoteverymaximalMaxwell-independentsetofagivengraphGcontainsamaximalindependentsetofG.Ontheleftisadouble-banana-bar,whichconsistsofadouble-bananaandabarconnectingtwoverticesfromeachbanana.Noticethatthisdouble-banana-barisrigid,thuseverymaximalindependentsetinithas3jVj)]TJ /F4 11.955 Tf 17.93 0 Td[(6=18edges.OntherightwehaveamaximalMaxwell-independentsetofthedouble-banana-bar,whichhas3jVj)]TJ /F4 11.955 Tf 17.93 0 Td[(6=18edges.Thegureontherightisdependent,soeverymaximalindependentsetofithassizelessthan3jVj)]TJ /F4 11.955 Tf 17.93 0 Td[(6=18.Sotherightgurecannotcontainamaximalindependentsetofsize3jVj)]TJ /F4 11.955 Tf 17.93 0 Td[(6. ofJacobsandHendrickson's([ 51 ])pebblegameford=2,andndsamaximalMaxwell-independentset(itmaybeneithertheminimumsizedonenorthemaximumsizedone).ThetechniquesdevelopedinChapter 3 simplifytheproofsofcorrectnessforthesealgorithms. 2.2.2MainResultofChapter 4 :GeneralConstructionSchemeforNucleation-Free,IndependentGraphswithImpliedNon-EdgesThisworkisanextension([ 16 ])ofjointresultsappearinginCCCG09[ 15 ].FromanalyzingknownexamplesofinsufciencyofMaxwell'scountingcondition,wendoutthatallthoseexampleshaveimpliednon-edgesandimpliednon-edgesplayanimportantroleintheinsufciencyofMaxwell'scountingcondition.However,animportantobservationinthoseexamples(e.g.Figures 2-3 2-4 )isthatalthoughsomeimpliednon-edgeslieinsiderigidcomponentsasopposedtorigidinducedsubgraphs,thesetroublesomedouble-impliednon-edgesexistduetothepresenceofarigidinducedsubgraphsomewhereinthegraph.I.e.,thoseexamplessatisfythefollowingproperty: 32

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Nucleationproperty.AgraphGhasthenucleationpropertyifitcontainsanon-trivialrigidsubgraph,i.e.,arigidsubgraphinisolation,whichwecallarigidnucleus.Here,weusetrivialtorefertographswith4orfewervertices.Ifagraphdoesnothaveanynucleus,wecallitnucleation-free.Notethatin2D,everyimpliednon-edgeinfactliesinsideanucleation,asastraightforwardconsequenceofLaman'sTheorem.In3D,providedagraphhasnucleationproperty,thereisapotentialmethodofovercomingtheobstacleofimpliednon-edgesusing(3,6)-sparsityfordetectingdependence[ 94 ]:recursivelyidentifynucleations,addnon-edgesinthosegraphstocompletethem,andthencheck(3,6)-sparsityinotherpartsofthegraph.However,fornucleation-freegraphs,theapproachin[ 94 ]collapsestosimple(3,6)-sparsitycheck,leadingtothesecondobstacletotheproblemofcombinatorialcharacterizationsof3Drigidity.Inparticular,inanucleation-freedependentgraph,theapproachin[ 94 ]wouldfailinthatitcannotdetecttheimpliednon-edges,sinceitreliesonanucleationasastartingpoint.Theexistenceofnucleation-freedependentgraphsindicatesthatagapexistsbetweenmodule-rigidityproposedin[ 94 ]andtruerigidity.Thus,tobetterunderstandrigidity,weneedtounderstandtheobstacleposedbynucleation-freegraphswithimpliednon-edges.Asarststeptowardsthisgoal,itisnaturaltoask: Problem2(Generalinductiveconstructionschemesfornucleation-freegraphswithimpliednon-edges.). Howdoweconstructgeneralfamiliesofnucleation-freegraphsthathaveimpliednon-edges?InChapter 4 ,weprovidegeneralinductiveconstructionschemesandprooftechniquesansweringProblem 2 .Wegivesystematicclassicationsofproofingredientsneededforourprooftechniques,andthoseingredientscanbegeneralized,mixedandmatchedtogenerateandvalidateconstructionschemes.Inaddition,wegiveseveral 33

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examplegraphsthatsatisfythestartinggraphrequirementsforourgeneralinductiveconstructionschemesforProblem 2 .Asabyproductofoneofourschemes,wendaninductivemethodtoconstructnucleation-free,independentgraphs.Itshouldbenotedthatthereareveryfewgeneralinductiveconstructionforindependentgraphsin3D.TheonlyknownonesarevertexsplitandHennebergconstructions([ 105 110 ]),whicharedistinctlydifferentandcannotgenerallymimicourinductiveconstruction.Furtherconsequences.Asmentionedearlier,weshowagapexistsbetweenmodule-rigidityproposedin[ 94 ]andtruerigidity.Thenextstepforabetterunderstandingofrigidityshouldbetondnotionsthatdetectimpliednon-edgesanddependenceinnucleation-freegraphs.Asanotherconsequenceofourwork,weshowtherstgeneralfamiliesofexamplesofexible3Drigiditycircuitswithnonucleations.Incontrast,allrigiditycircuitsin2Darerigid.Untilnow,exiblerigiditycircuitswithoutnucleationwereonlyavailablein4D,butnotin3D(seediscussionsearlierandin[ 35 ]):Theonlyknownnon-rigidcircuitsinthe3DrigiditymatroidarisefromamalgamationsofcircuitsforcedbyMaxwell'scountingcondition.Ourresultimplies,inaddition,thatLovasz'characterization[ 65 ]of2Drigidityviacoveringscannotbeextendedto3D.Note:Nucleation-freerigiditycircuitswithimpliednon-edgeshavebeenconjecturedandwrittendownbymany([ 48 103 ]).However,tothebestofourknowledgewearethersttogiveproofs.Inparticular,in[ 103 ],Tayclaimedaclassofexiblerigiditycircuitswithoutanynuclei.Oneofhisexamples,n-butteries,inwhichheclaimedexistenceofimpliednon-edges,isthesameasourwarm-upexamplegraphs,ringofroofs.ButhisproofattempthasaseriousgapwhichwewilldescribeindetailinAPPENDIX. 2.2.3MainResultofChapter 5 :aConcrete,PurelyCombinatorialDenitionofaClosureOperatorthatGivesanAbstractRigidityMatroidWiththeexistenceofnucleation-free,dependentgraphs,weknowtherecursiveversionofMaxwell-independenceproposedin[ 94 ]willfailtodetecttheirdependency, 34

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Figure2-6. Agureshowingminimum-sizedmaximalMaxwell-independentsetcanbeabadboundonrank.ThegraphconsistsofnK5)]TJ /F4 11.955 Tf 11.95 0 Td[(1'swhereallK5)]TJ /F4 11.955 Tf 11.95 0 Td[(1'shaveanon-edgeincommon.ThisgraphisMaxwell-independentsoitminimum-sizedMaxwell-independentsetisofsize9nwhiletherankofthegraphonis8n+1.SothedifferencebetweenthesizeofthismaximalMaxwell-independentsetandtherankwillben)]TJ /F4 11.955 Tf 11.96 0 Td[(1. althoughthedependenceofclassicalexamplessuchasdouble-bananaandCrapo'shingestructures(seeFigure 2-3 )arecorrectlydetected.Soitisnaturaltoask: Problem3. Isthereacombinatorialnotionthatdetectsthedependencyinthenucleation-freedependentgraphsconstructedinChapter 4 ?Moreimportantly,betterupperboundsthanthesizeofmaximalMaxwell-independentsetaredesirablesinceboundsgivenbyMaxwell-independentsetscanbearbitrarilybad,evenforminimum-sizedMaxwell-independentset.Forexample,inFig. 2-6 ,wehaven(K5)]TJ /F4 11.955 Tf 13.06 0 Td[(1)'seachofwhichsharesapairofverticeswithoneanother.ThewholegraphisMaxwell-independent,thusanymaximalMaxwell-independentsethassizeequaltothenumberofedgesinthegraph(9n)whiletherankofn-bananais8n+1.Thusinn-banana,thedifferencebetweenthesizeofminimumMaxwell-independentsetandtherankisn)]TJ /F4 11.955 Tf 11.96 0 Td[(1.Lastbutnottheleast,weknowindependencedenesamatroid,whichisakeypropertyforgenericrigidity.Tobetterunderstandrigidityandgivebetterrankestimates,itisofgreatinteresttondacombinatorialnotionthatdenesamatroidandappliestogeneralgraphs. 35

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Sofar,manyresultsconcerningthematroidpropertyofthegenericrigiditymatroidhavebeengivenford=2.Forexample,Yemini-Lovasz[ 65 ]gaveacountingformulaequaltotherankthegenericrigiditymatroidin2dimensions.Crapo[ 23 ]gaveanalternativecharacterizationofindependentsetsofthe2-dimensionalgenericrigiditymatroid.AshorterproofofLaman'sTheorembasedonCrapo'scharacterizationwasgivenbyTay[ 102 ].Dressgavetwoconjecturesabouttherankofd-dimensionalrigiditymatroidin[ 26 ]usingarankfunctiononcoversofthegraph.However,therstconjecturewasprovedtobefalseford=3in[ 47 ],inwhichaweakerconjectureofCrapoandTay[ 24 ]thattherankfunctionisamatroidfunctionisalsodisproved.Fordimensiond3,nopurelygraph-basedmatroidconceptscharacterizingthegenericrigiditymatroidhavebeendevelopedsofar,leavingthecombinatorialcharacterizationofthegenericrigiditymatroidalong-standingopenproblem.InChapter 5 ,wegiveacombinatorialgraph-basednotionandshowthatitdenesamatroidwhoserankisabetterrankestimatethatsizeofmaximalMaxwell-independentsetsforgenericrigidity.Thematroidisalsoprovedtobeanabstractrigiditymatroid.Note:Ourclosureoperator,calledGEM-closure(GradedExchangeMaxwell-closure),isdenedpurelybyinspectingedges,non-edgesandverticesofthegivengraph.I.e.,unlikethedenitionofagenericrigiditymatroidofagraph,itisnotnecessarytospecifyarepresentationwithanunderlyingeldandvectorspace;andnoalgebraicoperationsarenecessary.Note2:OurGEM-closureyieldsamatroid,whichisdistinctlydifferentfromthegradedsparsitymatroidintroducedin[ 61 ],whichisdenedonhypergraphsGandhasasparsitycountconditiononaseriesofpropersubgraphsonG.FurtherConsequences.In[ 35 ],thefollowingconjecturewasmade: Conjecture2.1(MaximalConjecture[ 35 ]). In3dimensions,everyindependentsetinanyabstractrigiditymatroidisalsoindependentinthegenericrigiditymatroid. 36

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OurresultwilleitherdisprovethisconjectureorshowthatGEM-independenceisequivalenttogenericindependenceprovidedourConjecture 5.1 istrue.WhenConjecture 5.1 istrue,theindependentsetsintheGEM-matroidalsoanswerProblem 3 inthepositive.Note3:ourGEM-matroidisaconcreteexampleofanabstractrigiditymatroid.Itspeciesaclosureoperatorthatcanbeusedtouniquelycomputetheclosureofanygivengraph.Moreover,GEM-matroidisdifferentfromexistingexamplesofabstractrigiditymatroidsuchas1-extendableabstractrigiditymatroidin[ 71 ],whichisonlydenedforspecialtypesofgraphs.AcomparisonofGEM-matroidwithgenericrigiditymatroid,abstractrigiditymatroidand1-extendableabstractrigiditymatroidisgiveninTable 2-1 2.2.4MiscellanyInChapter 6 ,welistsomepartialresultsandopenproblems.Oneofthepartialresultinvolvesaspecialtypeofcongurationspace,calledconvexCayleycongu-rationspace,wheretheparameterstodescribethecongurationspacearechosentobenon-edgesofthegraph.WeshowaequivalencebetweenasquaredCayleycongurationspaceandtheconvexityofacorrespondinggraph.Anotherpartialresultinvolvesglobalrigidity,specically,showingoneofthesufcientconditionforglobalrigiditycanimplyanotherwell-knowsufcientconditionforglobalrigidity.InChapter 7 ,wegiveaconclusion. 37

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Table2-1. AcomparisonofGEM-matroidwithotherrigidity-relatedmatroidconcepts. XXXXXXXXXXXXMatroidProperty Concreteanduniqueforagivengraph? Needalgebratodeneindependence(notpurelycombinatorial)? Denedforgeneralgraphs? Genericrigiditymatroid Yes Yes Yes Abstractrigiditymatroid No No Yes 1-extendableabstractrigiditymatroid N/A No No GEM-matroid Yes No Yes 38

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CHAPTER3MAXWELL-INDEPENDENCEInthischapter,wegivearesult(Theorem 3.1 )inSection 3.1 thatanswerstoProblem 1 ford=3inthepositive,assketchedinSection 2.2.1 .InSection 3.2 ,wealsorelateourboundstoexistingboundsandconjectures.IntheconcludingSection 3.3 ,weposeopenproblems. 3.1MainResultandProofInthissection,westateandgivetheproofofthefollowingmaintheorem.NotethatSections 3.1 and 3.2 dealexclusivelywithd=3andweuserank(G)todenotetherankofthe3-dimensionalgenericrigiditymatroidofgraphG. Theorem3.1. LetMbeamaximalMaxwell-independentsubgraphofagraphG=(V,E)andIbeamaximalindependentsetofthe3-dimensionalgenericrigiditymatroidofG.ThenjE(M)jjIj,whereE(M)denotestheedgesetofM.Intheremainderofthissection,wegivetheproofofTheorem 3.1 .Theproofrequiresafewdenitionsandintermediateresults. Denition3.1. TheMaxwellcountforagraphG=(V,E)in3dimensionsis3jVj)-229(jEj.GissaidtobeMaxwell-rigidin3dimensions,ifthereexistsaMaxwell-independentsubsetE?EsuchthattheMaxwellcountofG?=(V,E?)isatmost6.Asexceptions,j-cliques(j2)areconsideredtobeMaxwell-independentandMaxwell-rigid.AsubgraphG0=(V0,E0)inducedbyV0VissaidtobeacomponentofG,ifitisMaxwell-rigid.Inaddition,G0iscalledavertex-maximalcomponentofG,ifitisMaxwell-rigidandthereisnopropersupersetofV0thatalsoinducesaMaxwell-rigidsubgraphofG.Acomponentwith2verticesconsistsofasingleedgeofthegraph,andwecallitanedgecomponent,ortrivialcomponent.Othercomponentsarecallednon-trivialcomponents.Thefollowingconceptsofcoversandinclusion-exclusionformulaeoncoversfrom[ 22 46 48 64 66 85 94 ]areimportantfortheproofofTheorem 3.1 .Acoverof 39

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agraphG=(V,E)isacollectionXofpairwiseincomparableinducedsubgraphsG1,...,GmofG,eachwithatleasttwovertices,suchthat[Gi2XE(Gi)=E,whereE(Gi)istheedgesetofsubgraphGi.V(Gi)denotesthevertexsetofGi.LetGi[Gjdenotethegraph(V(Gi)[V(Gj),E(Gi)[E(Gj))andGi\Gjdenotethegraph(V(Gi)\V(Gj),E(Gi)\E(Gj)).GivenagraphGwithacoverX=fG1,...,Gmg,weuseH(X)todenotethesetofallpairsofverticesfu,vgsuchthatV(Gi)\V(Gj)=fu,vgforsome1i
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Lemma3.1. (a) GivenMaxwell-rigidgraphsM1andM2,ifV(M1)\V(M2)consistsoftwoverticesuandvandfu,vg62E(M1)[E(M2),thenM1[M2isalsoMaxwell-rigid. (b) GivenMaxwell-independentgraphMandtwoMaxwell-rigidsubgraphM1andM2ofM,ifjV(M1)\V(M2)j3,thenM1[M2isalsoMaxwell-rigid. Proof. (a) LetN1beaMaxwell-independentsubgraphofM1with3jV(M1)j)]TJ /F4 11.955 Tf 19.27 0 Td[(6edgesandN2beaMaxwell-independentsubgraphofM2with3jV(M2)j)]TJ /F4 11.955 Tf 19.26 0 Td[(6edges.WeshownextthatN1[N2isMaxwell-independent.SupposeN1[N2isMaxwell-dependent.ThenthereexistsN0N1[N2suchthatN0hasMaxwellcountlessthan6.SincebothN1andN2areMaxwell-independent,itisclearthatN0*N1andN0*N2.LetN0=N01[N02suchthatN01N1andN02N2.ThenN01andN02bothhaveMaxwellcountatleast6.TomaketheirunionhaveMaxwellcountlessthan6,N01andN02mustshareatleasttwovertices.SinceV(N1)\V(N2)consistsoftwoverticesuandv,weknowV(N01)\V(N02)consistsofatmosttwoverticesuandv.Sincefu,vg=2E(N1)[E(N2),itcanbeseenthatinordertomakeN0ofMaxwellcountlessthan6,atleastoneofN01andN02willhaveMaxwellcountlessthan6,whichtogetherwiththefactthatN01N1andN02N2violatesMaxwell-independenceofN1orN2.HenceN1[N2isMaxwell-independent.NoticethatN1[N2hasenoughedgestobeMaxwell-rigidandthusM1[M2isalsoMaxwell-rigid. (b) SinceMisMaxwell-independent,weknowbothM1[M2andM1\M2areMaxwell-independent.ThencanwecalculatetheMaxwellcountofM1[M2asfollows.Weknow(1)M1andM2eachhaveMaxwellcount6and(2)M1\M2hasMaxwellcountatleast6sinceM1\M2isMaxwell-independentandhasatleast3vertices.ThustheMaxwellcountofM1[M2isatmost6.TogetherwiththefactthatM1[M2isMaxwell-independent,weknowM1[M2isMaxwell-rigid. ThisfollowingpropositiongivesausefulpropertyofacoverofaMaxwell-independentsubgraphbyvertex-maximalcomponents. Proposition3.1. LetMbeaMaxwell-independentgraph.LetX=fe1,...,ek,M1,M2,...,MmgbeacoverofMbyvertex-maximalcomponents,wheree1,...,ekareedgecomponentsandM1,M2,...,Mmarenon-trivialcomponents.Then 41

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(a) Xisa2-thincoverofM. (b) Xisstrong2-thin. Proof. (a) Edgecomponentsdonotaffectthecoverbeing2-thinornot.LetMiandMjbetwonon-trivialvertex-maximalcomponentsinM.SupposeMiandMjshareatleast3vertices.ThenfromLemma 3.1 ( b ),weknowMi[MjisMaxwell-rigid,violatingthefactthatMiandMjarevertex-maximalcomponents. (b) Again,edgecomponentsdonotaffectthecoverbeingstrong2-thinornot.LetMiandMjbetwonon-trivalvertex-maximalcomponentsinM.IfMiandMjsharetwoverticesbutdonotshareanedge,thenfromLemma 3.1 ( a ),Mi[MjisMaxwell-rigid,whichviolatesthevertex-maximalpropertyofMiandMj. Nextweprovealemmaaboutthestructureofa2-thincoverofaMaxwell-independentgraph.Werstneedthefollowingdenitionof2-thincomponentgraph. Denition3.4. GivengraphG=(V,E),letX=fG1,G2,...,Gmgbea2-thincoverofGbycomponentsofG.The2-thincomponentgraphCXofG(componentgraphforshort)isdenedasfollows.V(CX)=Vcomponent(CX)[Vedge(CX),whereVcomponent(CX)consistsofcomponentnodesCGi,oneforeachcomponentGiinX;andVedge(CX)consistsofedgenodesCe,oneforeachedgeesharedbyatleasttwocomponentsinX.TheedgesinE(CX)areoftheform(CGi,Ce),whereCGi2Vcomponent(CX),Ce2Vedge(CX),ande2EisasharededgeofGi.Figure 3-1 showshowtoobtaina2-thincomponentgraphfromagraphandacoverbyitsvertex-maximalcomponents.Notethatcomponentssharingonlyverticesarenon-adjacentinthecomponentgraph.Edgecomponentshavedegreezeroandbecomedisconnectednodesinthecomponentgraph.SeeFigure 3-1 .Lemma 3.2 ( b )belowstatesanimportantpropertyofcomponentgraphsofMaxwell-independentgraphs.Specically,these2-thincomponentgraphsareinspiredbytheconceptof 42

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Figure3-1. Thegureontheleftrepresentsthevertex-maximalcomponentsofagraph.Ontherightsideisits2-thincomponentgraph,wherecirclesrepresentcomponentnodesandsquaresrepresentedgenodes.Notethatthe2-thincomponentgraphmaynotbeconnected. partialm-trees(alsocalledtree-widthmgraphs)andHennebergconstructions[ 35 ],whichwedenebelow. Denition3.5. Letmbeapositiveinteger.Thena2-thincomponentgraphiscalledageneralizedpartialm-treeifitcanbereducedtoanemptygraphbyasequenceofthefollowingtwooperations:(i)removalofacomponentnodeofdegreeatmostmand(ii)removalofanedgenodeofdegreeone.Nowwearereadytostatethelemma. Lemma3.2. IfMisaMaxwell-independentgraphandXisa2-thincoverofMbycomponentsofM,then (a) thecomponentnodesofanysubgraphofthe2-thincomponentgraphCXhaveaveragedegreestrictlylessthan4. (b) anysubgraphofthe2-thincomponentgraphCXofMisageneralizedpartial3-tree. Proof. 43

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(a) FirstweremovealledgecomponentsofMandshowtheremainderofthecomponentgraphhasaveragedegree<4.LetKXbeanysubgraphofthe2-thincomponentgraphCX.LetKdenoteKX'scorrespondingsubgraphinM.LetfM1,...,MngbeXrestrictedtoK.LetViandEibethesharedvertexandsharededgesetsofcomponentMiofK,i.e.,ViandEiaresharedbyothercomponentsMjofK.LetVsandEsbetheentiresetsofsuchsharedverticesandsharededgesinK.LetneandnvdenotethenumberofcomponentsMiofKthatshareeandvrespectively.SincetheMaxwellcountofeachMiis6(theyareallnon-trivial),theMaxwellcountofKcanbecalculatedasfollows:Xi6)]TJ /F4 11.955 Tf 11.96 0 Td[(3Xv2Vsnv+Xe2Esne+3jVsj)-222(jEsj=Xi(6)]TJ /F4 11.955 Tf 11.96 0 Td[(3jVij+jEij)+3jVsj)-222(jEsjSupposetheMaxwellcountofKis6.Wehave 6n)]TJ /F4 11.955 Tf 11.96 0 Td[(63XijVij)]TJ /F15 11.955 Tf 17.93 11.35 Td[(XijEij)]TJ /F4 11.955 Tf 17.93 0 Td[(3jVsj+jEsj(3)ConsideranysharedvertexvinVs.DenotebyCvf1,...,ngthesetofindicesofcomponentscontainingv.Inthisproof,sincethecontextisclear,werefertoMj,j2Cvasacomponentcontainingv.ThecollectionofallnvcomponentsofKmeetingatvformsasubgraphC.SinceKisMaxwell-independent,CisalsoMaxwell-independent.LetwjvbethenumberofsharededgesincidentatvincomponentMjandsvbethenumberofsharededgesthatareincidentatv.ThentheMaxwellcountofCcanbecomputedasfollows: therearenvcomponents,whichcontributes6nv; vissharedbynvcomponents,andthecontributionis)]TJ /F4 11.955 Tf 9.3 0 Td[((3nv)]TJ /F4 11.955 Tf 11.95 0 Td[(3); eachsharededgeinacomponentMjcontributes1totheMaxwellcount,andaltogetherthesharededgescontribute(Pj2Cvwjv))]TJ /F3 11.955 Tf 11.95 0 Td[(sv foreachsharededgee=fu,vg,vertexucontributes)]TJ /F4 11.955 Tf 9.3 0 Td[(3[(Pj2Cvwjv))]TJ /F3 11.955 Tf 11.95 0 Td[(sv] forthesetofsharedverticesthatarenotpartofanysharededgeinC,theircontributionis)]TJ /F4 11.955 Tf 9.3 0 Td[(foranon-negativenumber;ThustheMaxwellcountofCis:3nv)]TJ /F4 11.955 Tf 11.95 0 Td[(2[(Xj2Cvwjv))]TJ /F3 11.955 Tf 11.95 0 Td[(sv]+3)]TJ /F4 11.955 Tf 11.96 0 Td[(SinceCisMaxwell-independent,weknow:3nv)]TJ /F4 11.955 Tf 11.95 0 Td[(2[(Xj2Cvwjv))]TJ /F3 11.955 Tf 11.95 0 Td[(sv]+3)]TJ /F4 11.955 Tf 11.96 0 Td[(6 44

