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Geometric Under-Constraints

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Title:
Geometric Under-Constraints
Creator:
Gao, Heping
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[Gainesville, Fla.]
Florida
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University of Florida
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english
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1 online resource (131 p.)

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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 Members:
Sahni, Sartaj
Ungor, Alper
Davis, Timothy A.
Agbandje-McKenna, Mavis
White, Neil L.
Graduation Date:
12/19/2008

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Subjects / Keywords:
Algebra ( jstor )
Coordinate systems ( jstor )
Mathematical theorems ( jstor )
Polytopes ( jstor )
Space observatories ( jstor )
Tetrahedrons ( jstor )
Topological theorems ( jstor )
Triangle inequalities ( jstor )
Vertices ( jstor )
Zeolites ( jstor )
Computer and Information Science and Engineering -- Dissertations, Academic -- UF
algebraic, configuration, connected, convex, distance, edge, extreme, framework, graph, helix, henneberg, linkage, minor, sampling, triangle, underconstraint, zeolite
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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:
We define and study exact, efficient representations of realization spaces Euclidean Distance Constraint Systems (EDCS). These are graphs with distance assignments on the edges (frameworks) or graphs with distance interval assignments on the edges. Each representation corresponds to a choice of non-edges or Cayley parameters. The set of realizable distance assignments to the chosen parameters yields a parametrized configuration space. We initialize a systematic and graded program of combinatorially characterizing graphs with configuration spaces of different geometry and algebraic complexity. Our notion of efficiency is based on the convexity and connectedness of the configuration space, as well as algebraic complexity of sampling realizations: that is, sampling the configuration space and obtaining a realization from the sample (parametrized) configuration. Significantly, we give purely graph-theoretic, forbidden minor characterizations that capture the class of graphs that always admit efficient configuration spaces and the possible choices of representation parameters that yield efficient configuration spaces for a given graph. We completely characterize EDCS that have connected, convex and efficient configuration spaces, based on precise and formal measures of efficiency. It should be noted that our results do not rely on genericity of the EDCS. Some of our proofs employ an unusual interplay of classical analytic and algebraic results related to positive semi-definiteness of Euclidean distance matrices, and Cayley-Menger conditions, with recent forbidden minor characterizations and algorithms related to realizability of EDCS. We further introduce a novel type of restricted edge contraction or reduction to a graph minor, a strategy that we anticipate will be useful in other situations. We study the class of 1-dof Henneberg-I graphs in order to take the next step in a systematic and graded program of combinatorial characterizations of efficient configuration spaces. We prove an algebraic theorem that makes combinatorial classification meaningful. We give the graph characterization according to the classification. We prove our results are tight and our definitions are robust. Our results have immediate CAD applications. We give preliminary results and conjectures for two natural extensions: which 2D EDCS have configuration space with at most two connected components and which 3D EDCS have connected configuration space. Finally, we discuss two application problems: characterizing configuration space of packing Zeolite and Helix. We give a surprising configuration space description theorem for Zeolite problem. We show that our novel simulation of Helix packing via geometric constraint solving provides quality and efficiency guarantees that other methods do not. ( en )
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In the series University of Florida Digital Collections.
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Includes vita.
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Includes bibliographical references.
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This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis:
Thesis (Ph.D.)--University of Florida, 2008.
Local:
Adviser: Sitharam, Meera.
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RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2010-12-31
Statement of Responsibility:
by Heping Gao.