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Since0,weknow3nv)]TJ /F4 11.955 Tf 11.96 0 Td[(2[(Xj2Cvwjv))]TJ /F3 11.955 Tf 11.96 0 Td[(sv]3SummingoverallsharedverticesinVs,wehave:3Xv2Vsnv)]TJ /F4 11.955 Tf 11.96 0 Td[(2Xv2Vs[(Xj2Cvwjv))]TJ /F3 11.955 Tf 11.95 0 Td[(sv]3jVsjSincePv2Vsnv=PijVij,Pv2Vs(Pj2Cvwjv)=2PijEijandPv2Vssv=2jEsj,weknow3XijVij)]TJ /F4 11.955 Tf 17.94 0 Td[(4XijEij)]TJ /F4 11.955 Tf 17.93 0 Td[(3jVsj+4jEsj0Plugginginto( 3 ),wehave:6n)]TJ /F4 11.955 Tf 11.95 0 Td[(63XijVij)]TJ /F15 11.955 Tf 17.93 11.35 Td[(XijEij)]TJ /F4 11.955 Tf 17.93 0 Td[(3jVsj+jEsj3XijVij)]TJ /F4 11.955 Tf 17.93 0 Td[(4XijEij)]TJ /F4 11.955 Tf 17.93 0 Td[(3jVsj+4jEsj+3(XijEij)-223(jEsj)3(XijEij)-222(jEsj)SincejEsj1 2PijEij,wehave:6n)]TJ /F4 11.955 Tf 11.95 0 Td[(63 2XijEijWenowobservethatthecomponentnodesinKXmusthaveaveragedegreestrictlylessthan4.Otherwise,PijEij4n,leadingtoacontradictionthat6n)]TJ /F4 11.955 Tf 11.95 0 Td[(63 24n=6n.Thisproves( a ). (b) Thisfollowsimmediatelyfrom( a ). 45

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NextweestablishaconditiononthecoverofaMaxwell-independentgraphsuchthattheIErankcountinDenition 3.2 givesanupperboundonrank(M).Thisconditioniscalledanindependenceassignment. Denition3.6. GivenagraphG=(V,E)andacoverX=fG1,...,GmgofG,wesay(G,X)hasanindependenceassignment[I;fI1,...,Img],ifthereisanindependentsetIofGandmaximalindependentsetIiofeachoftheGi's,suchthatIrestrictedtoGi,(denotedIji),iscontainedinIiandforanye2H(X),eismissingfromatmostoneoftheIi'swhosecorrespondingGicontainse.WhenXisclear,wealsosaythereisanindependenceassignmentforG.ThenextlemmashowstheexistenceofanindependenceassignmentforMaxwell-independentgraphs. Lemma3.3. IfMisMaxwell-independentandXisa2-thincoverofMbycomponentsofM,then(M,X)hasanindependenceassignment. Proof. (ofLemma 3.3 ).Infact,wecanconstructanindependenceassignmentifthe2-thincomponentgraphofMisageneralizedpartial9-tree.FromLemma 3.2 ( b ),weknowthatanysubgraphofthe2-thincomponentgraphCXofMisageneralizedpartial3-tree,whichisautomaticallyageneralizedpartial9-tree.LetM1,M2,...MnbethecomponentnodesofMlistedinreverseorderfromtheremovalorderinDenition 3.5 .Weuseinductiontoprovethatthereisalwaysanindependenceassignmentfor(M,X).IfXhasonlyonecomponent,itisclearthatwecanndanindependenceassignment.Supposethereisanindependenceassignment[Ik;Iki1ik]forasubgraphCkXofCXcontainingthecomponentnodesM1,M2,...,Mk.AfteraddingMk+1toformCk+1X,weneedtondIk+1,whichisamaximalindependentsetofk+1Si=1Mi,andIk+1ifor1ik+1suchthat[Ik+1;Ik+1i1ik+1]isanindependenceassignment.FirstwetakeIk+1i:=Ikifor1ikandletSbethesetofedgesofMk+1thataresharedbyothercomponents.SincejSj9,Sisindependentford=3,becausefor 46

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d=3,aminimum-sizegraphthatisnotindependentwillhaveatleast10edges.ThuswecanextendStoamaximalindependentsetIk+1k+1ofMk+1.NowletIk+1:=Ik[Ik+1k+1,then(1)Ik+1spansalledgesink+1Si=1Mi,and(2)everyedgeeinIk+1thatissharedbyatleasttwocomponentsinM1,M2,...Mk+1ismissinginatmost1oftheIk+1i'ssharinge,since(a)[Ik;Iki1ik]isanindependenceassignmentforCkXand(b)Ik+1k+1containsallsharededgesofMk+1.IfIk+1isalreadyindependent,wehaveourindependenceassignment.OtherwisewecanremoveaminimumnumberofedgesfromIk+1untilitisindependent. ThefollowingtheoremgivesanalternativecombinatorialupperboundonrankofrigiditymatroidofMaxwell-independentgraphs. Theorem3.2. LetMbeaMaxwell-independentgraphandX=fe1,...,ek,M1,M2,...,Mmgbea2-thincoverofMbycomponentsofM.ThenPmi=1rank(Mi))]TJ /F15 11.955 Tf 9.3 8.97 Td[(Pfu,vg2H(X)\E(M)(nfu,vg)]TJ /F4 11.955 Tf -482.62 -23.9 Td[(1)+krank(M). Proof. WhenXisa2-thincover,wecanapplyLemma 3.3 andobtainthat(M,X)hasanindependenceassignment.FirstweremovealledgecomponentsofMtoobtainanewgraphM0.NowtheexistenceofanindependenceassignmentdirectlyimpliesthatPmi=1rank(Mi))]TJ /F15 11.955 Tf -415.05 -14.94 Td[(Pfu,vg2H(X)\E(M)(nfu,vg)]TJ /F4 11.955 Tf 11.95 0 Td[(1)rank(M0).Nextweconsidertheedgecomponentse1,...,ek.Ifweaddthecontributionsofallofthemtobothsidesoftheinequality,thelefthandsidebecomesPmi=1rank(Mi))]TJ /F15 11.955 Tf -432.27 -14.94 Td[(Pfu,vg2H(X)\E(M)(nfu,vg)]TJ /F4 11.955 Tf 12.09 0 Td[(1)+k,andtherighthandsidebecomesjIM0j+k,whichisatleasttherankofM,sinceE(M)=E(M0)[fe1,...,ekg. NowwearereadytoproveTheorem 3.1 47

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Proof. (ofTheorem 3.1 )First,noticethatifMitselfisindependent,wearedone.Similarly,ifMisMaxwell-rigid,thenwehavejE(M)j=3jVj)]TJ /F4 11.955 Tf 18.07 0 Td[(6rank(G)=jIj,hencewearedone.LetIMwithjIMj=rank(M)beamaximalindependentsetofM.Withoutlossofgenerality,letIMI.LetA:=InIM.Thusspan(E(M))\A=;.Herespan(E(M))meansthelinearspanofthoserowsoftherigiditymatrixR3(G)correspondingtoE(M).ConsideracoverX=fe1,...,ek,M1,M2,...,MmgofMbythecompletecollectionofvertex-maximalcomponents,wheree1,...,ekareedgecomponentsandM1,M2,...,Mmarenon-trivialcomponents.Nextweshowthatforeachedgefu,vginA,thereexistsatleastonenon-trivialcomponentMisuchthatu2Miandv2Mi.SinceA\E(M)=;,e=fu,vg2AisnotanedgecomponentofM.HenceifuandvlieinsideanycomponentofM,thecomponentmustbenon-trivial.Sincefu,vg62M,M[fu,vgisnotMaxwell-independent,andhencethereissomeMaxwell-rigidsubgraphM0ofMwithu2M0andv2M0.SinceXisthecompletecollectionofvertex-maximalcomponents,weknowthereexistsatleastonenon-trivialcomponentMisuchthatu2Miandv2Mi.DenotebyAithesetofedgesofAbothofwhoseendpointsareinMi.Hence jAjmXi=1jAij(3)TakeH(X)andnfu,vgasdenedearlierinthesection.Weget jE(M)j=kXi=11+mXi=1jE(Mi)j)]TJ /F15 11.955 Tf 47.45 11.36 Td[(Xfu,vg2H(X)\E(M)(nfu,vg)]TJ /F4 11.955 Tf 11.95 0 Td[(1)=k+mXi=1jE(Mi)j)]TJ /F15 11.955 Tf 47.45 11.36 Td[(Xfu,vg2H(X)\E(M)(nfu,vg)]TJ /F4 11.955 Tf 11.96 0 Td[(1) (3) SinceeachMiisMaxwell-rigid,addinganye2AiintoMicausesthenumberofedgesinMitoexceed3jV(Mi)j)]TJ /F4 11.955 Tf 19.44 0 Td[(6andinturnindicatestheexistenceofatruedependence.However,Ai\span(Mi)=;,sincespan(E(M))\Ai=;.Itfollowsthat 48

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MiwasalreadydependentevenbeforeAiwasadded.I.e.,toobtainanindependentsetinMi,atleastjAijedgesmustberemovedfromMi.Sowehave jE(Mi)jrank(Mi)+jAij(3)Plugging( 5 )into( 5 ),wehave jE(M)jmXi=1rank(Mi))]TJ /F15 11.955 Tf 41.48 11.36 Td[(Xfu,vg2H(X)\E(M)(nfu,vg)]TJ /F4 11.955 Tf 11.95 0 Td[(1)+mXi=1jAij+k(3)FromProposition 3.1 ( a ),weknowthatthecoverXis2-thin.ThenwecanapplyTheorem 3.2 andobtainthefollowing: mXi=1rank(Mi))]TJ /F15 11.955 Tf 41.47 11.35 Td[(Xfu,vg2H(X)\E(M)(nfu,vg)]TJ /F4 11.955 Tf 11.96 0 Td[(1)+krank(M)=jIMj.(3)Then,using( 5 )and( 5 ),weobtainthatjE(M)jjIMj+mXi=1jAij(using( 5 )and( 5 ))jIMj+jAj(using( 5 ))=jIj,whichprovesTheorem 3.1 NotethattheproofofTheorem 3.1 usesacoverbythecompletecollectionofvertex-maximalcomponents.Thisnotonlyimplies2-thinnessofthecover,butalsostrong2-thinness.However,2-thinness(Proposition 3.1 ( a ))issufcientforprovingTheorem 3.1 .Strong2-thinnessisusedinSection 3.2 3.2AlternativeUpperBoundsUsingIECounts 3.2.1RelationtoKnownBoundsandConjecturesUsingIECountsDecompositionofgraphsintocoversisanaturalwayofapproachingacombinatorialcharacterizationof3-dimensionalrigidity.Sofar,theinclusion-exclusion(IE)count 49

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methodforcovershasbeenusedbymanyintheliterature(see[ 22 46 48 64 66 85 94 ]).Themostexploreddecompositionsarethe2-thincovers.WedenedtwotypesofrankIEcountsinDenition 3.2 ,withIErankbeingusedintheproofofTheorem 3.1 .OurTheorem 3.3 belowinSection 3.2.2 ,willshowthatforaspecic,notnecessarilyindependentcover,aslightlydifferentinclusion-exclusioncountisequaltoIErankcount,whichinturngivesarankupperboundforMaxwell-independentgraphs.BesidestheIErankcount,otherIEcountshavealsobeenexploredintheaforementionedliterature.In1983,Dressetal[ 27 104 ]conjecturedthattheminimumoftheIEfullcounttakenoverall2-thincoversisanupperboundontherankofthe3-dimensionalgenericrigiditymatroid.However,thisconjecturewasdisprovedforgeneralgraphsbyJacksonandJordanin[ 45 ].AlthoughDress'conjectureisfalse,theIEfullcountcanbeanupperboundoftherankifthecoverisspecial:itisshownin[ 48 ]thattheminimumoftheIEfullcounttakenoverallindependent2-thincoversisanupperboundontherank.Here,anindependent2-thincoverXisoneforwhichtheedgesetgivenbythepairsinthesharedpartH(X)isindependent.Itisalsoshownthattoachievetheupperbound,thecoversneednotbeindependent,butcanbeobtainedasiterated,orrecursiveversionofindependentcovers.WehavenoexampleswhereourboundinTheorem 3.3 isbetterthantheabovementionedboundfrom[ 48 ],whichwasconjecturedtobetightwhenrestrictedtonon-rigidgraphsandcoversofsizeatleast2.Henceanysuchexampleswouldbecounterexamplestotheirconjecture.However,ourformulaprovidesanalternativewayofcomputingarankupperboundusingnotnecessarilyindependentcovers.InSection 3.2.3 ,weusethesameIEfullcountoveranotherspecialcover,whichisaspe-cicnon-iterated,non-independentcover,toobtainrankboundsonMaxwell-dependent 50

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graphs.Again,wehavenoexampleswhereourboundisbetterthantheabovementionedboundin[ 48 ],whichwasconjecturedtobetight.Henceanysuchexampleswouldbecounterexamplestotheirconjecture.Ourboundgivesanalternativemethodusingaspecic,non-iterated,notnecessarilyindependentcoverby(proper)vertex-maximalcomponents.However,thecatchisthatthesecoversmaynotexistforgeneralgraphs. 3.2.2AlternativeUpperBoundsforMaxwell-IndependentGraphsInthissection,wegivealternativecombinatorialboundsontherankofthegenericrigiditymatroidofMaxwell-independentgraphsin3dimensions.NoticethatifMisaMaxwell-independentgraphwithacoverX=fe1,...,ek,M1,M2,...,Mmgbyvertex-maximalcomponents,thenH(X)=H(X)\E(M)andthusPmi=1rank(Mi))]TJ /F15 11.955 Tf -442.36 -14.94 Td[(Pfu,vg2H(X)(nfu,vg)]TJ /F4 11.955 Tf 11.95 0 Td[(1)+k=IErank(X)rank(M).However,whenagraphMisMaxwell-rigid,thereisasinglevertex-maximalcomponentnamelyMitself,sotheaboveboundisuninteresting.Inthiscase,weusethecoverofMbypropervertex-maximalcomponents: Denition3.7. GivengraphG=(V,E),aninducedsubgraphispropervertex-maximal,Maxwell-rigidifitisMaxwell-rigidandtheonlygraphthatproperlycontainsthissub-graphandisMaxwell-rigidisGitself.Sincethecollectionofpropervertex-maximalcomponentsmaynotbea2-thincoverevenforMaxwell-independentgraphs,Theorem 3.2 doesnotdirectlyapply.ThefollowingtheoremdealswithcasesthatarerelativelyminorvariationsofTheorem 3.2 Theorem3.3. LetMbeaMaxwell-independentgraphandX=fe1,...,ek,M1,M2,...,MmgbeacoverofMbypropervertex-maximalcomponents.Thenwehave: 1. IfXisstrong2-thin,thenPmi=1rank(Mi))]TJ /F15 11.955 Tf 12.62 8.96 Td[(Pfu,vg2H(X)(nfu,vg)]TJ /F4 11.955 Tf 13.07 0 Td[(1)+k=IErank(X)rank(M). 2. IfXis2-thinbutnotstrong2-thin,Xconsistsentirelyoftwonon-trivialcompo-nentsMiandMjinXs.t.M=Mi[Mjandhencerank(Mi)+rank(Mj))]TJ /F8 11.955 Tf -406.9 -14.45 Td[(rank(Mi\Mj)rank(M). 51

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3. Otherwise,thereexisttwonon-trivialcomponentsMiandMjinX,s.t.M=Mi[Mjandhencerank(Mi)+rank(Mj))]TJ /F8 11.955 Tf 12.62 0 Td[(rank(Mi\Mj)rank(M). Proof. 1. WhenXisstrong2-thin,weknowH(X)=H(X)\E(M)andthusPmi=1rank(Mi))]TJ /F15 11.955 Tf 12.62 8.97 Td[(Pfu,vg2H(X)(nfu,vg)]TJ /F4 11.955 Tf 12.72 0 Td[(1)+k=IErank(X).ThenitfollowsfromTheorem 3.2 thatIErank(X)rank(M). 2. WhenXis2-thinbutnotstrong2-thin,weknowthereexisttwopropervertex-maximalcomponentsMiandMj,s.t.Mi\Mjhastwoverticesbutnoedge.FromLemma 3.1 ( a ),weknowMi[MjisMaxwell-rigid.SinceMiandMjarebothpropervertex-maximal,weknowV(M)=V(Mi)[V(Mj).SinceMisMaxwell-independent,weknowE(M)=E(Mi)[E(Mj).Sincethecoveris2-thin,noothernon-trivialvertex-maximalcomponentcanexist.HenceMiandMjaretheonlytwonon-trivialcomponentsinXanditfollowsthatrank(Mi\Mj)=0andhencerank(Mi)+rank(Mj))]TJ /F1 11.955 Tf 9.3 0 Td[(rank(Mi\Mj)rank(M). 3. WhenXisnot2-thin,i.e.,thereexistMiandMjsuchthattheirintersectionhasatleast3vertices.FromLemma 3.1 ( b ),weknowtheunionofMiandMjisalsoMaxwell-rigid.SinceMiandMjarebothpropervertex-maximal,weknowV(M)=V(Mi)[V(Mj).SinceMisMaxwell-independent,weknowE(M)=E(Mi)[E(Mj).Itremainstoshowthatrank(Mi)+rank(Mj))]TJ /F1 11.955 Tf 9.3 0 Td[(rank(Mi\Mj)rank(Mi[Mj).Toshowthis,wecanstartfromamaximalindependentsetIofMi\Mj,andexpandittomaximalindependentsetsIiofMiandIjofMj.ItisclearthatIi[IjspansthegraphMi[Mj,andhencerank(Mi)+rank(Mj))]TJ /F1 11.955 Tf 9.3 0 Td[(rank(Mi\Mj)=jIi[Ijjrank(Mi[Mj). 3.2.3RemovingtheMaxwell-IndependenceConditionWenowgiverankboundsforMaxwell-dependentgraphsusingtheIEfullcount.Westartwiththefollowingsimplebutusefulpropertyofedge-sharing,Maxwell-rigidsubgraphs. Lemma3.4. GivengraphG=(V,E),letG1andG2betwosubgraphsofGs.t.G0=G1\G2consistsoftwoverticesu,vandanedgee=fu,vg. (a) IfG1isavertex-maximalcomponentofGandthereisaMaxwell-independentsubgraphM1ofG1s.t.jE(M1)j=3jV(M1)j)]TJ /F4 11.955 Tf 18.5 0 Td[(6ande62M1,theneverymaximalMaxwell-independentsubgraphofG2containse. (b) IfG1isapropervertex-maximalcomponentofGandthereisaMaxwell-independentsubgraphM1ofG1s.t.jE(M1)j=3jV(M1)j)]TJ /F4 11.955 Tf 20.27 0 Td[(6ande62M1, 52

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thenoneoffollowingholds:(1)V=V(G1)[V(G2),or(2)everymaximalMaxwell-independentsubgraphofG2containse. Proof. (a) SupposethereisaMaxwell-independentsubgraphM2ofG2suchthatfeg[E(M2)isMaxwell-dependent.ThentheremustbeasubgraphM02ofM2suchthatM02hasMaxwellcount6.ThenitfollowsfromLemma 3.1 ( a )thatM02[G1isalsoMaxwell-rigid,acontradictiontothevertex-maximalityofG1. (b) Statementfollowsfrom( a )andthepropervertex-maximalityofG1. Nextwegivetwosimilartheoremswithsimilarproofs.Thersttheorem,Theorem 3.4 ,givesarankboundforgraphsforwhichthecompletecollectionofvertex-maximalcomponentsformsa2-thincover.Thesecond,Theorem 3.5 ,concernspropervertex-maximalcomponents. Theorem3.4. ForagraphG=(V,E),ifthecompletecollectionX=fe1,...,ek,G1,G2,...,Gmgofvertex-maximalcomponentsformsa2-thincover,thenIEfull(X)isanupperboundonrank(G),i.e.,mXi=1(3jV(Gi)j)]TJ /F4 11.955 Tf 17.94 0 Td[(6))]TJ /F15 11.955 Tf 27.04 11.36 Td[(Xfu,vg2H(X)(nfu,vg)]TJ /F4 11.955 Tf 11.96 0 Td[(1)+krank(G). Proof. Werstconsiderthecasewheretherearenoedgecomponents.First,weshowthatthecoverXisstrong2-thin.Supposenot,thenthereexistsfu,vg2H(X)suchthatfu,vg=2E.SupposefurtherthatGiandGjbothcontainuandv.FromLemma 3.1 ( a ),weknowGi[GjisMaxwell-rigid,contradictingthefactthatGiandGjarevertex-maximal,Maxwell-rigid.HencethecoverXisstrong2-thinandIEfull(X)canberewrittenasmPi=1(3jV(Gi)j)]TJ /F4 11.955 Tf 17.93 0 Td[(6))]TJ /F15 11.955 Tf 35.31 8.97 Td[(Pfu,vg2H(X)\E(nfu,vg)]TJ /F4 11.955 Tf 11.96 0 Td[(1).Weneedthefollowingclaim(whichisalsousedforprovingTheorem 3.5 ). Claim3.4.1. ForagraphG=(V,E),ifthecompletecollectionX=fG1,G2,...,Gmgof(proper)vertex-maximalcomponentsformsastrong2-thincover,thenthereisa 53

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maximalMaxwell-independentsubgraphMofGs.t.IEfull(X)=jE(M)jandhenceIEfull(X)rank(G). Proof. WeshowtheclaimforthecasewhereXconsistsofvertex-maximalcomponents.However,alongtheway,wepointouttheslightdifferencesforthecasewhereXconsistsofpropervertex-maximalcomponents,makingtheclaimapplicablealsotoTheorem 3.5 .WerstconstructasubgraphMGwithjE(M)jequaltoIEfull(X)asfollows.For1im,denotebyNiamaximumsizedMaxwell-independentsubgraphofGi.ThenfromLemma 3.4 ( a ),weknowthatforanyedgee2H(X),thereisatmostoneNi,suchthatfeg[E(Ni)isMaxwell-dependent.(Note:fromLemma 3.4 ( b ),evenifXisacoverbycompletecollectionofpropervertex-maximalcomponents,whentherearenotwocomponentsGiandGjs.t.V=V(Gi)[V(Gj),itstillholdsthatforanyedgee2H(X),thereisatmostoneNi,suchthatfeg[E(Ni)isMaxwell-dependent.)Thus,edgesofcomponentGicanbedividedintofourparts: Pi1:thesetofedgeseinH(X)\E(Ni)thatarepresentineachE(Nj)forwhichGjcontainse; Pi2:thesetofedgeseinH(X)\E(Ni)forwhichthereisexactlyoneNjwheree2GjnNj,i.e.,feg[E(Nj)isMaxwell-dependent; Pi3:thesetofedgeseinH(X)nE(Ni),andpresentinallotherNj's,whereGjcontainse. Pi4:E(Gi)nH(X).LetPk=SiPik.NowweconstructMasfollows.First,letV(M):=V(G).ThenweconstructtheedgesetE(M)byremovingalledgesinP2andP3frommSi=1E(Ni).ThusNi=Mji[Pi2,whereMjidenotesMrestrictedtoGi.NownotethatjE(M)j=mPi=1(3jV(Gi)j)]TJ /F4 11.955 Tf 18.2 0 Td[(6))]TJ /F15 11.955 Tf 24.79 8.97 Td[(Pfu,vg2P1(nfu,vg)]TJ /F4 11.955 Tf 12.09 0 Td[(1))]TJ /F15 11.955 Tf 34.09 8.97 Td[(Pfu,vg2P2[P3(nfu,vg)]TJ /F4 11.955 Tf 9.3 0 Td[(1),whichisexactlyIEfull(X),sinceXisstrong2-thin.Inthefollowingweshowthat 54

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thisnumberisatleastrank(G)byshowingthatMisamaximalMaxwell-independentsubgraphofGandusingTheorem 3.1 (I) MisMaxwell-independent.Supposenot,thenwecanndaminimalsubgraphM0MthatisMaxwell-dependent.SinceMispickedinsuchawaythateveryMjiisMaxwell-independent,weknowM0cannotbeinsideanyGi.BecauseM0isminimal,weknowthereexistsM00M0that(1)containsallverticesofM0and(2)isMaxwell-independentwithMaxwellcount6.ThenM00isacomponentthatisnotcontainedinanyGi,sinceM0isnotinsideanyGi,andremovinganedgefromM0doesnotmakeitinsideanyGieither.ThatisacontradictiontothefactthatG1,...,Gmisthecompletecollectionofvertex-maximalcomponentsofG.(Note:thiscontradictionwouldholdevenifXisacoverbycompletecollectionofpropervertex-maximalcomponents.) (II) MisamaximalMaxwell-independentsubgraphofG.Inordertoshowthis,werstnoticethatforeverye2Pi2,everymaximalMaxwell-independentsubgraphN0iofGicontainse,whichfollowsfromthestatementsthat(1)thereexistsaGjs.tfeg[E(Nj)isMaxwell-dependentand(2)Lemma 3.4 ( a ).(Note:fromLemma 3.4 ( b ),evenifXisacoverbycompletecollectionofpropervertex-maximalcomponents,whentherearenotwocomponentsG1andG2s.t.V=V(G1)[V(G2),itstillholdsthatforeverye2Pi2,everymaximalMaxwell-independentsubgraphofGicontainse.)Supposethereisanedgee2EnE(M)suchthatE(M)[fegisMaxwell-independent.Then(E(M)[feg)ji(whichdenotesE(M)[fegrestrictedtoGi)isalsoMaxwell-independent.Sincee2Giforsomei,weknowe2Pi2,Pi3orPi4.InfacteveryedgePj2forsomejisalsoinPi3forsomei,withoutlossofgenerality,wechooseacomponentisuchthate2Pi3orPi4.Noticethatthereisanextensionof(E(M)[feg)jiintoamaximalMaxwell-independentsubgraphM0iofGi,whichmustcontainalledgesinPi2asshowninthepreviousparagraph,i.e.,E(M0i)contains(E(M)[feg)ji[Pi2.SinceMji[Pi2=Ni,weknowE(M0i)hassizelargerthanE(Ni),whichisacontradictiontothefactthatNiisamaximumsizedMaxwell-independentsubgraphofGi.HenceMismaximalMaxwell-independent.ThusweknowMisamaximalMaxwell-independentsetofG.FromTheorem 3.1 ,weknowjE(M)jrank(G).Asnoticedbefore,theIEfullcountofthecoverXisequaltojE(M)j,hencewehavemPi=1(3jV(Gi)j)]TJ /F4 11.955 Tf 17.93 0 Td[(6))]TJ /F15 11.955 Tf 30.04 8.97 Td[(Pfu,vg2H(X)(nfu,vg)]TJ /F4 11.955 Tf 11.96 0 Td[(1)rank(G). ReturningtotheproofofTheorem 3.4 ,werstnoticethatClaim 3.4.1 completestheproof,whentherearenoedgecomponentsinthecoverX. 55

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Withedgecomponentsinthecover,noticethateachedgecomponentcontributes1tothelefthandsidebutcontributesatmost1totherighthandside.Thustheinequalitystillholds. ThenexttheoremextendstheboundinTheorem 3.4 tocoversbypropervertex-maximalcomponents. Theorem3.5. ForagraphG=(V,E),ifthecompletecollectionX=fe1,...,ek,G1,G2,...,Gmgofpropervertex-maximalcomponentsformsa2-thincover,thentheIEfullcountofthecoverXisanupperboundonrank(G),i.e.,mXi=1(3jV(Gi)j)]TJ /F4 11.955 Tf 17.94 0 Td[(6))]TJ /F15 11.955 Tf 27.04 11.36 Td[(Xfu,vg2H(X)(nfu,vg)]TJ /F4 11.955 Tf 11.96 0 Td[(1)+krank(G). Proof. WhenGisnotMaxwell-rigid,theproofisthesameasinTheorem 3.4 .WhenGisMaxwell-rigid,werstshowthetheoremforthecasewheretherearenoedgecomponents.Therearetwofurthercases: Case1. ThereexisttwocomponentsGiandGjs.t.V(G)=V(Gi)[V(Gj).Inthiscase,allothernon-trivialcomponentsinthecovercanonlybeK3orK4.Foreveryedgeeinthosecomponents,weknow(1)ife2Gi[Gj,thenecontributesto0toboththelefthandsideandrighthandsideoftheinequality;and(2)ife62Gi[Gj,thenecontributesto1tothelefthandside,and0or1totherighthandsideoftheinequality.ThusifwecanshowthatIEfullcountonGi[GjisanupperboundontherankofGi[Gj,thenthetheoremholds.NotethatIEfullcountonGi[Gjisequalto3jVj)]TJ /F4 11.955 Tf 16.18 0 Td[(7,andfromtheaxiomC5ofabstractrigiditymatroid(see[ 35 ]),weknowGi[Gjisnotrigidandthusrank(Gi[Gj)isatmost3jVj)]TJ /F4 11.955 Tf 18.29 0 Td[(7.HenceIEfullcountonGi[GjisanupperboundontherankofGi[Gj. Case2. ForanytwocomponentsGiandGj,wehaveV(G)6=V(Gi)[V(Gj).Inthiscase,weknowthecoverisstrong2-thin,sinceotherwise,thereexisttwo 56

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componentsG1andG2whoseintersectionisapairofverticeswithoutanedge.FromLemma 3.1 ( a ),weknowG1[G2isMaxwell-rigid.SincebothG1andG2arepropervertex-maximalcomponents,weknowV(G)=V(G1)[V(G2),acontradiction.Next,weapplyClaim 3.4.1 ofTheorem 3.4 tocompletetheproofofTheorem 3.5 wheretherearenoedgecomponents.Nowwecanconsiderthecasewithedgecomponentsinthecoverandnoticethateachedgecomponentcontributes1tothelefthandsidebutcontributesatmost1totherighthandside.Thustheinequalitystillholds. Remark:(I)Infact,inTheorems 3.4 and 3.5 ,whenGisnotMaxwell-rigidorGhasatleast3non-trivialcomponentsinthestrong2-thincoverX,itturnsoutthatwedonotneedTheorem 3.1 toshowthattheIEfullcountofthecoverXisanupperboundonrank(G).ThisisbecausewecanshowthatMconstructedinTheorem 3.4 isinfactamaximum-sizeMaxwell-independentsubgraphofG.OtherwisewecanndamaximalMaxwell-independentsubgraphM0suchthatjE(M0)j>jE(M)j.ThentheremustbesomeisuchthatjE(M0)jij>jE(M)jij.WeknowPi2isMaxwell-independentineveryMaxwell-independentsetofCiandsinceM0jiisMaxwell-independent,henceE(M0)ji[Pi2isalsoMaxwell-independentwithsizegreaterthanE(M)ji[Pi2,whichisE(Ni).ThatisacontradictiontothefactthatNiisamaximumsizedMaxwell-independentsubgraphofCi.(II)WecanusethemaximumsizedMaxwell-independentsubgraphMconstructedinTheorems 3.4 and 3.5 totestMaxwell-rigidity. 3.3OpenProblems 3.3.1ExtendingRankboundtoHigherDimensionsThedenitionofmaximalMaxwell-independentsetextendstoalldimensions,leadingtothefollowingconjecture. Conjecture3.1. Foranydimensiond,thesizeofanymaximalMaxwell-independentsetgivesanupperboundontherankofthegenericrigiditymatroidofagraphG. 57

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Moreover,thedenitionof2-thincomponentgraphscanalsobeextendedtoddimensions. Denition3.8. GivenG=(V,E),letX=fG1,G2,...,Gmgbea(d)]TJ /F4 11.955 Tf 12.02 0 Td[(1)-thincoverofG,i.e.,jV(Gi)\V(Gj)jd)]TJ /F4 11.955 Tf 12.02 0 Td[(1forall1i
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However,whileidentiedbody-hingeframeworksaccountforseveralcomponentnodessharinganedge(aswehavehere),generalizedbody-hingestructuresmayadditionallyhavesharededgesthathavecommonvertices,hencethegeneric,identiedbody-hingecountsmaynotapply. 3.3.2StrongerVersionsofIndependenceEvenforMaxwell-independentgraphs,therankboundsofourTheorem 3.1 canbearbitrarilybad.Evenasimpleexampleof2bananaswithoutthehingeedgehasasinglemaximalMaxwell-independentsetofsize18(whichistheboundgivenbyallofourtheorems),butitsrankisonly17.Anotherexampleistheso-calledn-banana:itisformedbyjoiningnK5'sonanedgeandthenremovingthatsharededge.Inthen-banana,thewholegraphisMaxwell-independent,soitselfistheuniquemaximalMaxwell-independentset.ThismaximalMaxwell-independentsetexceedstherankofthe3-dimensionalgenericrigiditymatroidofn-bananabyn)]TJ /F4 11.955 Tf 11.96 0 Td[(1.Theorem 3.3 givealternativeupperboundsforMaxwell-independentgraphs.(Infact,Theorem 3.3 leadstoarecursivemethodofobtainingarankboundbyrecursivelydecomposingthegraphintopropervertex-maximalcomponents.Asoneconsequence,itgivesanalternative,muchsimplerproofofcorrectnessforanexistingalgorithmcalledtheFrontierVertexalgorithm(rstversion)thatisbasedonthisdecompositionideaaswellasotherideasinthischaptersuchasthecomponentgraph[ 85 ].)AnaturalopenproblemistoimprovetheboundinTheorem 3.1 directlybyconsideringothernotionsofindependencethatarestrongerthanMaxwell-independence.(Algorithmsin[ 85 94 ]suggestandusestrongernotionsthanMaxwell-independence,butthealgorithmsusuallyusesomeversionofaninclusion-exclusionformula.TheydonotprovideexplicitmaximalsetsofedgessatisfyingthestrongernotionsofMaxwell-independence.Neitherdotheyprovethatallsuchsetsprovidegoodbounds.) 59

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3.3.3BoundsforMaxwell-DependentGraphsUsing2-ThinCoversWhileTheorem 3.3 givesastrongrankboundforMaxwell-independentgraphs,Theorem 3.4 andTheorem 3.5 givemuchweakerboundsforMaxwell-dependentgraphsbecauseacollectionof(proper)vertex-maximal,Maxwell-rigidsubgraphsmaybefarfrombeinga2-thincover.Forexample,inFigure 3-2 wehave3K5'sandtheneighboringK5'sshareanedgewitheachother.Therearetwovertex-maximal,Maxwell-rigidsubgraphs,eachofwhichconsistsof2K5'swithasharededge. Figure3-2. Acoverofvertex-maximalcomponentsthatisnot2-thin.ThecirclesareK5'sandthetwolargerellipsesarevertex-maximal,Maxwell-rigidsubgraphsthatformthecover. Whilemanyother2-thincoversexist,thecompletenessaswellas(proper)vertex-maximalityareimportantingredientsintheproofsofthesetheorems.Onepossibilityistouse2-thincoversthatareasubcollectionof(proper)vertex-maximal,Maxwell-rigidsubgraphs.Anotheristousecollectionsofnotnecessarilyvertex-maximal,butMaxwell-rigidsubgraphsinwhichnopropersubcollectionof2ormoresubgraphshasaMaxwell-rigidunion.AnothernotionthatcanbeusedinvolvesthefollowingdenitionofstrongMaxwell-rigidity: Denition3.9. AgraphG=(V,E)isstrongMaxwell-rigidifforallmaximalMaxwell-independentedgesetsE0E,wehavejE0j=3jV(E0)j)]TJ /F4 11.955 Tf 17.93 0 Td[(6.ItistemptingtousetheapproachinTheorem 3.4 toshowthattheIEfullcountforacoverbyvertex-maximal,strongMaxwell-rigidsubgraphsisanewupperboundontherank.Weconjecturethe2-thinnessofthecover,whichisacrucialpropertyexploredinprovingTheorem 3.4 60

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Conjecture3.4. Anycoverofagraphbyacollectionofvertex-maximal,strongMaxwell-rigidsubgraphsisa2-thincover.However,theideaintheproofofTheorem 3.4 willnotworkbecausethesetM,constructedintheproofofTheorem 3.4 thatisofsizeequaltotheIEfullcount,cannowbeofsmallersizethananymaximalMaxwell-independentsetofGasintheexampleofFigure 3-3 .Example(Figure 3-3 ):thereareveringsofK5's,whereeachringconsistsof7K5's.Inthegraph,everyK5isavertex-maximalstrongMaxwell-rigidsubgraph,andtheIEfullcountforthecoverXis(35)]TJ /F4 11.955 Tf 10.82 0 Td[(6)(65+1))]TJ /F4 11.955 Tf 10.81 0 Td[(55)]TJ /F4 11.955 Tf 10.82 0 Td[(10=244.Herethe(65+1)isthenumberofK5'sand55+10isthetotalnumberofsharededges.Butifwetake9edgesineveryK5exceptTsuchthatthemissingedgesarenotshared,thenweobtainasetM0thatisMaxwell-dependent.FromM0wedroponeedgeeofTandaddonemissingedgeftotheK5thatsharesewithT.ThenwegetasetM00thatisaminimum-sizemaximalMaxwell-independentsetofG.ThesizeofM00is(69)]TJ /F4 11.955 Tf 12.21 0 Td[(5)5=245,where69)]TJ /F4 11.955 Tf 12.86 0 Td[(5isthenumberofedgesineachring,notcountingtheedgesinTthatareunsharedinthatring.HenceintheFigure 3-3 example,theIEfullcountislessthanthesizeofanymaximalMaxwell-independentset,sothelattercannotbeusedasabridginginequalityasinTheorem 3.4 .However,theIEfullcountdoesseemtogiveadirectupperboundontherank(itisequaltotherank)henceadifferentproofideamightyieldtherequiredboundonrank. 3.3.4AlgorithmsforVariousMaximalMaxwell-IndependentSetsSofartheemphasishasbeentondgoodupperboundsonrankandTheorem 3.1 showsthattheminimum-sizemaximalMaxwell-independentsetofagraphGisatleastrank(G).Anaturalopenproblemistogiveanalgorithmthatconstructsaminimum-size,maximalMaxwell-independentsetofanarbitrarygraph. 61

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Figure3-3. AcounterexampletoshowthatIEfullcountofcoverXbyvertex-maximal,strongMaxwell-rigidsubgraphsturnsouttobesmallerthanthesizeofanymaximalMaxwell-independentset.StartwithaK5,denotedT.Eachof5pairsofedgesofTisextendedintoaringof7K5's,whereeachringisformedbyclosingachainofK5'swheretheneighboringK5'sshareanedge(bold)witheachother.Ineachofthe5rings,everyK5sharesanedgewitheachofitstwoneighboringK5'sandthesetwoedgesarenon-adjacent.Notethatinthegure,onlyoneoftheveringsisshown. NotethatMaxwell-rigidityrequiresthemaximumMaxwell-independentsettobeofsize3jVj)]TJ /F4 11.955 Tf 14.95 0 Td[(6.AlthoughthemaximumMaxwell-independentsetistriviallyasbigastherank(andisnotdirectlyrelevanttondinggoodboundsonrank),coversbyMaxwell-rigidcomponentshaveplayedaroleinsomeofthetheoremsabove(Theorems 3.3 3.4 3.5 )thatgiveusefulboundsonrank.RecallthatHendrickson[ 36 ]givesanalgorithmtotest2-dimensionalMaxwell-rigiditybyndingamaximalMaxwell-independentsetthatisautomaticallymaximumford=2.WhileanextensionofHendrickson[ 36 ]to3dimensionsgivenin[ 64 ]ndssomemaximalMaxwell-independentset,itisnotguaranteedtobemaximum(orminimum).ThusanotherquestionofinterestiswhethermaximumMaxwell-independentsetscanbecharacterizedinsomenaturalway. 62

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CHAPTER4NUCLEATION-FREEGRAPHSWITHIMPLIEDNON-EDGESInthischapter,wegivegeneralconstructionschemes(assketchedinSection 2.2.2 )fornucleation-free,independentgraphswithimpliednon-edges.InSection 4.1 ,wegiveawarm-upexampleofgraphswithnonucleationandgivetwodifferentprooftechniquestoshowtheexistenceofimpliednon-edges.InSection 4.2 ,wegivestraightforwardextensionsofourwarm-upconstructionandprooftechniques.InSection 4.3 ,wegiveasignicantlymorepowerfulinductiveconstructionscheme,roof-addition,fornucleation-free(independent)graphswithimpliednon-edges.TheindependenceisshowninTheorem 4.8 andisastand-aloneresultforinductiveconstructionofindependentgraphs.Theothertwoproperties(nucleation-freeandpresenceofimpliednon-edges)areshowninTheorem 4.9 .Inparticular,inObservation 4.4 welistandshowthepropertiesofseveralexamplestartinggraphsfortheroof-additionscheme.InTheorem 4.10 howwecaninductivelyusestartinggraphstogeneratenucleation-free,independentgraphswithimpliednon-edges.InSection 4.4 (Theorem 4.11 ),weshowhowtoobtainnucleation-free(minimally)dependentgraphs.ThenweexhibitafamilyofgraphsinCorollary 4.11.1 toshowthatthealgorithmof[ 94 ]i.e.,characterizationofmodule-rigidityisdistinctfromtruerigidity.InSection 4.5 wegiveopenproblems. 4.1Warm-UpExample:RingsofRoofsInthissection,wegiveawarm-upfamilyofexamplestomotivateconstructionschemesandprooftechniquesforProblem 2 .Therstprooftechnique,whichwecallex-sign,istodirectlyshowtheexistenceofimpliednon-edges;thesecondprooftechnique,whichwecallrank-sandwichtechnique,needstheindependenceofthegraph.Therearetwoingredientsintherank-sandwichprooftechniquethatgeneratestand-aloneresults:(1)showtheindependenceofthegraph(2)showtherankupper 63

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boundofthegraphwithimpliednon-edgesadded(whichinturncanbeshownusingtwodifferentmethods).Asnotedearlier,[ 103 ]claimstheexistenceofimpliednon-edgesinthisfamilyessentiallywithoutproof.Thebasicbuildingblockforthiswarm-upfamilyofnucleation-freegraphsisaroof(calledabutteryin[ 103 ]).Roof.AroofisagraphobtainedfromK5,thecompletegraphonvevertices,bydeletingtwonon-adjacentedges.Thesearecalledthehingenon-edgesofaroof.A3DrealizationofaroofisillustratedinFig. 4-1 .Intheterminologyof[ 96 ],thisisasingle-vertexorigamiovera4-gon.Example(Ringsofroofs):Aringofroofsisconstructedasfollows.Exactlytworoofsshareahingenon-edge,asinFig. 4-2 .Eachroofthenshareshingenon-edgeswithatmosttwoothers.Suchachainofsevenormoreroofsisclosedbackintoaring,asdepictedinFig. 5-1 .WewilldenotebyRkaringofkroofs.Rkisrigidfork6.ItiseasytoseethatthereisnonucleusinRkfork7.Wewillshowthatimpliednon-edgesexistinRkforanyk7,usingtwodifferentprooftechniques. Figure4-1. Aroof:aK5(completegraphon5vertices)withtwonon-adjacentedgesmissing.Ontheleftwegivethegeometricstructureoftheroofinspace.Ontherightwegiveaschematicofaroof:thebar-and-jointstructureisnotshownforclarity,butthepositionshingesareschematicallydepictedwithai,bi,ak,bk.Thevertexcimayormaynotbeshowninaschematicuseoftherooflater. 64

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Figure4-2. Connectingtworoofs.Ontheleftistherealgeometricconnection,whileontherightisaschematicshowinghowthetworoofsareconnectedviaahinge Figure4-3. Aringof7roofs:connecting7roofsinthemannershowninFig. 4-2 andwecanseeeachroofcanbethusconnectedtoatmosttwoothers.Suchachainofsevenroofsisclosedbackintotheringshownhere.Ontheleftisthegeometricstructureoftheringandontherightistheschematicofthering. 4.1.1Flex-SignTechniqueforExistenceofImpliedNon-EdgesinRingsofRoofsInthissection,wegiveourrstprooftechnique,calledex-signtechnique,fortheexistenceofimpliednon-edgesinringsofroofs.Thisprooftechniquereliesontheinnitesimalpropertiesofsingle-vertexorigamisfrom[ 96 ],togetherwithexpansion/contractionpropertiesofconvexpolygons[ 18 ]andpointedpseudo-triangulations[ 95 ],appliedtothesimplestpossiblecaseofa4-gon.TheseresultsshowthattheroofrealizationsintheringframeworkRk(p)fromFig. 4-4 havetheexpansion/contractionpropertiesstatedinthecaption.Weshowthatthehingenon-edgesareimpliedin 65