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Copyright Gao, Heping. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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IamgratefulforthehelpIreceivedinwritingthisdissertation.Firstofall,Ithankmyadvisor(Prof.MeeraSitharam)forherguidanceandsupport.Withoutnumerousdiscussionsandbrainstormswithher,theresultspresentedinthisdissertationwouldneverhaveexisted.IamgratefultoAlperUngor,MavisA.Mckenna,NeilWhite,SartajSahniandTimDavisfortheirguidanceandencouragementduringmyyearsattheUniversityofFlorida(UF).IamthankfultoallmycolleaguesinProf.Sitharam'sgroup,includingAndrewLomonosov,MohamadTari,SenthilNathanGandhiandYongZhou.Theyprovidedvaluablefeedbackformyresearch.IthankthehelpfulpeopleintheComputerandInformationScienceandEngineering(CISE)Departmentfortheirhelpinmyresearchwork.Lastbutnotleast,Iamgratefultomyfamiliesfortheirlove,encouragement,andunderstanding.Itwouldbeimpossibleformetoexpressmygratitudetotheminmerewords. 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFFIGURES .................................... 8 ABSTRACT ........................................ 12 CHAPTER 1INTRODUCTION .................................. 14 1.1Introduction ................................... 14 1.2Motivation .................................... 15 1.3Problems .................................... 20 2GRAPHSWITH2DCONNECTEDCONFIGURATIONSPACES ....... 22 2.1QuestionsandContributions .......................... 22 2.2NoveltyandRelatedWork ........................... 23 2.3Theorems .................................... 24 2.3.1Basics .................................. 24 2.3.2GraphswithConnected,Convex,LinearPolytope2DCongurationSpace ................................... 25 2.3.2.1Graphsandtheir\single-interval"non-edges ........ 27 2.3.2.2Graphswithgenericallycompletelinearpolytopecongurationspaces ............................. 35 2.3.2.3FullcharacterizationofCayleyparametersthatyieldalinearpolytope2Dcongurationspace ........... 36 2.3.2.4CharacterizationofEDCSwithdistanceintervals ..... 36 2.4ConclusionsandFutureWork ......................... 37 3UNIVERSALLYINHERENTSQUARECONVEXCONFIGURATIONSPACES 40 3.1Question ..................................... 40 3.2NoveltyandRelatedWork ........................... 41 3.3GraphRealizabilityImpliesSquareConvexCongurationSpaces ...... 42 3.4UniversallyInherentConnectedCongurationSpaceImplies2-Realizabilityand3-Realizability ............................... 44 3.5ConclusionsandFutureWork ......................... 48 5

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............................ 51 4.1QuestionsandContributions .......................... 51 4.1.1Question1:GraphswithEcientCongurationSpaces ....... 51 4.1.2Question2:ParametersthatYieldEcientCongurationSpacesforaGivenGraph ............................ 53 4.2NoveltyandRelatedWork ........................... 53 4.3Results ...................................... 55 4.3.1DenitionandBasicPropertiesofSimple1-DofHenneberg-IGraphs 55 4.3.2Simple1-DofHenneberg-IGraphswithEcientCongurationSpaces 59 4.3.2.1Combinatorialmeaningofcongurationspaceboundary 60 4.3.2.2Forbiddenminorcharacterizationfor1-PathTriangle-FreeSimple1-DofHenneberg-Igraphs .............. 67 4.3.2.3Tightnessoftheforbiddenminorcharacterization ..... 78 4.3.2.4Graphcharacterizationfor1-PathSimple1-DofHenneberg-Igraph ............................. 82 4.3.3CharacterizingParameterChoices:AllBaseEdgesYieldEquallyEcientCongurationSpaces ..................... 84 4.4ConclusionsandFutureWork ......................... 92 5FUTURETHEORETICALWORK ......................... 93 5.1CongurationSpacewithTwoConnectedComponents ........... 93 5.1.1QuestionsandContributions ...................... 93 5.1.2Dierencebetween2D2CCSand2DConnectedCongurationSpaces 94 5.1.3GraphCharacterizationforaSubclassofGraphs ........... 96 5.1.4ExtensionofGraphCharacterizationandObstacles ......... 102 5.2GraphsWith3DConnectedCongurationSpace .............. 106 5.2.1QuestionsandContributions ...................... 106 5.2.2Extensionof2DConnectedCongurationSpaceDoesNotHoldfor3D .................................... 107 5.2.2.1Lemmasstandforboth2Dand3D ............. 107 5.2.2.2Modiedconjecturein3D .................. 109 5.2.3PromisingWaytoCharacterize3DConnectedCongurationSpaces 114 6APPLICATIONS ................................... 118 6.1CompletionandCongurationSpaceforZeoliteGraphs ........... 118 6.1.1DenitionofZeoliteGraphsandConstraintSystems ......... 119 6.1.2CongurationSpaceDescriptionforZeoliteGraphs ......... 120 6.2HelixPackingviaConstraintSolving ..................... 122 6.2.1IntroductionandMotivation ...................... 122 6.2.2Denitions ................................ 122 6.2.3ACompleteListofExtremeCongurationTypes .......... 125 6.3Conclusion .................................... 125 6