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Figure4-4. All3typesofroofscouldoccurinthewarm-upconstruction,andtheprooftechniqueinSection 4.1.2 canbeappliedtoalltypes.However,thegureontheleftisaroofwhosebaseiscrossingintheshapeofabuttery.TheprooftechniqueinSection 4.1.1 doesnotapplytothistypeofroof.Thegureinthemiddleisaconvexorexpansive-contractiveroof,i.e.,ifoneofitshingenon-edgeshasacontractivemotion,thentheotherisforcedtohaveanexpansivemotion.Thegureontherightisapointedpseudo-triangularorexpansive-expansiveroof.Itstwohingenon-edgesmoveineitherbothexpansiveorbothcontractivefashion. anyringframeworkRk(p)ofk7roofsconsistingof1convexandk)]TJ /F4 11.955 Tf 12.89 0 Td[(1pointedpseudo-triangularroofs.Thisexpansion/contractionpropertyissimilartothesqueezein[ 103 ](seeAPPENDIX),i.e.,anon-trivialexormotionalongthehingenon-edge. Lemma4.1. ForallringframeworksRk(p)ofk)]TJ /F4 11.955 Tf 12 0 Td[(1,pointedpseudo-triangularroofsandoneconvexroof,thehingenon-edgesareimplied. Proof. Assumethatthereexistsaninnitesimalmotionalongonehingenon-edgefa,bg,andwithoutlossofgenerality,assumethatmotionisexpansive.Thentheincrease/decreasepatternsofthetwohingenon-edgesofpointedpseudo-triangularroofs,i.e.,expansive-expansiveandtheconvexroof,i.e.,expansive-contractive,whenfollowedalongtheringbacktothestartinghingenon-edge,implythatthemotionoffa,bgiscontractive,acontradiction. Nextweshowthatinfact,thehingenon-edgesareimpliedgenerically. Lemma4.2. TherearegenericframeworksRk(p)asinLemma 4.1 Proof. AnyframeworkRk(p)wheretherstroofisstrictlyconvexandtheremainderarestrictlypointedpseudo-triangularasinLemma 4.1 ,canbeviewedasapointinR33k. 66

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ThereisanopenneighborhoodaroundRk(p)consistingofframeworksthatcontinuetohavethesameconvex/pointedpseudo-triangularproperty.Thisshowsthatthehingenon-edgesareimpliedgenerically. CombiningLemmas 4.1 and 4.2 ,itfollowsthatthehingenon-edgesareimpliedintheringgraphRk. 4.1.2Rank-SandwichTechniquefortheExistenceofImpliedNon-EdgesinRingofRoofsInthissection,wegiveoursecondprooftechnique,calledrank-sandwichtechnique,toshowtheexistenceofimpliednon-edgesinringofroofsRk.Asmentionedearlier,thisprooftechniquehastwoingredientsthatareofinterestasstand-alonetechniques: showingtheindependenceofthegraphG(i.e.,numberofedgesistherank); showingthatasimplerankupperboundafteraddingpotentialimpliednon-edgesFasedgestoGisequaltothenumberofedgesinG.Togetherthisprovesthatthenon-edgesinFareimplied.Morespecically,weshowinSection 4.1.2.1 thatringsofroofsRkareindependent.ThenwegivetwodifferentargumentstoshowtherankupperboundonaringofroofsRkwithhingenon-edgesadded.Thosetwoargumentsare2-thincoverargumentinSection 4.1.2.2 andbody-hingeargumentinSection 4.1.2.3 4.1.2.1IndependenceofringofroofsWerstconsideraringoftetrahedra,i.e.,ringofK4'swhereneighboringK4'sshareanedge. Observation4.1. Aringofatleastsixtetrahedraisindependent.AHenneberg-IIconstructiononagraphGinR3istorstchooseavertexsetW=fw1,w2,w3,w4gof4verticesofGsuchthatthereisatleast1edgeinducedbyW,deleting1edgebetweentheverticesofW,andthenaddinganewvertexvandfouredges(v,w1),(v,w2),(v,w3),(v,w4).From[ 105 ],weknowthefollowing: 67

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Theorem4.1([ 105 ]). Henneberg-IIconstructionspreserveindependenceofagraphin3D.CombiningObservation 4.1 andTheorem 4.1 ,weobtainthefollowing: Theorem4.2. AringofroofsRkisindependentfork6. 4.1.2.2Rankupperboundusing2-thincoverargumentHere,wegiveourrstargumentoftherankupperboundofringsofroofsRkwithimpliednon-edgesadded.First,weneedthefollowingconcepts.ForanygraphG,independentedgesetsofGdeneamatroidofGinRd,whichiscalledthegenericrigiditymatroidofG.TherankofGinRdistherankofitsgenericrigiditymatroidinRd.Wehavethefollowinglemma. Lemma4.3. Therankofthe3DrigiditymatroidofaringofroofsRkdoesnotchangeifweaddallhingenon-edges.Toprovethis,weneedtointroduceafewmoreconcepts.AcoverofagraphG=(V,E)isacollectionXofpairwiseincomparablesubsetsofV,eachofsizeatleasttwo,suchthatSX2XE(X)=E,whereE(X)istheedgesetinducedbyX.AcoverX=fX1,X2,...,XngofGis2-thinifjXi\Xjj2forall1i
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whereG?[Xi]denotesthesubgraphofG?inducedbyXi.Remark:Both2-thincoversandvariantsoftherighthandsideofEquation( 4 )haveappearedinothercontexts,forexampleinthecontextoftheDress'3Drigidityconjectures[ 26 47 99 ],includingonecounterexample.Theyalsoappearusingdifferentterminology,forexample,inthecontextofalgorithmsforgeometricconstraintdecompositionbasedonthenotionofmodule-rigiditymentionedearlier[ 84 94 ],aswellasalgorithmsforisolatingandefcientlysolvingtheso-calledwell-formedsystemincidenceconstraintsbetweenstandardcollectionsofrigidbodies[ 86 91 ]. Proof. (ofLemma 4.3 )Afteraddinghingenon-edgesintoaringRkofkroofs,wewillgetaringCkofkK5's.DenotethoseK5'sbyfC1,C2,...,CkgandletX=fV(C1),V(C2),...,V(Ck)gbeacoverofCk.NotethatXisa2-thincover.Sincealledgesinthesharedpart(V,S(X))aredisjoint,(V,S(X))mustbeindependent.ApplyingTheorem 4.3 ,wehave:rank(Ck)nXirank(Ck[V(Ci)]))]TJ /F15 11.955 Tf 25.89 11.36 Td[(Xfa,bg2S(X)(d(a,b))]TJ /F4 11.955 Tf 11.95 0 Td[(1)=9k)]TJ /F3 11.955 Tf 11.96 0 Td[(k=8k.ByTheorem 4.2 ,weknowaringRkofkroofsisindependent.ThustherankRkisequalitsnumberofedges,whichisexactly8k.HenceafteraddingallhingestoRk,therankdoesnotchange. TheproofofLemma 4.3 showshowthe2-thincoverargumentisusedtocompletethesecondingredientneededtoshowtheexistenceofimpliednon-edges:therankupperboundforRkafterthehingenon-edgesareaddedisequaltothenumberofedgesinRk,which,bytheindependenceofRkshowninTheorem 4.2 ,isequaltotherankofRk.Generally,ifagraphGsatisesthefollowingconditions: thereisa2-thincoverX=fX1,X2,...,XngofGsuchthatsharedpart(V,S(X))isindependent; rank(G)=PXi2Xrank(G?[Xi]))]TJ /F15 11.955 Tf 11.95 8.96 Td[(Pfa,bg2S(X)(d(a,b))]TJ /F4 11.955 Tf 11.96 0 Td[(1),whereG?=G[S(X); 69

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thenthenon-edgesinS(X)areimplied.ThesewilllaterbeusedasstartingrequirementsongraphsforinductiveconstructionsinScheme 4.4 later. 4.1.2.3Rankupperboundusingbody-hingeargumentInthissection,wegiveasecondargumentoftherankupperboundofringsofroofsRkwithimpliednon-edgesadded.Forcompleteness,werstobservethefollowing: Observation4.2. LettheringframeworkRk(p)begeneric,thenforalli,therigiditymatricesofeachofthebananaframeworksBi(pi)areindependent,andthustheBi(pi)'sarerigid,wherepiistherestrictionofptotheverticesintheithroofRi.TogetherwiththefollowingresultbyTay[ 100 ]andWhiteandWhiteley[ 109 ]onbody-hingestructures,wecancompletetheproof. Theorem4.4. If8ik,theithbananaBi(pi)isrigid,thentheframeworkBk(p)isequivalenttoabody-hingeframeworkandisguaranteedtohaveatleastk)]TJ /F4 11.955 Tf 13.12 0 Td[(6independentinnitesimalmotions.Observation 4.2 andTheorem 4.4 showthattherankupperboundforaringwithhingenon-edgesaddedisequaltothenumberofedgesinthering,therebyshowingthatthehingenon-edgesareimplied. 4.2NaturalExtensionsofWarm-UpExampleInthissection,weextendthewarm-upexampletogivegeneralconstructionsofnucleation-freegraphswithimpliednon-edges.Wewilllistsomeconstructionschemeswhosecorrectnessfollowsfromtheex-signortherank-sandwichprooftechnique.First,wecanextendthebuildingblocktoothergraphsthatsatisfytheconditionsinLemma 4.1 andobtainthefollowingconstructionscheme: Scheme4.1(Flex-signRing). Inputgraphs:G1,G2,...,GnOutputgraph:aringgraphconsistingofG1,G2,...,Gnsuchthat(1)neighboringsubgraphsGiandGi+1shareahingenon-edgefai,bigwithGnandG1sharingfan,bng(2)eachhingenon-edgeissharedbyexactly2graphsGiandGi+1orG1andGn. 70

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Theorem4.5. IftheinputgraphsG1,G2,...,GnintheFlex-signRingscheme(Scheme 4.1 )havethefollowingproperty:(1)eachoftheGi'sisnucleation-freeand(2)oneoftheGi'scanberealizedasexpansive-contractivestructureandtheremainingGi'scanberealizedasexpansive-expansivestructure,thentheoutputgraphisanucleation-freegraphwithimpliednon-edges.TheproofforTheorem 4.5 followsdirectlyfromtheex-signprooftechniqueinSection 4.1.1 .RingsofroofsinSection 4.1 areexamplesofTheorem 4.5 .However,theseriousdisadvantageoftheex-signtechniqueisthatthereisnootherknownexample.Second,wecanapplyHenneberg-IIconstructionsonexistinggraphstoobtainthefollowingscheme: Scheme4.2(HennebergExtenderRing). Inputgraph:AringgraphHconsistingofG1,G2,...,Gnsuchthat(1)neighboringsubgraphsGiandGi+1shareahinge(ai,bi)withGnandG1sharing(an,bn)(2)eachhingeedgeissharedbyexactly2graphsGiandGi+1orG1andGn.Outputgraph:AringgraphGbyapplyingHenneberg-IIconstructionsonHasfollows:forGi,addavertexviandfouredges(vi,ai),(vi,bi),(vi,ai+1)and(vi,bi+1).Thenremovehingeedges(ai,bi). Theorem4.6. FortheHennebergExtenderRingscheme,ifthefollowingconditionshold:(1)theinputringgraphinindependent;(2)eachGiisrigid;and(3)eachGiisnucleation-freeafterremovingitstwohingeedges(ai,bi)and(ai+1,bi+1),thenScheme 4.2 outputsanucleation-free,independentgraphwithimpliednon-edges. Proof. Theproofissimpleandfollowstherank-sandwichprooftechnique:(1)Gisnucleation-free:since(a)afterremovingallhingeedges,Gi'sarenucleation-free,and(b)ifthereisanucleation,itmustincludeoneofthevi's,buteachviisincidentonlytoai,bi,ai+1andbi+1,whichisnotpartofanynucleation,sincetheveverticesdonotformatetrahedron;(2)Henneberg-IIconstructionalwayskeepsindependenceofthe 71

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graph,soGisindependent;(3)athin-coverargumentorabody-hingeargumentasinSection 4.1 caneasilyshowthe(ai,bi)'sareimplied. ExampleofHennebergextenderringscheme:Theringofroofsisagainanexample.Asanotherexample,wecanusemodiedoctahedralgraph,whichisaK6with3edgesmissing,asthebuildingblocksofthering.Fig. 4-5 showsonebuildingblockoftheoutputoftheHennebergextenderringschemefromaringofroofs.Notethatthereareseveralnon-isomorphicK6)]TJ /F4 11.955 Tf 12.78 0 Td[(3'sandonlytheoneinFig. 4-5 canbeobtainedusingHennebergextenderringschemefromaringofroofs. Figure4-5. AmodiedoctahedralgraphobtainedfromtheHennebergextenderringschemefromaringofroofs.Thedashedlinesarethetwohingenon-edges. Nextwedescribeanothercollectionsofstandardschemes,whichuse1)]TJ /F1 11.955 Tf 9.3 0 Td[(,2)]TJ /F1 11.955 Tf 12.62 0 Td[(and3)]TJ /F1 11.955 Tf 9.29 0 Td[(sumsandstandardinductiveconstructionstobuildontheexistingnucleation-freegraphswithimpliednon-edges. 72

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Denition4.1(k-sum). LetG1andG2betwographsthateachcontainsacompletegraphonkvertices,Kk,asapropersubgraph.ForanymatchingoftheverticesofthetwoKksbyidentifyingthematchedpairs,wecangetanewgraphG3.Wecallthisprocedureak-sumofG1andG2[ 8 89 ]. Denition4.2(Henneberg-Iconstruction). AHenneberg-IconstructiononagraphGinR3istorstchooseavertexsetW=fw1,w2,w3gof3verticesofGandthenaddanewvertexvandthreeedges(v,w1),(v,w2),(v,w3).From[ 105 ],weknowthatHenneberg-IconstructioninR3preservesindependence. Denition4.3(vertexsplit). GivenagraphGandavertexu,incidenttoverticesw1,w2,...,wn,thenavertexsplitofuoniedgesisanewgraphobtainedby(1)addinganewvertexv,(2)choosekedges(u,w1),...,(u,wk)incidenttouandremovethem,thenconnectvtow1,w2,...,wk,and(3)addanewedge(u,v)andiedgesfromvtoineighborsofu.From[ 110 ],weknowthatinR3,vertexspliton0,1,or2edgespreservesindependenceofthegraph.Wecanusetheaboveconceptstobuildonexistinggraphstoobtainanotherschemeasfollows: Scheme4.3(Standard-scheme). Inputgraphs:GraphsG1andG2;Outputgraph:Therearefourtypesofoutputgraphsasthefollowing: TypeI. AgraphGafterapplying1)]TJ /F8 11.955 Tf 9.3 0 Td[(,2)]TJ /F8 11.955 Tf 12.62 0 Td[(or3-sumonG1andG2; TypeII. AgraphGafterapplyingHenneberg-IconstructionsonG1; TypeIII. AgraphGafterapplyingHenneberg-IIconstructionsonG1; TypeIV. AgraphGafterapplyingvertexspliton0,1or2edgesonG1.ThenexttheoremprovesthecorrectnessoftheStandard-scheme. Theorem4.7. Forthestandard-scheme(Scheme 4.3 ),wehavethefollowing: TypeI. IfgraphsG1andG2arebothnucleation-free,independentwithimpliednon-edges,thentheir1-sumand2-sumarenucleation-free,independentwithimplied 73

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non-edges.Whenapplying3-sum,iftheK3intheintersectionofG1andG2isnotapartofaK4ineitherG1orG2,thentheoutputgraphGalsonucleation-free,independentwithimpliednon-edges. TypeII. LetG1beanucleation-free,independentwithimpliednon-edges.ApplyHenneberg-Iconstructionbyaddinganewvertexvandthreeedges(v,w1),(v,w2),(v,w3),suchthatw1,w2andw3isnotpartofaK4inG1.ThentheoutputgraphGisanucleation-free,independentwithimpliednon-edges. TypeIII. LetG1beanucleation-free,independent.ApplyHenneberg-IIconstructionasfollows:chooseavertexsetW=fw1,w2,w3,w4gG1suchthat(1)thereisatleast1edgeinducedbyWand(2)WdoesnotinduceaK4,thenwedelete1edgebetweentheverticesofW,andaddanewvertexvandfouredges(v,w1),(v,w2),(v,w3),(v,w4).ThentheoutputgraphGisnucleation-freeandindependent. TypeIV. IfG1isnucleation-free,independent,thenapplyingvertexspliton0,1,or2edgesonG1outputsanucleation-freeandindependentgraphG. Proof. (OfTheorem 4.7 )WeprovethethreetypesofoutputgraphsforScheme 4.3 asthefollowing:TypeI.IfG1andG2arebothindependent,then(1)itfollowsbyinspectingthemotionspaceofG(rightnullspaceofthegenericrigiditymatrixofG)asacombinationofthemotionspacesofG1andG2thatthe1-sumand2-sumofG1andG2arebothindependent;and(2)itfollowsfromthepropertiesofabstractrigiditymatroid[ 35 ](especiallyaxiomC6)thatthe3-sumofG1andG2isindependent.Moreover,itiseasytoseethattheimpliednon-edgesinG1andG2remainimpliedintheresultinggraphs.For1-sumand2-sum,theresultinggraphcannotcontainanynucleationbysimplycountingthenecessarynumberofedgesforagraphtoberigid.Ifa3-sumcreatesanucleation,thentheremustbetwosubgraphsG01ofG1andG02ofG2thatarerigidandbothproperlycontaintheK3sharedbyG1andG2.Theonlyrigidsubgraphsavailableinnucleation-free,independentgraphsaretrivialonesandinthiscase,onlyK4.Thusif 74

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thesharedK3isnotapartofaK4inG1ornotapartofaK4inG2,theresultinggraphisnucleation-free.TypeII.IfG1isnucleation-free,independentwithimpliednon-edges,thenafterapplyingHenneberg-Iconstruction,theoutputgraphGisindependentsinceHenneberg-Iconstructionpreservesindependence.ToshowGisnucleation-free,weonlyneedtoshowthattheaddedvertexvisnotpartofanynucleation.Supposeotherwise,thenviseitherin(1)aK5with1edgemissingor(2)anucleationwithatleast6vertices.For(1)tohappen,weneedw1,w2andw3tobepartofaK4,whichisfalse.If(2)istrue,thenweknowthatinG1,therewasanucleationwithatleast5vertices,contradiction.TypeIII.IfG1isnucleation-free,independent,thenafterapplyingHenneberg-IIconstruction,theoutputgraphGisindependentsinceHenneberg-IIconstructionpreservesindependence.ToshowGisnucleation-free,weonlyneedtoshowthattheaddedvertexvisnotpartofanynucleation.Supposeotherwise,thenviseitherin(1)aK5with1edgemissingor(2)anucleationwithatleast6vertices.For(1)tohappen,weneedWtoinduceaK4,whichisfalse.If(2)istrue,thenweknowthatinG1,therewasanucleationwithatleast5vertices,contradiction.TypeIV.IfG1isnucleation-free,independent,thenafterapplyingvertexspliton0,1,or2edgesonG1,wehaveanindependentgraphG.Thisgraphisalsonucleation-freesincetheaddedvertexconnectsonlytoverticesthatwereincidenttoacommonvertexofG1. Remark:TheproofofTheorem 4.7 (TypeIandII)followsfromtherank-sandwichprooftechniquebuttherankupperboundingredientisnotneededsinceimpliednon-edgesareinheritedfromtheconstituentgraphs.WithTheorem 4.7 ,wecaninductivelyapply1-sumsand2-sumsandtheresultinggraphsarealwaysnucleation-free.Butfor3-sums,ifbothK3'sweidentifywerepartofaK4intheoriginalgraphs,theresultinggraphhasa 75

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nucleation.Moreover,ifwedo3-sumsonnucleation-freerigidgraphs,thenwecanstillobtainindependentgraphs,butthosegraphsarenotnucleation-free. 4.3Roof-Addition:GeneralInductiveConstructionforNucleation-Free,IndependentGraphwithImpliedNon-EdgesInthissection,weintroduceapowerful,newgeneralinductiveconstructionscheme,calledroof-addition,forinductivelyconstructingavarietyofarbitrarilylargenucleation-free,independentgraphswithimpliednon-edges.Thesegraphscannotbeconstructedusingtheschemessofar.However,weshowthattheschemesintheprevioussectionpreservethestartinggraphpropertiesneededtoapplytheroof-additionschemeofthissection.Thusthenew,roof-additionschemecanbefreelycombinedwiththeschemesintheprevioussection.First,Theorem 4.8 showsthatanewinductiveconstructiongivesindependentgraphs.Theothertwoproperties(nucleation-freeandpresenceofimpliednon-edges)imposefurtherrequirementsontheinputgraphsfortheroof-additionScheme 4.4 .TheproofofTheorem 4.9 usestherank-sandwichtechniquetoshowthattheroof-additionconstructionontheappropriatestartinggraphsresultinnucleation-freegraphswithimpliednon-edges. 4.3.1InductiveConstructionforIndependentGraphsInthissection,weintroduceaninductiveconstructionforindependentgraphs. Scheme4.4(Roof-addition). Inputgraph:GraphHwithatleastonenon-edge.Outputgraph:AnewgraphGobtainedin2steps. Step1 (graphcuttingalongnon-edgefa,bg).Splitaintotwoverticesa1anda2andsplitbintotwoverticesb1andb2.DistributeedgesofHincidenttoabyassigningthemtoa1anda2inanarbitrarymanner.DistributeedgesofHincidenttobbyassigning 76

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Figure4-6. AschematicshowinghowtoapplygraphcuttingofGonfa,bg themtob1andb2inanarbitrarymanner.Fig. 4-6 showstheprocedureofgraphcutting. Step2 (roofpasting).TaketworoofsR1andR2sharingahingenon-edgefu,vg.Theotherhingenon-edgeofR1isidentiedwiththeverticesa1andb1,andtheotherhingenon-edgeofR2isidentiedwiththeverticesa2andb2.Denotethenon-hingeverticesofR1andR2ascandc0respectively.Fig. 4-7 showshowtoapplyroof-additionschemeonagivengraphH. Nextweshowthattheroof-additionschemecanbeappliedtoinductivelyconstructindependentgraphs. Theorem4.8(Roof-additiongivesindependentgraphs). IfHisindependent,thentheroof-additionscheme(Scheme 4.4 )outputsanindependentgraphG.BeforegivingtheproofofTheorem 4.8 ,werstnoticethatroof-additionisdifferentfromexistinginductivemethodssuchasHennebergconstructionandvertexsplitthatgenerateindependentgraphs. Observation4.3. GivengraphH,thenthegraphGgeneratedbyroof-additiononH,cannotbegeneratedbyanycombinationofHennebergconstructionsand/orvertexsplitonHwithedgeremovalintheend. 77

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Figure4-7. Aschematicshowinghowtoinductivelyconstructindependentgraphsbyroof-addition.Firstweidentifyanon-edgepairfa,bg,cuttingitbysplittingaandbanddistributeincidentedgesamongeachsplitvertexanditscounterpartinanarbitraryfashion.Thenweregardtwonewnon-edgesastwohingesandaddtworoofsbetweenthem. Proof. Letfa,bgbethenon-edgepairofHonwhichthetworoofsareaddedtoobtainG.WeknowtheedgesonaandbareredistributedinordertoobtainG.First,applyingonlyHenneberg-IconstructionsonHdoesnotchangetheincidenceoftheoriginaledgesofH,thusthiscannotgenerateG.IfweonlyapplyHenneberg-IIconstructiononH,sincefa,bgisanon-edgeofH,weneedtoselecttwootherverticesw,x6=a,bonwhichanincidentedgeisremoved.ApplyingHenneberg-IIconstructionscanneveraddbackthisedge(w,x)andthustheresultinggraphwillbedifferentfromtheoutputgraphGofScheme 4.4 ,whichhastheedge(w,x).IfweonlyapplyvertexsplitonH,therearetwocases:(1)weapplyavertexsplitonavertexw6=a,b.Thiswillleavethenewlyaddedvertexw0multiplepathstoV(H)nfa,bgwithoutpassingaorb,whereV(H)denotesthevertexsetofH.ThisgenerallyisnotthesameastheoutputgraphofScheme 4.4 .(2)Weapplyavertexsplitonaorb.Thismeansoneneighbortofawillbeconnectedtothenewlyaddedvertexx.Sincevertexsplitwillalsoaddanedgebetweena1anda2,inorderthemakethe 78

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resultinggraphthesameastherequiredoutputgraph,weneedtoapplyedgeremovalon(a1,a2).However,removing(a1,a2)meanstheedge(w,x)willnotberemoved,thustheresultinggraphisdifferentfromtheoutputgraphofScheme 4.4 .Whencombiningtheabovethreeconstructions,itisobviousthattherststepisavertexsplit.Thecase(1)aboveisstillgenerallydifferentfromourScheme 4.4 duetothereasonmentionedabove.Forcase(2),evenifwerstapplyvertexsplitofaandbonzeroedgesandmanagetouseHenneberg-IIconstructiontoadduandv,theremainingtwoverticescandc0cannotbeaddedusinganycombinationofHennebergconstructionsandvertexsplit. Notehoweverthatknowninductivemethods,suchasvertexsplitand/orHennebergconstructions,maybeabletoobtainGwithadifferentindependentinputgraphH0.Forexample,althougharingofroofsRkcannotbeobtainedusingHenneberg-IIconstructionsfromaringofk)]TJ /F4 11.955 Tf 13.22 0 Td[(2roofs,RkcanbeobtainedusingHenneberg-IIconstructionsfromaringofktetrahedra.Intheremainderofthissection,wegivetheproofofTheorem 4.8 .First,weneedthefollowingdenition: Denition4.4. LetG=(V,E)beagraphwithnverticesfv1,...,vng.Anysetofscalarssi,j=sj,idenedforeachedge(vi,vj)inEiscalledastressforG.Moreover,wesays=(...,si,j,...)isaself-stressvectorforframeworkG(p)ifforanyvertexpointpiofG(p),thefollowingstressbalancevectorequationatpiholds:Xjsi,j(pi)]TJ /F23 11.955 Tf 11.95 0 Td[(pj)=0I.e.,sR(p)=0,whereR(p)istherigiditymatrixofG(p).Eachsi,jiscalledascalarself-stressassociatedwiththebarij.Notethatthestressbalanceequationatpi(boldfacerepresentsthecoordinatepositionofavertex)isusedtorefertoasystemofdscalarequations,oneforeachcoordinateofthepointpi.Wesometimesrestrictourselvestoequationscorresponding 79

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tosomespeciedsubsetorsubspaceofthecoordinatebasisatpi.Werefertothisasaspecicprojectionofthestressbalanceequationatpj.Inaddition,wesometimesrestrictourselvestotermssi,j(pi)]TJ /F23 11.955 Tf 11.98 0 Td[(pj)ofaspecicsubsetofthestressbalanceequationatpi,correspondingtothebarsbetweenpiandasubsetofpoints,pj. Proof. (ofTheorem 4.8 )WewillconstructaspecicframeworkofGfromanygivengenericframeworkH(p)ofHasspeciedbelowandshowninFig. 4-8 wesuperposea2ontopofa1andb2ontopofb1.Weassumewithoutlossthatthehingesareparalleltotheyaxis.WecalltheothertwohingepointsAandB. thefourhingepointsA,B,a1,b1lieonthesameplane,withoutloss,thexy-plane,andA,B,a1,b1formasquare. theremainingtwopointscandc0lieonalineperpendiculartoandpassingthroughthecenterofthesquareformedbyA,B,a1,b1.Noticethatthislineisparalleltothezaxis.Assumethatthenewframeworkisdependent,i.e.,thereexistsanon-zeroself-stressvectoronedgesoftheframeworksuchthatforeachpointpi,thestressbalanceequationatpiholds.Wewillshowthatbysimplyrestrictingthisself-stressvectortotheedgesofH,wegetanon-zeroself-stressvectoronagenericframeworkH(p)ofHthatisobtainedbygluingtogetheroridentifyingthepointsa1anda2,andsimilarlythepointsb1andb2.ButweknewHisindependent.Thuswedrawacontradiction,therebyprovingthetheorem. Acloserlookatthestressbalanceequationsgivesusthefollowingclaims. Claim4.8.1. Theprojectionofsc,A(c)]TJ /F9 11.955 Tf 12.26 0 Td[(A)ontheplanexoyisequaltotheprojectionofsc,b1(c)]TJ /F9 11.955 Tf 11.2 0 Td[(b1)ontheplanexoy.Thesamehappensbetweensc0,B(c0)]TJ /F9 11.955 Tf 11.2 0 Td[(B)andsc0,a2(c0)]TJ /F9 11.955 Tf 11.21 0 Td[(a2).Theremainingtwostressesonchavethesamemagnitudeandsodotheremainingstressesonc0. 80

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Figure4-8. Aschematicshowingtheframeworks,i.e.,includingthepositioningoftheverticesforthelasttworoofsfortheproofofTheorem 4.8 .ThetworoofsarevieweddifferentfromadifferentperspectivethanFig. 4-7 .Notethatthetwopointsa1anda2representdifferentverticesthatarecoincidentandsimilarlyb1andb2representdifferentverticesthatarecoincident. Proof. Considerallstressequationstermsatc.Allthosefourtermsadduptozero.Inparticular,ifweprojectthetermsontheplanexoy,thentheirprojectionslieontwolinesthatareperpendiculartoeachother.Thussumoftheprojectionsoneachlineshouldadduptozero,thusprovingtheclaim.SeeFig. 4-9 Figure4-9. Agureshowingthesymmetryoftheveedgesatc. 81

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Claim4.8.2. Considerthetermsofthestressbalanceequationata1anda2correspond-ingtothebarsinthetwoaddedroofs.Allthesetermstogetheradduptozero.Similarly,theanalogoustermsatb1andb2adduptozero. Proof. Wewillshowthepartabouta1anda2rst.Therearethreetermsforthestressbalanceequationata1restrictedtotheaddedgraph,andthreetermsforthestressbalanceequationata2restrictedtothe(n+2)ndroofaswell.Wewillshowthatfourofthesesixtermsadduptozeroi.e., sa2,c0(a2)]TJ /F9 11.955 Tf 11.96 0 Td[(c0)+sa1,c(a1)]TJ /F9 11.955 Tf 11.96 0 Td[(c)+sa1,B(a1)]TJ /F9 11.955 Tf 11.96 0 Td[(B)+sa2,B(a2)]TJ /F9 11.955 Tf 11.96 0 Td[(B) (4) =0andtheremainingtwotermsalsoadduptozero, sa2,A(a2)]TJ /F9 11.955 Tf 11.95 0 Td[(A)+sa1,A(a1)]TJ /F9 11.955 Tf 11.96 0 Td[(A)=0.(4)Toshow( 4 ),weneedtoinspectthesixstressequationtermsatB,especiallythefourtermscorrespondtoedgesthatareincidenttoc,c0,a1anda2.ReectionsymmetryoftheoctahedronandClaim 4.8.1 togethershowthatthesixtermsinequation( 4 )-i.e.,threetermsinthestressbalanceequationata2andthreetermsinthestressbalanceequationata1-areobtainedasthereectionoffourcorrespondingtermsofthestressbalanceequationatB.Thusifwecanshowthatthelatter4termsadduptozero,thentheformer6termsadduptozeroaswell,therebyshowingEquation( 4 ). Moreprecisely,denotebys0Bthestressvectorrestrictedtothesespecic4edgesatB.DenotebyR0Bthecorresponding4by3submatrixoftherigiditymatrix(whoserowvectorsspecifythecorresponding4barsatB).Theirproducts0BR0Bisavectorwhoseentriesarethe4termsatB.Considertheplanecontainingthetwolinescc0andAb2, 82

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Figure4-10. AgureshowingthefourofthesixtermsthatadduptozeroatBandtheirreectionthroughtheplanepassingcc0andAa2(thetwodashedline)tothefourtermsin( 4 ). i.e.,thedashedlinesshowninFigure 4-10 .DenotebyLthereectionacrossthisplane.Nownoticethattheentriesoftheproducts0BR0BLarepreciselythe4termsinEquation( 4 ).Theproofaboutb1andb2isessentiallythesameasabove,theonlydifferenceisthatin( 4 ),wehaveonlyfourtermsinsteadofsix.Wecanstillusethereectionsymmetryandobtaintheexactsameresult. Thusweknowthestressbalanceequationtermsata1restrictedtoHanda2restrictedtoHadduptozero.Thesameresultcanbeobtainedatb1andb2.Soifweremovethetwoaddedroofsandusethesamestressbalanceequationoneachedge,weknowthatthestressbalanceequationtermsadduptozerooneverypointofthegenericframeworkofH.Thustherestrictedself-stressvectorofthenewframeworkisaself-stressvectoronagenericframeworkofH.Nextwewillshowthisself-stressvectorisnon-zero. Claim4.8.3. Supposesisanon-zeroself-stressvectorforG(whichhasthetwoaddedroofs).Denotebys0self-stressvectorsrestrictedtoH,afterweremovethelasttwo 83

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roofsandgluea2witha1andb2withb1.Thens0isanon-zeroself-stressvectorforH(p). Proof. Wearguetheexistenceofnon-zerostressintheoriginalframeworkforallthefollowingcases: i Ifthereisanon-zerostressat(c,b1)(resp.(c,a1),(c0,b2),(c0,a2)),thenthestressequationtermsofthetwoaddedroofswillalwayshaveazprojectionatb1(resp.a1,b2,a2).SotheoriginalframeworkH(p)musthaveanon-zerostresstocanceloutthiszprojection.WhenthereisnoedgeinHthatisincidenttob1(resp.a1,b2,a2),weknowthestressat(c,b1)(resp.(c,a1),(c0,b2),(c0,a2))hastobezero. ii Ifthereisanon-zerostressat(c,A)(resp.(c,B),(c0,A),(c0,B)),thenfromClaim 4.8.1 ,weknowthereisanon-zerostressat(c,b1)(resp.(c,a1),(c0,b2),(c0,a2)),whichisequaltoCase i iii Ifthereisanon-zerostressat(A,b1)(resp.(B,a1),(A,b2),(B,a2)),theneither(1)thereisanon-zerostressat(c,b1)(resp.(c,a1),(c0,b2),(c0,a2)),whichisCase i ,or(2)thestressat(c,b1)(resp.(c,a1),(c0,b2),(c0,a2))iszero.When(2)happens,thestressequationtermsofthetwoaddedroofswillalwayshaveayprojectionatb1(resp.a1,b2,a2).Sotheoriginalframeworkmusthaveanon-zerostresstocanceloutthisyprojection.OrifthereisnoedgeinHthatisincidenttob1(resp.a1,b2,a2),thenthestressat(c,b1)(resp.(c,a1),(c0,b2),(c0,a2))hastobezero. iv Ifthereisanon-zerostressat(B,b1)(resp.(A,a1),(B,b2),(A,a2)),thenthereare3furthercases.(1)Thereisanon-zerostressat(c,b1)(resp.(c,a1),(c0,b2),(c0,a2)),whichisCase i .(2)Thestressat(c,b1)(resp.(c,a1),(c0,b2),(c0,a2))iszero,thereisanon-zerostressat(A,b1)(resp.(B,a1),(A,b2),(B,a2)).ThisisjustCase iii .(3)Thestressesat(c,b1)(resp.(c,a1),(c0,b2),(c0,a2))and(A,b1)(resp.(B,a1),(A,b2),(B,a2))arebothzero.Thenthestressequationtermsofthetwoaddedroofswillalwayshaveaxprojectionatb1(resp.a1,b2,a2).SotheoriginalframeworkH(p)musthaveanon-zerostresstocanceloutthisxprojection.OrifthereisnoedgeinHthatisincidenttob1(resp.a1,b2,a2),thenthestressat(c,b1)(resp.(c,a1),(c0,b2),(c0,a2))hastobezero. ReturningtotheproofofTheorem 4.8 ,usingClaims 4.8.1 4.8.2 and 4.8.3 ,wehavefoundanon-zeroself-stressfortheoriginalframeworkofH.I.e.,thegenericframeworkofHisdependent.Contradiction. 84

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4.3.2Roof-AdditionGivesNucleation-Free,IndependentGraphswithImpliedNon-EdgesInthissection,weshowthattheroof-additionschemegivesnucleation-free,independentgraphswithimpliednon-edgesinthefollowingtheorem. Theorem4.9(Roof-additiongivesnucleation-free,independentgraphswithimpliednon-edges). LetHbeanindependentgraphsatisfyingthefollowing:thereisa2-thincoverX=fX1,X2,...,XngofHsuchthat(1)thesharedpart(V,S(X))isindependent;and(2)rank(H)=PXi2Xrank(H?[Xi]))]TJ /F15 11.955 Tf 12.62 8.96 Td[(Pfu,vg2S(X)(d(u,v))]TJ /F4 11.955 Tf 10.47 0 Td[(1),whereH?=H[S(X).Letfa,bg2S(X)beanon-edgeofHforwhichthegraphcuttingoperationofScheme 4.4 isappliedtoHalongfa,bginthefollowingmanner.(1)ForanycoveringsubgraphH[Xj]ofHthathasatleastonesharednon-edge,ifoneofitsedgesisassignedtoa1orb1(resp.a2orb2),thenallofitsedgesincidenttoaorbareassignedtoa1orb1(resp.a2orb2).(2)Aftergraphcutting,thegraphisnucleation-free.NotethatsuchaXmaynotexist.Whenitexists,wecallHastartinggraphforTheorem 4.9 .ApplyinggraphcuttingonastartinggraphHintheabovemannerresultsinanoutputgraphGthatisnucleation-free,independentwithimpliednon-edges. Proof. (ofTheorem 4.9 )Theproofusestherank-sandwichprooftechnique.First,theindependenceofGisguaranteedbyTheorem 4.8 anditisalsoclearthatGisnucleation-free.WewillusetherankupperboundargumentinSection 4.1.2.2 toshowtheexistenceofimpliednon-edgesinG.Infact,wewillshowthatwheneveragraphcuttingpreservesallcoveringsubgraphsthatcontainasharednon-edge(itisclearthatthegraphcuttinginTheorem 4.9 isonesuchcutting),thentherankupperboundargumentworks.Whenwesayacoveringsubgraphispreserved,wemeanaftergraphcutting,itscorrespondingsubgraphdoesnotcontainbotha1anda2(orb1andb2).WecanndacoverX0ofGbymodifyingX.WeuseV(G)(resp.V(H))todenotethevertexsetofgraphG(resp.H)andE(G)(resp.E(H))todenotetheedgesetofgraphG(resp.H). 85

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WewillstartfromtheconditionjE(H)j=rank(H)=PXi2Xrank(H?[Xi]))]TJ /F15 11.955 Tf -404.56 -14.94 Td[(Pfu,vg2S(X)(d(u,v))]TJ /F4 11.955 Tf 12.91 0 Td[(1)andthenobtainarankIEcountonG?(i.e.,G[S(X0))toshowthatrank(G?)isequaltojE(G)j.First,wenoticethattherearethreetypesofcoveringsubgraphsofHinX: (i) trivialcoveringsubgraphs,i.e.,edges.WedenotethissetasX0. (ii) non-trivialcoveringsubgraphsH[Xi]thatarepreservedinX0andwedenotethosecoveringsubgraphsasthesetX1. (iii) non-trivialcoveringsubgraphsH[Xj]thatarenotpreservedinX0andwedenotethosecoveringsubgraphsasthesetX2.I.e.,thesecoveringsubgraphsfallapartsothattheiredgesbecometrivialcoveringsubgraphsinX0,exceptthoseedgesthatarepresentinsomecoveringsubgraphofX1.WenotethatthesecoveringsubgraphsonlyhaveedgessharedbyothercoveringsubgraphsinX.LetdXi(u,v)(i=1or2)bethenumberofsetsXiinXisuchthatfu,vgXi.LetS(Xi)(i=1or2)bethesetofallpairsofverticesfu,vgsuchthatXj\Xk=fu,vgforsomeXj2XiandXk2Xi.LetL:=S(X)nS(X1).ThenwecanrewritetherankconditiononHasfollows:jE(H)j=rank(H)=XXi2Xrank(H?[Xi]))]TJ /F15 11.955 Tf 26.24 11.35 Td[(Xfu,vg2S(X)(d(u,v))]TJ /F4 11.955 Tf 11.96 0 Td[(1)=XXi2X1rank(H?[Xi]))]TJ /F15 11.955 Tf 27.57 11.35 Td[(Xfu,vg2S(X1)(dX1(u,v))]TJ /F4 11.955 Tf 11.95 0 Td[(1))]TJ /F15 11.955 Tf 18.13 11.36 Td[(Xfu,vg2L(dX1(u,v)+dX2(u,v))]TJ /F4 11.955 Tf 11.95 0 Td[(1)+XXi2X2rank(H?[Xi])+XXi2X01Itisclearthattheonlynon-trivialcoveringsubgraphsbesidesthetwoaddedroofsinX0areinX1.SinceintherankIEcount,thecontributionoftrivialcoveringsubgraphsisequaltotheirnumberofedges,wewillshow: 86

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XXi2X2rank(H?[Xi])+XXi2X01)]TJ /F15 11.955 Tf 18.13 11.36 Td[(Xfu,vg2L(dX1(u,v)+dX2(u,v))]TJ /F4 11.955 Tf 11.96 0 Td[(1)=Xe2E(H)nE(X1)1, (4) whereE(X1)denotesthesetofedgesinducedbyvertexsetsofX1.First,sinceHisindependent,weknoweverycoveringsubgraphinX2isindependent.ThuseveryedgeinX2thatisnotinLcontributes1tothetothelefthandsideofEquation( 4 ).Forfu,vg2L,werstnotethateveryfu,vgisanedge,sincethecoveringsubgraphsinX2donotcontainsharednon-edgesinS(X).Anotherkeyobservationisthatforeveryfu,vg2L,(1)dX1(u,v)=1whenfu,vgisinsomecoveringsubgraphofX1,sinceotherwisefu,vg2S(X1);and(2)dX1(u,v)=0whenfu,vgisnotinanycoveringsubgraphofX1.When(1)happens,fu,vgisanedgeinsomecoveringsubgraphofX1anditisclearthatdX2(u,v)isequaltothenumberofXi'sthatareinX2andcontainfu,vg.I.e.,theseedgescontributezerotothelefthandsideofEquation( 4 ).When(2)happens,fu,vgisnotanedgeinanycoveringsubgraphofX1andeachfu,vgcontributes1tothelefthandsideofEquation( 4 ),sincedX1(u,v)=0andeverycoveringsubgraphinX2isindependent.ThusthetotalcontributionofthelefthandsideofEquation( 4 )isequaltothenumberofedgesofX2thatdonotappearinX1.HencewehaveEquation( 4 )andmoreimportantly,thefollowing: jE(H)j=rank(H)=XXi2X1rank(H?[Xi]))]TJ /F15 11.955 Tf 27.57 11.36 Td[(Xfu,vg2S(X1)(dX1(u,v))]TJ /F4 11.955 Tf 11.95 0 Td[(1)+Xe2E(H)nE(X1)1 (4) 87

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ToapplyrankupperboundargumentonG,weneedtoshowthatrank(G)=PXi2X0rank(G?[Xi]))]TJ /F15 11.955 Tf 12.62 8.96 Td[(Pfu,vg2S(X0)(d(u,v))]TJ /F4 11.955 Tf 11.86 0 Td[(1).SinceGisindependent,allweneedistoshowthatjE(G)j=PXi2X0rank(G?[Xi]))]TJ /F15 11.955 Tf 12.62 8.97 Td[(Pfu,vg2S(X0)(d(u,v))]TJ /F4 11.955 Tf 11.96 0 Td[(1).NextweturntothecoveringsubgraphsofX0.Therearetwotypes. (i) trivialcoveringsubgraphs,i.e.,edges.WedenotethissetasX00andweknowjX00j=Pe2E(H)nE(X1)1. (ii) non-trivialcoveringsubgraphsH[Xi]2X1thatarepreservedasG[Xi]inX0andtwoaddedroofsR1andR2.ItisclearthatthenumberofedgesofGcanbewrittenasfollows:jE(G)j=jE(H)j+rank(G[R1])+rank(G[R2])=jE(H)j+rank(G?[R1])+rank(G?[R2]))]TJ /F4 11.955 Tf 11.95 0 Td[(2PlugginginEquation( 4 ),wehave:jE(G)j=rank(G?[R1])+rank(G?[R2]))]TJ /F4 11.955 Tf 11.96 0 Td[(2+XXi2X1rank(H?[Xi]))]TJ /F15 11.955 Tf 27.56 11.36 Td[(Xfu,vg2S(X1)(dX1(u,v))]TJ /F4 11.955 Tf 11.95 0 Td[(1)+Xe2X001ThenalstepistobuildarelationshipbetweenPXi2X1rank(H?[Xi]))]TJ /F15 11.955 Tf 12.63 8.97 Td[(Pfu,vg2S(X1)(dX1(u,v))]TJ /F4 11.955 Tf 13.06 0 Td[(1)andPXi2X1rank(G?[Xi]))]TJ /F15 11.955 Tf 12.62 8.97 Td[(Pfu,vg2S(X0)(dX0(u,v))]TJ /F4 11.955 Tf 13.07 0 Td[(1).Wenoticethat(1)everycoveringsubgraphinX1ispreservedinX0,soPXi2X1rank(H?[Xi])=PXi2X1rank(G?[Xi]);and(2)fa,bgistheonlysharednon-edgethatischanged.Moreprecisely,fa,bgissplitintofa1,b1gandfa2,b2g.Togetherwiththefactthatfa1,b1gandfa2,b2garebothsharedbythetwoaddedroofs,weknowdX1(a,b)=dX0(a1,b1))]TJ /F4 11.955 Tf 12.33 0 Td[(1+dX0(a2,b2))]TJ /F4 11.955 Tf 12.31 0 Td[(1=dX0(a1,b1)+dX0(a2,b2))]TJ /F4 11.955 Tf 9.3 0 Td[(2.Letfw,tgbethenon-edgesharedbyR1 88