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................................... 127 REFERENCES ....................................... 128 BIOGRAPHICALSKETCH ................................ 131 7

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Figure page 1-1WhenparametersfortheEDCSin(a)arechosentobethetwodashednon-edges,wegetaconvex2Dcongurationspaceshownin(b).Thehorizontalaxisdenotes(v2;v4)andtheverticalaxisdenotes(v1;v4). ................. 16 1-2WhenparameterfortheEDCSin(a)ischosentobethedashednon-edge,wegetadisconnected2Dcongurationspaceshownin(b):therealizationp(v1)canlieineitherofthetwosolidarcsegmentsofthecirclelabeledp(v1),yieldingtwodisconnectedintervalsforthecongurationspaceonthenon-edge(v1;v3)asshownin(c).Theaxisin(c)denotes(v1;v3). ................ 16 1-3Figure(left)isa1-DofHenneberg-Igraphwhosecongurationspaceonthebasenon-edgehasintervalendpointsthatarenotalwaysquadraticallysolvable.Figure(right)ontheotherhandhasquadraticallysolvableendpoints.Fortheedgedistancesshow,theintervalsare[1 8p 8p 5p 5p ...................... 19 2-1Thegraphofonlysolidedgesisanunderconstrainedpartial2-treewhilethegraphofbothsolidanddashededgesiswellconstrainedandisa2-tree. ..... 26 2-2A2-Sumofveminimal2-Sumcomponents(markedbydashedcircles).The2-Sumcomponentinthemiddleisapartial2-treebuttheentiregraphisnotapartial2-tree.Theunionofthemiddlecomponentwithanyothercomponentisalsoa2-Sumcomponentbutnotminimal. ..................... 26 2-3Nonon-edgefexistssuchthat2f(G;)isalwaysconnected. ........... 27 2-4BaseCase1ofTheorem 2 .............................. 27 2-5BaseCase2ofTheorem 2 :Theverticesui:i=1;;mwherem1aretheonlyverticesotherthanv1andv2withdegreetwoandtheyareadjacenttobothv1andv2;f=(v1;v2)isnotanedgeofthegraph. .............. 28 2-6GraphGhasaconnected2Dcongurationspaceonf=(v1;v2)ifandonlyifforall1ik,thegraphGihasaconnected2Dcongurationspaceonf. .. 29 2-7Vertexv3isanarticulationvertexforv1andv2. .................. 29 2-8Casek=1,Subcasel2inproofofTheorem 2 .Thereareatleasttwodisjointpathsfromv1tov2;G[fisaminimal2-Sumcomponentcontainingbothv1andv2;andG[fisnotapartial2-tree. ...................... 30 2-9ForObservation 10 ,G's3Dcongurationspaceonnon-edgefisoneinterval,althoughG[ffghasaK5minor. ......................... 34 8

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10 .GraphGhasconnected3Dcongurationspaceonnon-edgef;GdoesnothaveK5minororK2;2;2minor;G[fhasaK2;2;2minorbutdoesnothaveaK5minor;inparticular,withoutcontractingedgefinG[fwecannotgetaK2;2;2minor. .............................. 34 2-11ForObservation 1 .Alltheminimal2-SumcomponentsofG[fcontainingfarepartial2-treesbut2f(G;[l(E);r(E)])isnotalwaysconnected. ...... 37 2-12Genericallygloballyrigidgraphin2D. ....................... 38 3-1DistanceassignmentforK2;2;3withoneedgeremoved:(a)K2;2;2withoneedgefremoved;(b)seeproofofLemma 6 :adistanceassignmenttoK222nfsuchthatthe3Dcongurationspaceonfisnotconnected. .............. 46 4-1Tree-DecomposableGraph:agraphGisTree-DecomposableifitcanbedividedintothreeTree-DecomposablesubgraphsG1,G2andG3suchthatG=G1[G2[G3,G1\G2=(fv3g;;),G2\G3=(fv2g;;)andG1\G3=(fv1g;;)wherev1,v2andv3arethreedierentvertices;asbasecases,apureedgeandatrianglearedenedtobeTree-Decomposable. .................. 55 4-2Henneberg-Igraph,Simple1-DofHenneberg-Igraphandextremegraph.(a)Henneberg-Igraph:(v1;v2)isthebaseedge;(b)Simple1-DofHenneberg-Igraph:(v1;v2)isthebasenon-edge;(c)Theextremegraphof(b)thatcorrespondstov7/(v5;v6);itisalsoaK3;3graph.Forboth(a)and(b),theHenneberg-Iconstructionscontain(v3/(v1;v2);v4/(v1;v2);v5/(v1;v3);v6/(v2;v4);v7/(v5;v6)). ..... 56 4-3SimpleHenneberg-Igraph:(a)Henneberg-Igraphwith(v1;v2)asbaseedge;(b)Simple1-DofHenneberg-Igraphwith(v1;v2)asbasenon-edge;(c)extremegraphof(b)thatcorrespondstov7/(v5;v6);itisalsoaC3C2graph.Forboth(a)and(b),theHenneberg-Iconstructionscontain(v3/(v1;v2);v4/(v1;v2);v5/(v3;v4);v6/(v1;v2);v7/(v5;v6)). .......................... 56 4-4Whenp(v7)andp(v8)arecoincident,distance(v5;v9)isnotafunctionof(v1;v2). 61 4-5ForLemma 7 .Newconstrainton(uk;wk)changestheintervalendpointsin2f(Gk;). ....................................... 63 4-6ForObservation 2 .(Left)showsextremeEDCScongurationsinE(G;)thatareinsomeproperintervalofI,butnotendpoints;(Middle)showsextremeEDCScongurationsthatareendpointsofintervalsinofI;and(Right)showsextremeEDCScongurationsthatareisolatedpointsinI.Thehorizontalaxisdenotes(f),theverticalaxisdenotesfunctionpuk;vk((f))=puk;vk((f)). ............................................. 66 4-7Allthe1-PathTriangle-FreeSimple1-DofHenneberg-Igraphswithlessthen8vertices;neither(d)nor(h)haslowsamplingcomplexityonbasenon-edge(v1;v2)whilealltheotherhave;both(d)and(h)haveaK3;3minorwhilealltheothersdonothave. ................................ 68 9

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8 ................. 70 4-9ProofofLemma 8 (2b). ............................... 71 4-10ForproofofLemma 9 ................................ 72 4-11Casesthatmustbedistinguishedindeterminingthedistanceintervalforf,giventhedistanceinterval[l(e);r(e)]fore.Thehorizontalaxisdenotes(f);theverticalaxisdenotes(e).Themeaningofthesymbolsare:min:l(e);max:r(e);1:rl(f);2:rr(f);3:ll(f);4:lr(f);5:min(f);6:max(f);e:(e);f:(f). ......................................... 76 4-12ForObservation 4 .ASimple1-DofHenneberg-IgraphGonbasenon-edge(v1;v2)whichisnotTriangle-FreebuthasasingleHenneberg-Iconstructionpathforv14onbasenon-edge(v1;v2);Ghascongurationspaceoflowsamplingcomplexityon(v1;v2);butG1hasaK3;3minorandG2hasaC3C2minor. ........ 79 4-13ForObservation 5 .ASimple1-DofHenneberg-Igraphonbasenon-edge(v1;v2)thathasoneHenneberg-Iconstructionpathonbasenon-edge(v1;v2);ithasacongurationspaceoflowsamplingcomplexityon(v1;v2)butithasaK5minorshownintheleftcircledsubgraph;ingeneral,itcanhaveaarbitrarycliqueasaminor. ....................................... 81 4-14ForObservation 6 .ASimple1-DofHenneberg-Igraphonbasenon-edge(v1;v2)thathasmorethanoneHenneberg-Iconstructionpathsonbasenon-edge(v1;v2);ithasacongurationspaceoflowsamplingcomplexityon(v1;v2);butithasbothK3;3andC3C2minors.Aside:(v3;v4),(v5;v6),(v1;v5)and(v2;v6)arealsobasenon-edgesandallofthemyieldcongurationspacesoflowsamplingcomplexity. ...................................... 81 4-15A1-Path1-DofHenneberg-Igraphthathaslowsamplingcomplexityonbasenon-edge(v1;v2);exactly1vertexnamelyv3isconstructedonv1andv2.SeeproofofTheorem 12 andObservation 7 ...................... 82 4-16ProofsofClaim 4 andinClaim 5 .Simple1-DofHenneberg-IgraphGhaslowsamplingcomplexityonbasenon-edge(v1;v2)whileGdoesnothavelowsamplingcomplexityonbasenon-edge(v3;v4);vertexv9istheonlyvertexdirectlyconstructedonv1andv2;triangle4(v9;v10;v1)correspondstothesecondHenneberg-Iconstructionfrom(v1;v2);in(a),(v1;v2)and(v3;v4)donotshareanyvertex;in(b)and(c),(v1;v2)and(v3;v4)shareavertex. ......................... 87 4-17ProofofClaim 5 :v1iscoincidentwithv3. ..................... 89 4-18ForproofofClaim 5 :v1isdierentfromv3andv4. ................ 90 5-1ProofforLemma 10 ................................. 95 10