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andR2.Hencewehave:jE(G)j=rank(G?[R1])+rank(G?[R2]))]TJ /F4 11.955 Tf 11.96 0 Td[(2+XXi2X1rank(G?[Xi]))]TJ /F15 11.955 Tf 39.58 11.35 Td[(Xfu,vg2S(X1)nfa,bg(dX1(u,v))]TJ /F4 11.955 Tf 11.96 0 Td[(1))]TJ /F4 11.955 Tf 9.29 0 Td[((dX1(a,b))]TJ /F4 11.955 Tf 11.96 0 Td[(1)+Xe2X001=rank(G?[R1])+rank(G?[R2]))]TJ /F4 11.955 Tf 11.96 0 Td[(2+XXi2X1rank(G?[Xi]))]TJ /F15 11.955 Tf 39.58 11.36 Td[(Xfu,vg2S(X1)nfa,bg(dX1(u,v))]TJ /F4 11.955 Tf 11.96 0 Td[(1))]TJ /F4 11.955 Tf 9.29 0 Td[((dX0(a1,b1)+dX0(a2,b2))]TJ /F4 11.955 Tf 11.96 0 Td[(2)]TJ /F4 11.955 Tf 11.96 0 Td[(1)+Xe2X001=rank(G?[R1])+rank(G?[R2]))]TJ /F4 11.955 Tf 11.96 0 Td[(1+XXi2X1rank(G?[Xi]))]TJ /F15 11.955 Tf 39.58 11.36 Td[(Xfu,vg2S(X1)nfa,bg(dX1(u,v))]TJ /F4 11.955 Tf 11.96 0 Td[(1))]TJ /F4 11.955 Tf 9.29 0 Td[((dX0(a1,b1))]TJ /F4 11.955 Tf 11.95 0 Td[(1))]TJ /F4 11.955 Tf 11.95 0 Td[((dX0(a2,b2))]TJ /F4 11.955 Tf 11.95 0 Td[(1)+Xe2X001=rank(G?[R1])+rank(G?[R2]))]TJ /F4 11.955 Tf 11.96 0 Td[((d(w,t))]TJ /F4 11.955 Tf 11.96 0 Td[(1)+XXi2X1rank(G?[Xi]))]TJ /F15 11.955 Tf 39.58 11.36 Td[(Xfu,vg2S(X1)nfa,bg(dX1(u,v))]TJ /F4 11.955 Tf 11.96 0 Td[(1))]TJ /F4 11.955 Tf 9.29 0 Td[((dX0(a1,b1))]TJ /F4 11.955 Tf 11.95 0 Td[(1))]TJ /F4 11.955 Tf 11.95 0 Td[((dX0(a2,b2))]TJ /F4 11.955 Tf 11.95 0 Td[(1)+Xe2X001=XXi2X0rank(G?[Xi]))]TJ /F15 11.955 Tf 27.59 11.36 Td[(Xfu,vg2S(X0)(dX0(u,v))]TJ /F4 11.955 Tf 11.96 0 Td[(1).Noticingthat(V(H),S(X))isindependent,weknowthat(V(G),S(X0))isindependent,sincetheonlydifferencesbetweenS(X)andS(X0)arethesplitoffa,bgandtheadditionofthesharednon-edgeofR1andR2,whichisdisjointfromtherestofS(X0).ThuswecanapplyTheorem 4.3 toobtainthatrank(G?)PXi2X0rank(G?[Xi]))]TJ ET BT /F1 11.955 Tf 227.35 -687.85 Td[(89

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Pfu,vg2S(X0)(dX0(u,v))]TJ /F4 11.955 Tf 12.33 0 Td[(1)=jE(G)j=rank(G).SinceGG?andGisindependent,weknowrank(G?)rank(G)andhencerank(G?)=rank(G).Itfollowsthateverynon-edgeinS(X0)isimplied.Theseinclude(1)allnon-edgesinS(X)exceptfa,bg;(2)fa1,b1gandfa2,b2g;and(3)thenon-edgefw,tgsharedbythetwoaddedroofsR1andR2. 4.3.3ExistenceandGenerationofStartingGraphsforTheorem 4.9 Inthissection,wegiveseveralconcreteexamplestartinggraphsforTheorem 4.9 .Weshowtheexistenceofseveralxed-sizebasestartinggraphsthatcannotbegeneratedbytheSchemes 4.1 4.4 .Then,weshowthattheschemesgivensofarcanbeinductivelyandfreelycombinedtogeneratestartinggraphs,andtherebygeneratealargevarietyofarbitrarilylarge,nucleation-free,independentgraphswithimpliednon-edges.Notethatforxed-sizebasestartinggraphs,theirindependenceisveriedbysymbolicallycomputingthegenericrankofarigiditymatrixofindeterminatesforframeworkscorrespondingtothoseexamplestartinggraphs.Additionally,wearenotrelyingonspecialpositionsofthejoints(vertices)andweareshowingthattherankismaximal,i.e.,equaltothenumberofedges,soinanycase,wedonothavetoworryaboutnumericalroundingissuesdistortingtherankcomputationsincesuchdistortionscanonlydecreasetherank. Observation4.4. Thefollowingaregraphsthatarenucleation-freewithimpliednon-edges,butarenotencompassedbytheschemesofthischapter. Ringofmodiedoctahedralgraphswithsize7,8,9,or10.AmodiedoctahedralgraphisdrawninFig. 4-11 Ringofmodiedicosahedralgraphswithsize7,8,9,or10.AmodiedicosahedralgraphisdrawninFig. 4-11 Two-body-sharingringswithicosahedra.TaketworingsR1andR2andcombinetheminsuchawaythattheysharetwoormorecoveringsubgraphsandonesharedcoveringsubgraphhastwodegrees-of-freedom,whiletheothersharedcoveringsubgraphshave1dofeach.SeeFig. 4-12 forasimpleexample.Wecallthistypeofclassabody-sharingring. 90

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Figure4-11. Twotypesofbuildingblocksofringgraphs.Ontheleftisanothermodiedoctahedralgraph.Thismodiedoctahedralgraph,whenusedasthebuildingblockofaringgraph,resultsinaringthatcannotbeobtainedusingHennebergconstructionsorallotherknowninductivemethodsfromaringofroofs.Ontherightisamodiedicosahedralgraphusedtoformaringthatisindependentwithimpliednon-edges.Thedashedlinesarethetwohingenon-edges. Four-body-sharingrings.SeeexampleinFig. 4-13 Proof. Weusetherank-sandwichprooftechnique.Therankupperboundargumentforallexamplesisclear.Theindependenceofthosegraphsisveriedbysymbolicallycomputingthegenericrankofarigiditymatrixofindeterminatesforframeworkscorrespondingtotheexamplegraphs(forexample,usingMaple).Asmentionedearlier,wedonothavetoworryaboutnumericalroundingissuesdistortingtherankcomputationsincesuchdistortionscanonlydecreasetherank. ToextendObservation 4.4 toarbitrarysizedrings,again,therankupperboundisstraightforward.Toshowindependence,wewouldrequireprovingageneralizedversionofroof-additionthatpermitsaddinggeneralpolyhedra. 91

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Figure4-12. TworingsoficosahedrasharetwoicosahedraasinFig. 4-11 .Eachringconsistsof7icosahedra,representedbyacircleinthegure.WedroptwoedgesfromI(namely(v1,v2)and(v3,v4)asinFig. 4-11 )andoneedge((v1,v2))fromallothericosahedra.Wechooseanon-edge(v9,v12)asanotherhingenon-edgeforallicosahedra.I.e.,Ihas3hingenon-edges,whereoneisanon-edgeoficosahedraandtheothertwoaredroppededges,whileallothericosahedrahavetwohingenon-edges,whereoneisanon-edgeofanicosahedronandtheotherisadroppededge.Thefullrankofthegraphisveriedbysymbolicallycomputingthegenericrankofarigiditymatrixofindeterminatesforframeworkscorrespondingtothegraph(forexample,usingMaple). Next,weshowhowwecaninductivelyapplyTheorem 4.7 andTheorem 4.9 togeneratearbitrarilylargenucleation-free,independentgraphswithimpliednon-edges. Theorem4.10. IfagraphHsatisesthestartinggraphconditionforTheorem 4.9 ,thenapplyingk-sumorHenneberg-IconstructionsasinTheorem 4.7 onHgivesastartinggraphforTheorem 4.9 .Ifweapplyroof-additiononHaccordingtoTheorem 4.9 ,wewillgetanoutputgraphGthatagainsatisesthestartinggraphconditionforTheorem 4.9 Proof. First,forK-sumsonastartinggraphHandanucleation-free,independentgraphG1,thenresultinggraphGisindependentandnucleation-freefromTheorem 4.7 .WecantakethecoverXofHandextendittoacoverofGbyaddingallpairsofverticesof 92

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Figure4-13. AFour-body-sharingring.Thegraphhasfourlargericosahedralgraphs(C1,C2,C3andC4asinFig. 4-11 )representedwithlargercirclesand24smallerroofsrepresentedwithsmallerellipses.Thedashedlinesinthegraphrepresenthingenon-edges.NotethatC1andC3aremodiedicosahedralgraphswhoseedges(v3,v4)and(v5,v6)aredroppedandusedashingenon-edges.Theyeachhave27edgesandthushave3degrees-of-freedom.C2andC4aremodiedicosahedralgraphswhosenon-edges(v4,v11)and(v5,v8)areusedashingenon-edges.Theyeachhave29edgesandthushave1degree-of-freedom.Thehingenon-edgesofroofsarementionedasearlier.Thegraphisclearlynucleation-freeandithas104verticesand304edges.Thefullrankofthegraphisveriedbysymbolicallycomputingthegenericrankofarigiditymatrixofindeterminatesforframeworkscorrespondingtothegraph(forexample,usingMaple).A2-thincoverargumentgivesusthatthedashedlinesareallimpliednon-edges. 93

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G1toX.ItiseasytocheckthatifXsatisestherankboundconditiononH,thenewcoversatisestherankboundconditiononG.Second,forHenneberg-IconstructionsonstartinggraphH,therearetwocases.(1)ThenewlyaddedvertexuhasallthreeneighborsinsideonecoveringsubgraphCi.Inthiscase,wecanextendCibyaddingutoitandmaintainallothercoveringsubgraphsinthecover.Therankboundconditioncertainlyholdsinthiscase.(2)Thethreeneighborsofuareindifferentcoveringsubgraphs.Inthiscase,wecanagaintakethecoverXofHandextendittoacoverofGbyaddingthreecoveringsubgraphsthatarethreeedgesincidenttou.ItiseasytocheckthatifXsatisestherankboundconditiononH,thenewcoversatisestherankboundconditiononG.Third,fromTheorem 4.8 ,weknowapplyroof-additiononstartinggraphHmaintainsindependence.Itisobviousthatnucleation-freenessisalsomaintained.Moreover,itiseasytocheckthatapplyingroof-additionaccordingtoTheorem 4.9 maintainsallotherpropertiesrequiredforstartinggraphs.Last,forallaboveschemes,itisclearthatthecuttinganddistributionspeciedinTheorem 4.9 canbeappliedonanyimpliednon-edge. WhileTheorem 4.10 givesusasufcientconditionforstartinggraphsforTheorem 4.9 ,itisnotnecessary.Inthissection,weshowedaninductivemethodtoconstructindependentgraphsinTheorem 4.8 andillustratedhowwecaninductivelyapplyTheorem 4.7 andTheorem 4.9 togeneratearbitrarilylargenucleation-free,independentgraphswithimpliednon-edges.Nextwewillturntoaconstructionschemefornucleation-freedependentgraphs,whichusestheimpliednon-edgesintheaboveconstructednucleation-free,independentgraphs. 4.4DependentGraphswithNoNucleusandFurtherConsequencesInthissection,weintroduceaconstructionscheme(Scheme 4.5 )fornucleation-freedependentgraphs.AnillustrationusingringsofroofsisgiveninFig. 4-14 94

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Figure4-14. Adouble-ringof14roofsconsistingoftworingsof7roofs.Thetworingsshareacommonnon-edge.Thegureontheleftshowsthedouble-ringofroofsandthegureontherightshowstheschematicofanypairofnucleation-free,independentgraphwithasharedimpliednon-edge.Sinceeachofthetwopartsimpliesthesharedhingenon-edge,asinthedouble-bananaexampleinFig. 2-2 ,thesharedhingenon-edgeisdouble-impliedandhencethecompositeofthetwopartsisdependent. Scheme4.5(Graph-combination). Inputgraphs:G1withimpliednon-edgefa1,b1gandG2withimpliednon-edgefa2,b2gOutputgraph:GraphGafterjoiningG1andG2byidentifyinga1witha2andb1withb2. Theorem4.11. Forgraph-combinationscheme(Scheme 4.5 ),ifG1andG2arebothnucleation-free,thenGisnucleation-freedependent.If,inaddition,G1[fa1,b1gandG2[fa2,b2garebothrigiditycircuits,thentheoutputgraphGisarigiditycircuit. Proof. (ofTheorem 4.11 ).WeknowbothG1andG2implythesharednon-edgeandtheyeachhave1dof.AfterglueingG1andG2,theremustbeadependenceintheoutputgraphG.IfinadditionG1[fa1,b1gandG2[fa2,b2garebothrigiditycircuits,toshowtheoutputgraphGisarigiditycircuit,wecandropanyedgefromGtoobtainG0andshowthatG0isindependent.Withoutlossofgenerality,wedropanedgee1fromG1toobtainG01.SinceG1[fa1,b1gisarigiditycircuit,weknowG01[fa1,b1gisindependent.Since 95

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G2[fa2,b2gisarigiditycircuit,ifwechooseanarbitraryedgee2onG2,then(1)G2nfe2g[fa2,b2gisindependent;and(2)thelinearspanoftheedgesinG2isequaltothelinearspanoftheedgesinG2nfe2g[fa2,b2g.ItfollowsthatthelinearspanofG0isequaltothelinearspanofG00:=G0nfe2g[fa2,b2g.Moreover,wecaneasilyseethatG00isa2-sumofG01[fa1,b1gandG2nfe2g[fa2,b2g,bothofwhichareindependent.SowecanuseasimilarargumentasintheproofofTheorem 4.7 toshowthatG00isindependent,whichinturnmeansG0isindependent.ThusitfollowsthattheoutputgraphGforScheme 4.5 isminimallydependent. Figure4-15. Abraceddouble-ringof14roofs:itconsistsoftworingsof7roofsandtwoextrabars.Thetworingsinthegraphareconnectedviaacommonnon-edge.Thegureontheleftshowsthegeometricstructureofthebraceddouble-ringandthegureontherightshowstheschematic.ThebraceddoubleringhasenoughedgestobeminimallyrigidbutisinfactexiblefromTheorem 4.11 SitharamandZhou[ 94 ]gaveseveralexampleswhereMaxwell'scountingconditionwasinsufcientforrigidity,allofwhichsatisfythenucleationproperty.Usingacombinatorialnotioncapturingtherecursivenatureofnucleation(calledmodule-rigidity),theyproposeanalgorithmwhichisatractablecharacterizationofgenericindependenceinalargeclassofsuchgraphsbyusingthepresenceofrigidnuclei.Ithasbeenanopenproblemwhetherthisalgorithmcanfailtodetect3Dindependenceandrigidity,i.e.whethermodule-rigiditycoincidesornotwith3Drigidity.Thefollowingcorollary 96

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Table4-1. Constructionschemesfornucleation-free,independentgraphswithimpliednon-edges.Here,Adenotesconstructionschemes,BdenotestheprooftechniquesandCdenotes2-thincover/body-hingeargument. PPPPPPPPPPPBA Flex-signRing HennebergextenderRing Standard-scheme Roof-addition Flex-sign Theorem 4.5 )]TJ ET q .398 w 465.99 -82.46 m 465.99 -64.53 l S Q BT /F25 14.346 Tf 471.97 -77.08 Td[()]TJ ET q .398 w 589.98 -82.46 m 589.98 -64.53 l S Q BT /F25 14.346 Tf 595.95 -77.08 Td[()]TJ ET q .398 w 721.51 -82.46 m 721.51 -64.53 l S Q q .398 w 3.99 -82.66 m 721.51 -82.66 l S Q q .398 w 3.99 -100.79 m 3.99 -82.86 l S Q BT /F24 14.346 Tf 20.73 -95.41 Td[(Rank-sandwich (i)Independence )]TJ ET q .398 w 323.69 -100.79 m 323.69 -82.86 l S Q BT /F24 14.346 Tf 329.67 -95.41 Td[((i)Theorem 4.1 (i)Theorem 4.7 (i)Theorem 4.8 (ii)Rankupper-bound )]TJ ET q .398 w 323.69 -118.72 m 323.69 -100.79 l S Q BT /F24 14.346 Tf 329.67 -113.34 Td[((ii)C (ii)Notnecessary (ii)C:Theorem 4.9 settlesthequestion(inthenegative),i.e.,therearegraphsthatarenotrigidbutaremodule-rigid. Corollary4.11.1. Theexiblebraceddouble-ringinFig. 4-15 ismodule-rigidaccordingtothedenitionin[ 94 ].Therefore,module-rigiditydoesnotcoincidewith3Drigidity. Proof. (ofCorollary 4.11.1 ).Whenagraphhasnonucleation,SitharamandZhou's[ 94 ]algorithmreducesto(3,6)-sparsitycheck.Forabraceddouble-ring,thegraphwillbedeclaredmodule-rigid.Ontheotherhand,fromTheorem 4.11 ,weknowadouble-ringisdependent,thusthebraceddouble-ringisalsodependent.Togetherwiththefactthatthebraceddouble-ringhasminimumnumberofedgestoberigid,weknowitsrank(whichissmallerthanitssizeduetodependence)cannotbeenoughtoberigid. 4.5ConclusionsandOpenProblemsInthischapterweprovidedgeneralinductiveconstructionschemesfornucleation-free(independent)graphswithimpliednon-edges,nucleation-freedependentgraphsandnucleation-freecircuits.Besidessettlingproblemsposedpreviouslyintheliterature,thisworkextendstherepertoireofusefulexamplesthatelucidatetheobstructionstoobtainingcombinatorialcharacterizationsof3Drigidity.Wehaveprovidedtwoprooftechniquesforshowingimpliednon-edgesinnucleation-freegraphs.Table 4-1 sumsuptheconstructionswehavepresentedinthischapter,andtheirassociatedprooftechniques.Oneopenproblemtoextendtheapplicationoftherstprooftechniqueistondothergraphsthatsatisfytheexpansion/contractionproperty.Anotherinterestingopen 97

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problemtoextendtheapplicationofourrank-sandwichprooftechniqueistondotherconstructionschemesforindependentgraphsandothertechniquesbesides2-thincoverargument/body-hingeargumenttoproverankupperbounds.Tocompleteourunderstandingofnucleation-freegraphswithimpliednon-edges,thenextstepistostudyexamplesextendingthoseinObservation 4.3 thatcannotbeobtainedbyanyofourconstructionschemes.Anotherinterestingopenproblemistoextendourinductiveconstructionforindependentgraphstoaninductiveconstructionforisostaticgraphs(independentandminimallyrigid).Inordertodothat,weneedtoaddtwomoreedgesintheroofpastingstep.Onepossiblewayistoaddtwomoreedges(c0,a1),(c,b2)(or(c0,b1),(c,a2)).WenotethatourcurrentmethodtoshowindependenceinTheorem 4.8 wouldfail,sinceClaim 4.8.3 failsinthatthereisanon-zerostressontheaddedpart.However,ifwecanshowthatifthereisagenericcircuitinthenewgraphG,thenthereisonef(w1,t1),...(wn,tn)gthatremainsadependenceforthenon-genericpositionpusedinTheorem 4.8 ,i.e.,thereexistsnon-zerostressesfs1,...,sngs.t.Pisi(p(wi))]TJ /F9 11.955 Tf 12.03 0 Td[(p(ti))=0,thenwecansimplyuseClaim 4.8.2 tocompleteourproof. 98

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CHAPTER5GEM(GRADEDEXCHANGEMAXWELL)-MATROIDInthischapter,wedeneanewmatroidtomoveclosertowardscombinatoriallycapturingrigidityin3dimensions.InSection 5.1 ,wegivemotivation,severalexamplesandthenthedenitionofGEM-closure.InSection 5.2 welistourresultsthatweresketchedinSection 2.2.3 andillustratedinTable 2-1 .Inparticular,weshowinSection 5.2.1 thattheGEM-closureiscomputableforanyedgeset.InSection 5.2.2 andSection 5.2.3 weshowthatGEM-independenceactuallydenesamatroidandalsoanabstractrigiditymatroid.InSection 5.2.4 weshowthatforanygraphG,everymaximalGEM-independentsetGhassizeatmosttheminimum-sizedmaximalMaxwell-independentsetofG. 5.1Motivation,ExamplesandDenitionsFornucleation-free,dependentgraphsmentionedinChapter 4 ,therecursiveversionofMaxwell-independenceproposedin[ 94 ]willfailtodetectdependence,thusindicatingtheexistenceofpropertiesofindependenceinthegenericrigiditymatroidthat[ 94 ]failstocapture.Oneofthemostsignicantpropertiesofindependenceinthegenericrigiditymatroidisthatindependentsetsdeneamatroid,whilethereisnoguaranteethat[ 94 ]givesamatroid.Intheremainderofthischapter,wecalltheclosureoperatordenedbythegenericrigiditymatroidasthegenericrigidityclosure.Wereferthereaderagaintotheaxioms(seeDenition 2.1 )thataclosureoperatormustsatisfyinordertodeneamatroid.Theclosureoperatorweseekshouldbedenedonlybasedongraphproperties,i.e.,vertexsetandedgesetofthegraph,notincludinganyalgebraicpropertiesassociatedwithgenericframeworksofthegraph,whichthegenericrigidityclosurerequires. 99

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Thefollowing2axiomsdeneapropertyofmatroidscalledtheabstractrigiditymatroidproperty([ 35 ]),formulatedtoexpressakeypropertyofthegenericrigiditymatroidinpurelycombinatorialterms. Denition5.1. LetVbeanitesetanddbeapositiveinteger.LetKdenotetheedgesofthecompleteundirectedgraphonV.AmatroidonK(seeDenition 2.1 )withclosureoperator<>iscalledand-dimensionalabstractrigiditymatroidforVifitsatisestheadditionalconditions: C5 IfE,FK,andjV(E)\V(F)j(K(V(E))[K(V(F))). C6 IfE,FKarerigidandjV(E)\V(F)jd,thenisrigid.Intuitively,axiomC5meansiftwoedgesetsEandFhavelessthandcommonendpoints,thentheclosureoftheiruniondoesnothaveanyedge(u,v)suchthatu2V(E)andv2V(F).AxiomC6meansiftworigidedgesetsEandFhaveatleastdcommonendpoints,thentheirunionisalsorigid.Theclosureoperatorweseekshouldsatisfytheabstractrigiditymatroidaxioms,sincethenwecouldeithergiveacounterexampletotheMaximalconjecturein[ 35 ](seeSection 2.2.3 ),orshowthatindependenceinourmatroidismorerestrictivethanindependenceinthegenericrigiditymatroid.If,inaddition,wecanshowthatanindependentsetinthegenericrigiditymatroidisalsoindependentinthematroidweseek,thenindependenceinourmatroidisatmostasrestrictiveasindependenceinthegenericrigiditymatroid,thuscompletingacombinatorialcharacterizationofthegenericrigiditymatroid.Beforewedeneourclosureoperator,weprovideafewmotivatingexamples.Therstexampleisaclassicnucleation-freegraphwithimpliednon-edges,i.e.,ringR7of7roofs.RecallthatinChapter 4 ,weshowedthatallhingenon-edgesinR7areimplied.I.e.,thegenericrigidityclosureofR7containsallthehingenon-edges.ThusforR7,ourclosureshouldalsocontainthehingenon-edges.NoticingthataxiomC4(exchangeaxiom)isanimportantaxiomcharacterizingmatroids,wecandesignourclosureonan 100

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edgesetEtohavetheabilitytoexchangeanon-edgefwithanexistingedgeeande2[ffg[Enfeg]canbeveriedinsomerecursivefashionwiththebasecaserelyingonanucleationpropagation,asin[ 94 ].ForR7,wecouldeasilyexchangeahingenon-edgewithanyexistingedge,causinganucleation(seeFigure 5-1 ).Noticingthatin3-dimensions,thesmallestnucleationwithimpliednon-edgesisabanana,i.e.,aK5withanedgemissing,thenucleationpropagationinthecaseofR7actuallyutilizestrulyrigidsubgraphs.Similarly,inourclosureoperator,wewouldwantrigidnucleations(inthegenericrigiditymatroid)inthenucleationpropagation.Sinceabanana,consistingof9edges,istherigidsubgraphwithfewestnumberofedgesthathasanimpliednon-edge,weincorporategraphsofatleast9edgesastherstnon-trivialcaseafterthebasecaseinourclosuredenition(Denition 5.2 ). AE B C DFigure5-1. AgureshowinghowtoobtainourclosureinR7,aringof7roofs.FigureAisaringR7whileFigureBisitsgenericrigidityclosure.WewantourclosureonAtobethesamegraphasB.Todothat,weexchangeanon-edgewithanedge,asdrawninFigureCandthenaddedgesbynucleationpropagationasin[ 94 ]:whenasubgraphisaK5with1edgemissing(thissubgraphisactuallyknowntoberigid),thenweaddthemissingedge.TheresultinggraphisFigureD. 101

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InChapter 4 ,wealsoconstructedothernucleation-freegraphswithimpliednon-edges.OneexampleisFigure 4-13 ,wheretherearefourrings.Asingleexchangeasintheringofroofswillnotworkonthatgraphsincenomatterwhichedge/non-edgepairwechoosetoexchange,thechosenedgecannotbeobtainedwiththenon-edgeaddedusingnucleationpropagationasin[ 94 ].Thisfailureisduetothefactthatonlyonepaircanbeexchangedatatimeforagivengraph,andtoxthat,inourclosure(Denition 5.2 below),wehavearecursivewayofaddingedgestoallowmultiplenon-edgestobeexchangedwithedges.Hence,ourclosureshouldbedenedinanestedstages/grades/levelsthat(1)multiplenon-edgescanbeexchangedwithedgesofthegraph,ortheclosureofthegraphonthepreviousstage/grade/level;and(2)ateachstageonlyonepairofedge/non-edgeisexchangedsothatpotentiallyanefcientalgorithmcalculatingtheclosurecanbefound.Intheremainderofthissection,wegivethedenitionandseveralexamples.Note:Inthischapter,wheretheprimaryfocusisonedgesets,weuseV(E)todenotethesetofendpointsofanedgesetE,anduseK(V(E))todenotethecompletegraphonV(E)(weuseKntodenoteacompletegraphonnvertices). Denition5.2. GivenanedgesetE,theGEM-closure(GradedExchangeMaxwell-closure)ofE,denoted[E],isdenedtobeaunionoverlevelsk(k8)andgradesm(m1)Sk,m[E]km,where[E]kmisdenedasfollows: 1. Unionovergradem.Foranylevelk,[E]kisdenedasSm[E]km. 2. Basecasefork=8.[E]8m:=Eforallm0. 3. Closureforlevelk=9,gradem=1.[E]91iscalledtheclosureofE(forlevel9,grade1),whichisdenedasfollows: (a) Singlestageinlevelk=9,gradem=1.WesayEyieldsanedgeeinasinglestage,denotedE!91eore21[E]91,ifeithere2[E]8,orthereexistsE0Es.t. 102

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i. 8f2E0,f62[E0nffg]8 ii. jE0j=3jV(E0)j)]TJ /F4 11.955 Tf 17.93 0 Td[(6,jE0j9; iii. e2K(V(E0)) (b) Mutliplestagesinlevelk=9,gradem=1.Forj>1,wesaye2j[E]91ifthereexistsE0j)]TJ /F6 7.97 Tf 6.59 0 Td[(1[E]91,s.t. i. 8f2E0,f62[E0nffg]8 ii. jE0j=3jV(E0)j)]TJ /F4 11.955 Tf 17.93 0 Td[(6,jE0j9; iii. e2K(V(E0))WesayEimpliesanedgeeinmultiplestages,denotedE!9,?1e,ifthereexistsj1,s.t.e2j[E]91.I.e.,mpliesisthetransitiveclosureofyields. (c) Theclosureoverstages,denoted[E]91,isdenedtobefejE!9,?1eg. 4. Closureforlevelk,gradem=1.Fork>9,[E]k1:=[E]k)]TJ /F6 7.97 Tf 6.58 0 Td[(1. 5. Closureforlevelk,gradem(recursiononm).Form2,k9,[E]kmiscalledtheclosureofE(forlevelk,gradem),whichisdenedasfollows: (a) Singlestageinlevelk,gradem.WesayEyieldsanedgeeinasinglestage,denotedE!kme,ore21[E]km,ifeithere2[E]km)]TJ /F6 7.97 Tf 6.59 0 Td[(1,orthereexistsE0Es.t.jE0j3jV(E0)j)]TJ /F4 11.955 Tf 17.93 0 Td[(6,jE0jkandthefollowingholds: i. 8f2E0,f62[E0nffg]km)]TJ /F6 7.97 Tf 6.58 0 Td[(1 ii. 9g2E0,s.t.g2[feg[E0nfgg]km)]TJ /F6 7.97 Tf 6.59 0 Td[(1 (b) Multiplestagesinlevelk,gradem.Forj>1,wesaye2j[E]91ifthereexistsE0j)]TJ /F6 7.97 Tf 6.58 0 Td[(1[E]km,s.t.jE0j3jV(E0)j)]TJ /F4 11.955 Tf 17.93 0 Td[(6,jE0jkandthefollowingholds: i. 8f2E0,f62[E0nffg]km)]TJ /F6 7.97 Tf 6.58 0 Td[(1 ii. 9g2E0,s.t.g2[feg[E0nfgg]km)]TJ /F6 7.97 Tf 6.58 0 Td[(1WesayEimpliesinlevelk,grademanedgeeinmultiplestages,denotedE!k,?me,ifthereexistsj1,s.t.e2j[E]km.I.e.,impliesisthetransitiveclosureofyields. (c) Theclosureoverstages,denoted[E]km,isdenedtobefejE!k,?meg.ThisdenitionofGEM-closurecanalsobewrittenasanalgorithminpseudocode.Table 5-1 describeshowwendinarecursivewaytheGEM-closureofanedgesetE.Forexample,Figure 5-2 showshowtoobtainclosureforlevelk,grade1.TheaforementionedFigure 5-1 showshowtoobtainclosureforlevelk,grade2.Alldashed 103

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Table5-1. IllustrationoflevelsandgradesintheGEM-closure(GradedExchangeMaxwell-closure)ofanedgesetE. XXXXXXXXXXXXXXXLevelkGradem m=1 ... m)]TJ /F29 14.346 Tf 14.35 0 Td[(1 m ... Sm k=8 E ... E E ... [E]8:=E k=9 [E]91!k,?2 ... [E]9m)]TJ /F31 9.963 Tf 7.75 0 Td[(1!9,?m [E]9m ... [E]9:=Sm[E]9m ... ... ... ... ... ... ... k [E]k)]TJ /F31 9.963 Tf 7.75 0 Td[(1!k,?2 ... [E]km)]TJ /F31 9.963 Tf 7.75 0 Td[(1!k,?m [E]km ... [E]k:=Sm[E]km ... ... ... ... ... ... ... Sk [E] [E] ... [E] ... [E] 104

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AE BClosureforEinlevel9,grade1Figure5-2. Agureshowinghowtoobtainclosureforlevel9,grade1.ThedashedlinesinFigureBrepresentedgesaddedintheclosureoftheFigureAinlevel9,grade1. AE BE[f(a,b)gnf(c,d)gFigure5-3. AgureshowingwhytheGEM-closureonanoctahedralgraphisacompletegraph.FigureBisyieldedbyFigureAinlevel9,grade2sinceedge(c,d)2[E[f(a,b)gnf(c,d)g]91andFigureBisapartial3-tree,whoseclosureforlevel9,grade1isacompletegraph.Thus[E]92isacompletegraphon6vertices. edgesFigure 4-13 theclosureofthegraphforlevelk,grade3.Figure 5-3 showstheGEM-closureofanoctahedralgraph.Figure 5-4 showstheGEM-closureofanicosahedralgraph.Figure 5-5 showstheGEM-closureofaringof6roofs.Note:ourGEM-closuredoesnotexplicitlyenforceamaximumnumberonm(althoughk)]TJ /F10 7.97 Tf 5.48 -4.38 Td[(jV(E)j2).Itisnotnecessaryforatleastoneedgetobeaddedtotheclosureineverygrade.ConsideranyoftheexamplesinFigures 5-1 5-3 5-4 and 5-5 wheretherearenoedgesaddedtotheGEM-closureingrade1.Thusitisnecessary 105

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Figure5-4. AgureshowingwhytheGEM-closureonanicosahedralgraphisacompletegraph.SinceFigureBisyieldedbyFigureAinlevel9,grade4,since(v1,v2)2[E[f(v1,v8)gnf(v1,v2)g]93,whichinturnisdueto(v2,v5)2[E[f(v1,v8),(v4,v6)gnf(v1,v2),(v2,v5)g]92and(v9,v10)2[E[f(v1,v8),(v4,v6),(v7,v8)gnf(v1,v2),(v2,v5),(v9,v10)g]91.FigureDisapartial3-tree,thustheclosureofEisacompletegraph. 106

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AE B C D E F GFigure5-5. AgureshowingwhytheGEM-closureonaringof6roofsisacompletegraph.StartingfromFigureA,wecanapplyGEM-closureforlevelk,grade7andthenFigureBthroughFwillappearintheGEM-closureofFigureA.FigureFisapartial3-tree,whoseGEM-closureforlevel9,grade1isguaranteedtobeacompletegraph. toshowthatGEM-closureprocessisniteandGEM-closureiscomputable,whichisprovedinSection 5.2.1 .Note2:intheremainderofthischapter,wesayE!meifE!kmeandkisclearfromthecontext. Denition5.3. AnedgesetEisGEM-independentifforalle2E,e62[Enfeg].WesayEisGEM-rigidif[E]isthecompletegraphonV(E). 107

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5.2TheoremsInthissection,welistandshowourresultsforGEM-closure.InSection 5.2.1 ,weshowthatGEM-closureiscomputable.InSection 5.2.2 ,weshowthatGEM-independencesatisesallfouraxiomsneededforanotiontodeneamatroid.InSection 5.2.3 ,weshowGEM-independenceinadditionsatisesthetwoextraaxiomsforamatroidtobeanabstractrigiditymatroid.InSection 5.2.4 ,werelateMaxwell-independencewithGEM-independence. 5.2.1GEM-ClosureIsComputableInthissection,weshowthatmembershipofanedgeintheGEM-closureisdecidableinthefollowingtheorem. Theorem5.1. GivenanedgesetE,[E]canbecomputedafternitelymanysteps,i.e.,GEM-Closureiscomputable. Proof. Wewillshowthefollowingclaims: Claim5.1.1. Foranykandm,thenumberofstagesisnomorethanthenumberofedgesin[E]km. Proof. Thisfollowsdirectlyfromthedenitionofstages(Denition 5.2 )oftheGEM-closure. Claim5.1.2. Foranylevelk,ife2[E]kmn[E]km)]TJ /F6 7.97 Tf 6.58 0 Td[(1,thenthegrademisnite. Proof. Frome2[E]m,weknowthereexistsE0[E]km)]TJ /F6 7.97 Tf 6.58 0 Td[(1s.t.jE0j3jV0j)]TJ /F4 11.955 Tf 17.93 0 Td[(6and 1. 8f2E0,f62[E0nffg]km)]TJ /F6 7.97 Tf 6.59 0 Td[(1 2. 9g2E0,s.t.g2[e[E0nfgg]km)]TJ /F6 7.97 Tf 6.58 0 Td[(1Sincee62[E]km)]TJ /F6 7.97 Tf 6.58 0 Td[(1,weknow9g2E0,s.t.g62[e[E0nfgg]km)]TJ /F6 7.97 Tf 6.58 0 Td[(2.DenoteE0nfggbyE1.Lete1:=eande2:=g,thenwehave:(1)fe1g[E1!k,?m)]TJ /F6 7.97 Tf 6.58 0 Td[(1e2,and(2)fe1g[E19k,?m)]TJ /F6 7.97 Tf 6.59 0 Td[(2e2. 108

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Frome22[fe1g[E1]km)]TJ /F6 7.97 Tf 6.59 0 Td[(1n[fe1g[E1]km)]TJ /F6 7.97 Tf 6.59 0 Td[(2,weknow,thereexistsE2ande3,s.t.(1)fe2g[E2!k,?m)]TJ /F6 7.97 Tf 6.59 0 Td[(2e3,and(2)fe2g[E29k,?m)]TJ /F6 7.97 Tf 6.59 0 Td[(3e3.Similarly,wehaveasetofmedgesfe1,e2,...,emg,andacollectionofm)]TJ /F4 11.955 Tf 12.3 0 Td[(1edgesetsfE1,E2,...,Em)]TJ /F6 7.97 Tf 6.58 0 Td[(1g,s.t.foranyeiandEi(1im)]TJ /F4 11.955 Tf 12.13 0 Td[(1),wehave(1)feig[Ei!k,?m)]TJ /F14 7.97 Tf 6.59 0 Td[(iei+1,and(2)feig[Ei9k,?m)]TJ /F14 7.97 Tf 6.59 0 Td[(i)]TJ /F6 7.97 Tf 6.58 0 Td[(1ei+1.Akeyobservationisthatifwetakeanorderingofei,Eiandei+1,thenallm)]TJ /F4 11.955 Tf 12.76 0 Td[(1orderingsaredifferent.Otherwisewehavetwoorderingthatarethesameandwouldbeobviouscontradiction.Moreover,weknowthesizeofeachEiisk)]TJ /F4 11.955 Tf 12.64 0 Td[(1,otherwisebyGEM-closure,oneoftheorderings'property(2)doesnothold.I.e.,ifEiistherstedgesetthathassizelessthank)]TJ /F4 11.955 Tf 12.28 0 Td[(1,thenfeig[Ei!k,?m)]TJ /F14 7.97 Tf 6.59 0 Td[(iei+1actuallybecomesfeig[Ei!?,k)]TJ /F6 7.97 Tf 6.59 0 Td[(1m)]TJ /F14 7.97 Tf 6.58 0 Td[(iei+1,whichcontradictsthepropertyfeig[Ei9k,?m)]TJ /F14 7.97 Tf 6.59 0 Td[(i)]TJ /F6 7.97 Tf 6.59 0 Td[(1ei+1.Thusthemaximumnumberoforderingsis)]TJ /F10 7.97 Tf 5.48 -4.37 Td[(jVj2()]TJ /F10 7.97 Tf 5.48 -4.37 Td[(jVj2)]TJ /F4 11.955 Tf 9.29 0 Td[(1))]TJ /F4 11.955 Tf 5.48 -4.86 Td[((jVj2))]TJ /F6 7.97 Tf 6.58 0 Td[(2k)]TJ /F6 7.97 Tf 6.59 0 Td[(1,whereVisthesetofendpointsofE,andthatisanitenumber.Sincethemaximumnumberofmisatmostthemaximumnumberoforderingsplusone,weknowmisanitenumber.Notethatwhenmachievesmaximumnumber,theclosureofEwillbeVVandwecanstoptheclosureprocessimmediately. CombiningClaims 5.1.1 and 5.1.2 andnoticingthatk)]TJ /F10 7.97 Tf 5.48 -4.38 Td[(jV(E)j2,weobtainthattheGEM-closureprocessiscomputableforeveryedgesetE. 5.2.2GEM-ClosureDenesaMatroid Theorem5.2. GEM-closuresatisesthematroidpropertyC1throughC3. Proof. C1 Bydenition. C2 WhencalculatingtheclosureofT,Risalwaysavailableanditsclosurewillbeincludedin[T]. 109

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C3 Bydenition,[T][[T]].Suppose[T]=[T]km.Thenweknow[T]k+qm+p=[T]kmforallp,q1.Supposefurther[[T]]=[[T]]k0m0.Thenbydenition,[[T]]=[T]k+k0m+m0,whichinturnisequalto[T]km=[T]. Next,weshowtheexchangeaxiomC4tocompletetheproofthatGEM-closuredenesamatroidonK(V(E)). Theorem5.3. Ife2[E[ffg]ande62[E],thenf2[E[feg]. Proof. Supposee2[E[ffg]bute62[E]andf62[E],andwewanttoshowf2[E[feg].Morespecically,assumee2[E[ffg]km.Thenwecanshowthatf2[E[e]km+1.AllweneedistondsomeGEM-independentsetE0[fgg[E[feg]km,s.t.g2[E0[ffg]km.Forthat,wecaneasilypickE0:=Eandg:=e,andtherestfollows. Note:GEM-closurehasbeendenedsothattheproofofC4followsdirectlyfromthedenition.SinceGEM-closuredenesamatroid,wecallthesizeofeverymaximalGEM-independentsetofanygivengraphGtheGEM-rankofG.Note2:SinceGEM-closuredenesamatroid,weknowanymaximalGEM-independentsetof[E]hassizeatmostjEj.ThusinordertocalculatetheGEM-closureofanedgesetE,wedonotneedtocalculatek>jEj. 5.2.3GEM-ClosureDenesanAbstractRigidityMatroidInthissection,weshowthatGEM-closuredenesanabstractrigiditymatroid.Recallthatweneedtoshow2moreaxiomsC5andC6.Theorem 5.4 showsC6andTheorem 5.5 usesClaim 5.5.1 toshowC5.Weadditionallyshow2corollariesconcerninginductiveconstructionofindependentandminimallyrigidgraphsbyHenneberg-IconstructionsandtheGEM-rankofcompletegraphs.First,weshowaxiomC6forGEM-closure. Theorem5.4. GivenGEM-rigidgraphsG1andG2,ifG0:=G1\G2hasatleast3vertices,then[G1[G2]isalsoGEM-rigid. 110

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Proof. LetG1:=(V1,E1),G2:=(V2,E2)andG0:=(V0,E0).ItissufcienttoshowthatthereexistanedgesetEin[G1[G2]where[E]=K(V1[V2).SinceG1andG2arebothGEM-rigid,weknow[E1]=K(V1)and[E2]=K(V2).WecanndanedgesetD1inK(V1)inthefollowingway:(1)thereexiststhreeverticesu,v,w2V0s.t.f(u,v),(u,w),(v,w)gD1and(2)foreveryothervertexx2V1,f(x,v),(x,v),(x,w)gD1.WecandothesametondanedgesetD2inK(V2).NowconsiderE:=D1[D2.ForanyedgeeinK(V1[V2),eitherealreadyinEoreisapartofaK5whoseother9edgesarealreadyinE.Thus[E]91=K(V1[V2). Next,weshowaxiomC5forGEM-closure. Theorem5.5. IfE,FK,andjV(E)\V(F)j<3,then[E[F](K(V(E))[K(V(F))). Proof. OfTheorem 5.5 .Werstshowthefollowingclaim. Claim5.5.1. Foranym2N,ifE!mebut(1)e62[E]m)]TJ /F6 7.97 Tf 6.58 0 Td[(1and(2)foranyf2E,e62[Enffg],thenthereexistsawitnesscollectionC:E!m)]TJ /F6 7.97 Tf 6.59 0 Td[(1E1!E2!...!es.t.foralledgeshin[E]m)]TJ /F6 7.97 Tf 6.59 0 Td[(1thatareinE(C)(theedgesinthewitnesscollectionC),thefollowingholds:betweenanyoftheendpointsu,vofhandanyoftheendpointsw,xofe,thereareatleast3vertex-disjointpathsin[E]m)]TJ /F6 7.97 Tf 6.59 0 Td[(1\E(C). Proof. Werstshowthecasewherek=9.Thebasecaseism=1.Thiscanbedoneasfollows.WhenE!91e,weknowE[eisaK5andtheclaimcertainlyholds.Nextwedoaninductiononm.Notethattheargumentholdsforanykiftheclaimsholdsform=1.Fortheinductionhypothesis,weassumethatforalledgesetsEandanyn
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BytheGEM-closuredenition,fromE!me,weknowthereexistsfgg[E0[E]m)]TJ /F6 7.97 Tf 6.58 0 Td[(1s.t.g62[E0]m)]TJ /F6 7.97 Tf 6.59 0 Td[(1butg2feg[E0.NowdenotebySjgi,jthesetofalledgesthatareintheithstageoffeg[E0inlevelk,gradem)]TJ /F4 11.955 Tf 12.4 0 Td[(1andarewitnessedgesofe.Wecallthesetofallthoseedgesparticipatingedges.Fromtheinductionhypothesis,weknowwecanchooseacollectionCforallparticipatingedges,s.t.ifgi,j1,gi,j2,...,gi,jlwereneededtoimplygi+1,k,thenthereareatleast3vertex-disjointpathsin[E]m)]TJ /F6 7.97 Tf 6.58 0 Td[(1betweentheendpointsofanyedgeinfgi,j1,gi,j2,...,gi,jlgandtheendpointsofgi+1,k.NextwewillshowthatfromcollectionC,wecanndawitnesscollectionC0ofe,s.t.thereareatleast3vertex-disjointpathsbetweentheendpointsofeandallparticipatingedgesh2[E]m)]TJ /F6 7.97 Tf 6.59 0 Td[(1\C0.Fromtheinductionhypothesis,weknowtheonlypossible2-separatorsaretheendpointsofsomegi,j,sinceallotherpairswillnotseparategi,jwithparticipatinggi)]TJ /F6 7.97 Tf 6.58 0 Td[(1,k.Therearetwocases: Case1 .IfinC,thereisno2-separator.ThenPistherequiredcollection. Case2 .Supposethereexistssomegi,joncollectionCs.t.(a)theendpointsofgi,jisa2-separatorofeandsomeotherparticipatingedgegi,t2[E]m)]TJ /F6 7.97 Tf 6.59 0 Td[(1,and(b)iisminimum,thenweknow(1)foralli9.Itshouldbenotedthatfork>9,[E]k1isequalto[E]k)]TJ /F6 7.97 Tf 6.58 0 Td[(1,thusthiscaseisequaltothemaximumms.t.[E]k)]TJ /F6 7.97 Tf 6.58 0 Td[(1mn[E]k)]TJ /F6 7.97 Tf 6.59 0 Td[(1m)]TJ /F6 7.97 Tf 6.58 0 Td[(1=6;.Form>1,wecanusetheaboveargumentintheinduction. NowwereturntoTheorem 5.5 .Weshowthetheoremontwolevelsofinduction.Therstinductionisonk,andthesecondinductionisonmwhilekisxed. 112

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Firstofall,thetheoremclearlyholdsforthecaseofk=8,wheretheclosureofanedgesetforlevel8isitself.Nextwesupposethetheoremholdsforlevellwherel
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showthatE[f(u,v),(u,w)gisGEM-independentandthenE[f(u,v),(u,w),(u,x)gisalsoGEM-independent. Corollary5.5.2. IfK=(V,E)isacompletegraph,thenitsGEM-rankisequalto3jVj)]TJ /F4 11.955 Tf 17.93 0 Td[(6. Proof. StartingfromK5,whichhasGEM-rank9bydenition,wecanapplyHenneberg-IconstructionstoobtainaGEM-independentsetofsize3jVj)]TJ /F4 11.955 Tf 17.08 0 Td[(6foreverycompletegraphK=(V,E). 5.2.4GEM-IndependenceV.S.Maxwell-IndependenceFromDenition 5.2 ,itisclearthateveryGEM-independentsetisMaxwell-independent.Wecanfollowtheproofof[ 13 14 ]toshowthatforanygraphG,everymaximalMaxwell-independentsetofGhassizeatleastGEMrankofG. Theorem5.6. LetMbeamaximalMaxwell-independentsubgraphofagraphG=(V,E)andDbeamaximalGEM-independentsetofG.ThenjE(M)jjDj,whereE(M)denotestheedgesetofM. Proof. (ofTheorem 5.6 )First,noticethatifMitselfisGEM-independent,wearedone.Similarly,ifMisMaxwell-rigid,thenwehavejE(M)j=3jVj)]TJ /F4 11.955 Tf 18.75 0 Td[(6jDj,hencewearedone.LetDMwithjDMj=rank(M)beamaximalGEM-independentsetofM.Withoutlossofgenerality,letDMD.LetA:=DnDM.Thus[E(M)]\A=;.Here[E(M)]meanstheGEM-closureofE(M).ConsideracoverX=fe1,...,ek,M1,M2,...,MmgofMbythecompletecollectionofvertex-maximalcomponents,wheree1,...,ekareedgecomponentsandM1,M2,...,Mmarenon-trivialcomponents.FollowingtheproofofProposition1in[ 13 ],itiseasytoshowthatforeachedge(u,v)inA,thereexistsatleastonenon-trivialcomponentMisuchthatu2Miandv2Mi.DenotebyAithesetofedgesofAboth 114

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ofwhoseendpointsareinMi.Hence jAjmXi=1jAij(5)TakeH(X)andneasdenedinSection2of[ 13 ].Weget jE(M)j=kXi=11+mXi=1jE(Mi)j)]TJ /F15 11.955 Tf 24.2 11.36 Td[(Xe2H(X)(ne)]TJ /F4 11.955 Tf 11.55 0 Td[(1)=k+mXi=1jE(Mi)j)]TJ /F15 11.955 Tf 24.2 11.36 Td[(Xe2H(X)(ne)]TJ /F4 11.955 Tf 11.55 0 Td[(1)(5)SinceeachMiisMaxwell-rigid,addinganye2AiintoMicausesthenumberofedgesinMitoexceed3jV(Mi)j)]TJ /F4 11.955 Tf 20.73 0 Td[(6andinturnindicatestheexistenceofaGEM-dependence.However,Ai\[Mi]=;,since[E(M)]\Ai=;.ItfollowsthatMiwasalreadyGEM-dependentevenbeforeAiwasadded.I.e.,toobtainaGEM-independentsetinMi,atleastjAijedgesmustberemovedfromMi.Sowehave jE(Mi)jGEM-rank(Mi)+jAij(5)Plugging( 5 )into( 5 ),wehave jE(M)jmXi=1GEM-rank(Mi))]TJ /F15 11.955 Tf 19.03 11.36 Td[(Xe2H(X)(ne)]TJ /F4 11.955 Tf 11.95 0 Td[(1)+mXi=1jAij+k(5)FromProposition1of[ 13 ],weknowthatthecoverXis2-thin.ThenwecanfollowtheproofofTheorem2of[ 13 ],andobtainthefollowing: mXi=1GEM-rank(Mi))]TJ /F15 11.955 Tf 19.03 11.36 Td[(Xe2H(X)(ne)]TJ /F4 11.955 Tf 11.95 0 Td[(1)+kGEM-rank(M)=jDMj.(5)Then,using( 5 )and( 5 ),weobtainthatjE(M)jjDMj+mXi=1jAij(using( 5 )and( 5 ))jDMj+jAj(using( 5 ))=jDj,whichprovesTheorem 5.6 115

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Note:theproofin[ 13 ]onlyusesthefactthat(1)independentsetsinthegenericrigiditymatroidactuallydeneamatroidand(2)everydependentsetinthegenericrigiditymatroidhasatleast10edgesand(3)themaximumsizeofanyindependentsetinthegenericrigiditymatroidisatmost3jVj)]TJ /F4 11.955 Tf 19.13 0 Td[(6.ItiseasytoseethatGEM-independencesatisesallthosethreeconditionsandhencetheprooffollows. 5.3OpenProblemsInSection 2.2.3 ,wementionedtheMaximalConjecturebyGraveretal[ 35 ]whichstatesthateveryabstractrigiditymatroidAin3DandhigherdimensionsismorerestrictivethanthegenericrigiditymatroidM.I.e.,everyindependentsetinAisindependentinM.Ifthatconjectureistrue,thentogetherwithresultthatGEM-independencedenesanabstractrigiditymatroid(Section 5.2.3 ),wehavethefollowing: Observation5.1. IftheMaximalConjecturein[ 35 ]holds,thenGEM-independenceimpliesindependence.Sofar,wedonothaveanexamplegraphthatisGEM-independentbutdependent.Aninterestingopenproblemistondadependentgraphthatafteraddingallimpliednon-edges,thereisstillnonucleus.Ifsuchagraphexists,thentheMaximalConjectureisfalse.InSection 5.2.4 ,weshowedthatMaxwell-independentsetsareallGEM-independent,andthusGEM-rankispotentiallyabetterrankupperboundontherankofthegenericrigiditymatroid.Thefollowingconjectureiscrucialforthisupperboundtohold: Conjecture5.1. IfEisindependent,thenEisGEM-independent.Wehavethefollowingobservation: Observation5.2. Conjecture 5.1 isequivalenttothefollowingstatement:ifEisindependent,then[E]isasubsetofthegenericrigidityclosureofE. 116

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Proof. (1)Theforwarddirection.SupposeConjecture 5.1 holdsandthereisanedgesetEthatisindependentandGEM-independent,but[E]containsanedgefthatisnotinthegenericrigidityclosureofE.ItfollowsthatE[ffgisindependentandthusf62E.Togetherwithf2[E],weknowE[ffgisGEM-dependent,acontradictiontothefactthatE[ffgisindependent.(2)Thereversedirection.SupposeforanyedgesetE,ifEisindependent,then[E]isasubsetofthegenericrigidityclosureofE.AssumefurtherthatEisindependentbutGEM-dependent.ThenweknowthereisamaximalGEM-independentF(Es.t.E[F].I.e.,foranye2EnF,e2[F].Since[F]isasubsetofthegenericrigidityclosureofF,weknowforanye2EnF,eisinthegenericrigidityclosureofF,acontradictiontotheassumptionthatEisindependent. WithConjecture 5.1 ,foranygraphG,GEM-rankofGisabetterupperboundonrank3(G)thanthesizeofamaximalMaxwell-independentsetofG.TogetherwithObservation 5.1 ,wehavethefollowing: Observation5.3. (i) IfConjecture 5.1 andtheMaximalConjecturein[ 35 ]arebothtrue,thentheGEM-matroidisacombinatorialcharacterizationofgenericrigidityin3D. (ii) Equivalently,ifConjecture 5.1 istrue,theneithertheMaximalConjecturein[ 35 ]isfalse,ortheGEM-matroidisacombinatorialcharacterizationofgenericrigidityin3D. 117

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CHAPTER6OTHERPARTIALRESULTSANDFURTHEROPENPROBLEMS 6.1ConvexCayleyCongurationSpaceandD-realizabilityInthissection,wegivearesultaboutaspecialparametrizedcongurationspace.First,wehavethefollowingdenition. Denition6.1. Givenlinkage(G,E),non-edgesetF,theCayleyCongurationspaceonFis dF(G,E):=fFj(G[F,E[F)hasarealizationind)]TJ /F8 11.955 Tf 11.95 0 Td[(spaceg(6)ThendG(;)denotesthesetofallrealizationsofGinEd.AsquaredCayleyCong-urationspace(2)dF(G,E)isthecongurationspaceofalinkageintermsofsquareddistances.In[ 8 ],theconceptofd-realizabilitywasintroduced: Denition6.2. AgraphG=(V,E)isd-realizableifforanyrealizationpofG,thereisacorrespondingrealizationqofGinEds.t.forany(u,v)2E,jjpu)]TJ /F9 11.955 Tf 11.96 0 Td[(pvjj2=jjqu)]TJ /F9 11.955 Tf 11.96 0 Td[(qvjj2.In[ 8 ],combinatorialcharacterizationsofd-realizablegraphshavebeengivenford2:treesare1-realizablegraphsandpartial2-treesare2-realizablegraphs.Ingeneral,agraphGisd-realizableifforanyrealizablelinkage(G,E),itscorrespondingassignmentmatrixcanbecompletedtoaEuclideandistancematrixwithrankatmostd+1.Alternatively,agraphGisd-realizableifandonlyifSi(2)iG(;)=(2)dG(;).Thefollowingtheoremrelates(2)dG(;)directlywiththed-realizabilityofG. Theorem6.1. AgraphGisd-realizableifanonlyif(2)dG(;)isconvex. Proof. Theonlyifpartcanbefoundin[ 89 ].Theifpartisnothardtoshow.Suppose(2)dG(;)convex,wewanttoshowthatS(2)iG(;)=(2)dG(;).Notethatconvexhullof1-rankstratumofEuclideandistancematrixcone(EDMC)givestheEuclideandistancematrixconeitself,sincewecanprojecttherealizationofanyEuclideandistancematrixDontothen-axisofEnandthen 118

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obtainn1-dimensionalrealizationsthatcorrespondston1-rankEuclideandistancematrices.ItiseasytoseethatDistheconvexcombinationofthen1-rankEuclideandistancematrices.Wealsoknowthattheconvexhullof1-rankstratumisinsidetheconvexhullofd+1-rankstratumofEDMC,thustheconvexhullofd+1-rankstratumofEDMCequalsEDMC.WeprojectthisstratumonG,whichis(Phi2)dG(;),andweknowtheprojectionofconvexobjectisalsoconvex.Hence(Phi2)dG(;)isconvex,andtheconvexhullof(2)dG(;)isitself.IfweswitchtheorderofconvexhullandprojectiononG,weknow(2)dG(;)equalstheprojectionofconvexhullofd+1-rankstratumofEDMConG,whichinturnequalsprojectionofEDMConG,i.e.,Si(2)iG(;). Itisnothardtoseethat(2)dG(;)isconvexifandonlyifforanylinkage(G,),andforanysectionofGintoH[F,(2)dH(H,F)isconvex.WecallgraphGsatisfyingthelatterconditionsadmitsauniversally,inherentlyconvexsquaredCayleycongurationspace.Aninterestingopenproblemrelatedtod-realizabilityandconvexCayleycongurationspaceisshowthatafteraddinggenericityinthedenitionofd-realizability,itstillholdsthatagraphisgenericallyd-realizableifanonlyif(2)dG(;)isconvex.AgraphGisgenericallyd-realizableifforallgenericedgeslinkages(G,),whenever(G,)isrealizable,(G,)isrealizableinddimensions.Accordingly,wecanaddgenericitytouniversallyinherentlyconvexsquaredCayleycongurationspaceofGind-dimensionalspace:ifforanygenericlinkage(G,),andforanysectionofGintoH[F,(2)dH(H,F)isconvex. 6.2ResultsaboutGlobalRigidityAsmentionedbefore,Hendricksonrst[ 36 ]gaveanecessaryconditionforagraphtobegloballyrigidinanydimension. Theorem6.2(Hendrickson). IfagraphGisgloballyrigidinddimensions,thenitis(d+1)-connectedandredundantlyrigid. 119

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AgraphGisredundantlyrigidifafterremovinganyedgeeofG,Gneisstillrigid.Hendricksonalsoshowedthathisconditionwassufcientin1Dand2D.Later,Connelly[ 20 ]gaveasufcientconditionforagraphtobegloballyrigidinanydimension. Theorem6.3(Connelly). AgraphGisgloballyrigidinddimensionsifithasagenericframeworkthathasastressmatrixofrankn)]TJ /F3 11.955 Tf 11.95 0 Td[(d)]TJ /F4 11.955 Tf 11.95 0 Td[(1.WewilldenoteConnelly'sconditionbyCC.CCwasprovedtobenecessarybyGortleretal[ 33 ].ThusHendrickson'scondition,HCforshort,shouldbeequivalenttoCCin1Dand2D.ItisinterestingtoseehowtoapplyCCtoshowHCissufcient. 6.2.1Hendrickson'sConditionin1DFirst,wewillapplyCCin1Dtoshowthatany2-connectedgraphisgloballyrigid.ItsufcestoshowthatthereexistsastressmatrixSforagenericframeworkofacycleofnverticessuchthatrank(S)=n)]TJ /F4 11.955 Tf 12.46 0 Td[(2.Letv1,v2,...,vnbethenverticesofthen-cycle,wherethenedgesare(vi,vi+1),1in)]TJ /F4 11.955 Tf 12.08 0 Td[(1and(vn,v1).Consideraconguration(notnecessarilygeneric)wheretheedgedistances(vi,vi+1),1in)]TJ /F4 11.955 Tf 12.24 0 Td[(1areassignedtobe1,andthedistance(vn,v1)isassignedtoben.ThenonestressmatrixSwillbethefollowing:266666666664n+1)]TJ /F3 11.955 Tf 9.3 0 Td[(n0...0)]TJ /F4 11.955 Tf 9.3 0 Td[(1)]TJ /F3 11.955 Tf 9.3 0 Td[(n2n)]TJ /F3 11.955 Tf 9.3 0 Td[(n0...0......0...0)]TJ /F3 11.955 Tf 9.29 0 Td[(n2n)]TJ /F3 11.955 Tf 9.3 0 Td[(n)]TJ /F4 11.955 Tf 9.3 0 Td[(10...0)]TJ /F3 11.955 Tf 9.3 0 Td[(nn+1377777777775Removingtherstrow,therstcolumn,thelastrowandthelastcolumn,wecanobtainthefollowingmatrixwhichhasfullrank: 120

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2666666666642n)]TJ /F3 11.955 Tf 9.3 0 Td[(n0...0)]TJ /F3 11.955 Tf 9.3 0 Td[(n2n)]TJ /F3 11.955 Tf 9.3 0 Td[(n0.........0...n2nn0...0)]TJ /F3 11.955 Tf 9.29 0 Td[(n2n377777777775ThusShasrankn)]TJ /F4 11.955 Tf 12.52 0 Td[(2,i.e.,forthen-cycle,thereexistsaframeworkwhichhasastressmatrixofrankequalton)]TJ /F3 11.955 Tf 11.03 0 Td[(d)]TJ /F4 11.955 Tf 11.03 0 Td[(1.Nextwewillapplythefollowingtheorem(Lemma5.8from[ 33 ]). Theorem6.4. GivengraphG,letUbethesetofframeworksofGthatachievethemaximalrankoftherigidmatrix.Lets0bethemaximalvalueofrankofstressmatrixoverallstressmatricesofallframeworksinU.Thenforallgenericframeworksp,themaximumachievablerankofthestressmatricesofG(p)isequaltos0.Sincen)]TJ /F4 11.955 Tf 12.99 0 Td[(2isalwaysamaximumachievablerankofanystressmatricesofaframeworkofn-cycle,inordertoapplyTheorem 6.4 ,weonlyneedtoshowthattheframeworkweconstructedaboveisinnitesimallyrigid.Thiscanbedonebyinspectingtherigiditymatrix.Thusitfollowsthatthereisastressmatrixofrankn)]TJ /F4 11.955 Tf 12.16 0 Td[(2foragenericframeworkofthen-cycle.UsingCC,wecouldobtainthatn-cycleisglobalrigid. 6.2.2OpenProblemsaboutGlobalRigidityOneinterestingproblemistoextendtheresultinChapter 6 ,Section 6.2.1 to2Dbyshowingthatevery3-connectedandredundantlyrigidgraphisgloballyrigidin2D.WecantrytoconstructaframeworkG(p)ofanygiven3-connectedandredundantlyrigidgraphG,andtheshowthatG(p)isinnitesimallyrigidwithastressmatrixofrankn)]TJ /F4 11.955 Tf 12.02 0 Td[(3.ThenwecouldapplyTheorem 6.4 toconcludethatGisgloballyrigid. 6.2.3DetectingtheInsufciencyofHendrickson'sConditionInthissection,wewilltrytoshowthatHendrickson'sconditionisinsufcientbyinspectingthestressmatrixofagenericframeworkof(d+1)-connectedandredundantlyrigidgraphinddimension.Notethatin[ 33 ],Section3.4,Gortleretal 121

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mentionedanapplicationoftheirtheoremforcertainbipartitegraphs,whichinspectsthestressmatrixofthosegraphsdirectly.Anotherpossibleprocedureisthefollowing:startingfromtherankdeciencyofastressmatrixofagenericframework,wewanttoprovesomeintermediateconditionholds.Thenwiththatcondition,wewanttoapplyatechniqueasin[ 19 ],torstshowtheexistenceofinnitesimallyexibleframeworks(sincewestartfromredundantlyrigidframeworks,theseinnitesimallyexibleframeworkscannotbegeneric),thenconstructtwoequivalentframeworksfromthatinnitesimalframework. 122

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CHAPTER7CONCLUSIONWemovetowardsacombinatorialcharacterizationofgenericrigidityin3-dimensionsandgivegoodupperboundsonrank.Ourresultsincludethefollowing.EverymaximalMaxwell-independentsethassizeatleasttherankofthegenericrigiditymatroidin3dimension.Combinatorialrankboundsusinginclusion-exclusionemployingthisresultarealsogiven.OurboundsarenotonlyapplicabletoMaxwell-independentgraphs,butalsoaspecialclassofnon-Maxwell-independentgraphs.Wehaveshownthatimpliednon-edgesexistingraphsthatdonothaveanynucleation.Wealsogivegeneralconstructionschemesfornucleation-free,independentgraphswithimpliednon-edgesandageneralconstructionschemesfornucleation-freedependentgraphs.Thesegraphshelpusunderstandthecauseoftruedependenceandgivebetterrankupperbounds.Wedeneanotion,i.e.,GEM-closure,thatappliestogeneralgraphsandgivesamatroid.Weshowthatthismatroidisalsoanabstractrigiditymatroid.WealsoobtainthattrueindependenceimpliesGEM-independenceandeverymaximalMaxwell-independenthassizeatlesstheGEM-rankofagraph.Therefore,wehaveabetterboundthantheonegivenbyMaxwell-independence. 123

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APPENDIX:ISSUEINTAY'SPAPER AsmentionedpreviouslyinSection 2.2.2 ,nucleation-freerigiditycircuitswithimpliednon-edgeshavebeenconjecturedandwrittendownbymany([ 48 103 ]).However,tothebestofourknowledge,wearethersttogiveproofs.Inparticular,in[ 103 ],Tayclaimedaclassofexiblerigiditycircuitswithoutanynuclei.Oneofhisexamples,n-butteries,inwhichheclaimedexistenceofimpliednon-edges,isthesameasourwarm-upexamplegraphs,ringofroofs.In[ 103 ],Taypresentedaproofoftheindependenceofringofroofs.HisproofisbasedontheProposition4.6inthepaper,whichshowstheexistenceofimpliednon-edgesinthering.However,theargumenthemadeintheproofisimpreciseatbest.InProposition4.6,hersttookachainofgraphsG1,G2,...Gnwhichisknowntobea3Drigiditycircuitasawhole,andthenheclosedthechainandsubsequentlyremovedthetwojoiningedges(p,q)and(r,s).Hestated,ifthestresses(pqandrs)onthosetwoedgescancelout,onecankeeppqxedandchangethevalueofrsbychangingthepositionofeithera1orb1(a1,b1arethetwoverticessharedbythersttwosubgraphsG1andG2ofthechain).However,hedidnotmentionhowtochangethepositionofa1orb1.Thusonecanndsomeexamplethatasmallchangeofthepositionwouldnotaffectthestressesonpqorrs.Forexample,inFig. A-1 ,wehaveachainofgraphswhereG1happenstobetheunionoftherestofthesubgraphsinthechain.Thewholegraphisrealizedinapositionwhereitissymmetricalongthelinea1b1.Ifwemovea1orb1alongthecurrentlinea1b1,thestressesonpqandrsshouldalwayscancelout.Anotherpotentialholeisthathedidnotspecifythestartingrealizationofthegraph.Soitispossiblethatinsomerealization,notnecessarilygeneric,ofsuchachaingraph,nomatterhowonealtersthepositionofa1orb1,thestressesofpqandrsremainsoppositeofeachoneinmagnitude.Forexample,inFig. A-2 ,a1andb1arenotadjacenttoporq.Thenweputverticesthatareadjacenttoporqatthesameposition(indicated 124

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FigureA-1. AcounterexampletoTay'sproof:therstsubgraphistheunionoftherestandweputtheminsuchapositionthatthechainissymmetricalonga1b1. asvinFig. A-2 ).Thenweput(p,v)insuchapositionthattheanglebetweenpvandpqis45degrees.Similarlymaketheanglebetweenqvandpqtobe45degrees.Thenitiseasytoobtainthatthestresseson(p,q)iszero.Dothesamefor(r,s).Nomatterhowwemovea1orb1,thestress,pq,isalwayszero,andsoisrs. FigureA-2. AnothercounterexampletoTay'sproof:weputallverticesadjacenttoporqatthesamepositionasvandmakesurepq,pv,qvformsaisoscelesrighttriangle.Itisnothardtoshowthatthestresspqisalwayszero.Wedothesamefortheneighborsofrandsthusthestressespqandrsalwayscancelout. 125

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BIOGRAPHICALSKETCH JialongChengwasborninYingcheng,Hubei,China.HereceivedhisB.E.degreeincomputerscienceandtechnologyin2008fromUniversityofScienceandTechnologyofChina(USTC).HewasawardedthePh.D.incomputerengineeringfromtheUniversityofFloridain2013. 134