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................... 96 5-3Case4:Gdoesnothave2D2CCSonnon-edgef. ................. 97 5-4ProofforTheorem 14 ................................ 100 5-5GraphGdoesnotadmit2D2CCSonnon-edgef=(v1;v2). ........... 102 5-6Proofforcase(f)inTheorem 15 .......................... 104 5-7Overconstraintsresultincongurationspaceof2connectedcomponents. .... 105 5-8ForproofofObservation 11 ............................. 111 5-9Threebasicsum-componentsofmaximally3-Realizablegraphs:(a)K4;(b)C5C2;(c)V8. ....................................... 113 5-10Giventwodierentvaluesfor(v3;v4),wecanndappropriatedistanceassignmenttothefouredgesofquadrilateral(v1;v2;v3;v4)suchthat(v1;v2)hastwodierentvalues. ......................................... 115 5-11Basecasesthatdonothold3Dconnectedcongurationspaces. ......... 115 5-12ProofforLemma 23 case(c)(Figure 5-11 (c)). .................. 116 6-1ZeolitegraphG6;7andone2Dcompletion. ..................... 119 6-2FirstrowandrstcolumnsubgraphofG6;7;itisapartial2-tree. ........ 120 6-3AnothercompletionforG6;7. ............................ 122 6-4Thealphahelixisatightlycoiled,rodlikestructurewhichhasanaverageof3.6aminoacidsperturn.ThehelixisstabilizedbyhydrogenbondingbetweenthebackbonecarbonylofoneaminoacidandthebackboneNHoftheaminoacidfourresiduesaway.Allmainchainaminoandcarboxylgroupsarehydrogenbonded,andtheRgroupsstickoutfromthestructureinaspiralarrangement.[Fromwiz2.pharm.wayne.edu/biochem/prot.html] ................. 123 6-5Twohelices ...................................... 123 6-6Bi-incidence ..................................... 124 6-7Eachtagrepresentsabipartitegraphwith(left,right)partshaving(i;j)verticesrespectively,wherei+1j6;i3:Thereshouldbe6edgesinthebipartitegraphandEachvertexcanhaveatmost3edgesincidentuponit. ....... 126 11

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Wedeneandstudyexact,ecientrepresentationsofrealizationspacesEuclideanDistanceConstraintSystems(EDCS).Thesearegraphswithdistanceassignmentsontheedges(frameworks)orgraphswithdistanceintervalassignmentsontheedges.Eachrepresentationcorrespondstoachoiceofnon-edgesorCayleyparameters.Thesetofrealizabledistanceassignmentstothechosenparametersyieldsaparametrizedcongurationspace.Weinitializeasystematicandgradedprogramofcombinatoriallycharacterizinggraphswithcongurationspacesofdierentgeometryandalgebraiccomplexity. Ournotionofeciencyisbasedontheconvexityandconnectednessofthecongurationspace,aswellasalgebraiccomplexityofsamplingrealizations,i.e.,samplingthecongurationspaceandobtainingarealizationfromthesample(parametrized)conguration.Signicantly,wegivepurelygraph-theoretic,forbiddenminorcharacterizationsthatcapturetheclassofgraphsthatalwaysadmitecientcongurationspacesandthepossiblechoicesofrepresentationparametersthatyieldecientcongurationspacesforagivengraph. WecompletelycharacterizeEDCSthathaveconnected,convexandecientcongurationspaces,basedonpreciseandformalmeasuresofeciency.ItshouldbenotedthatourresultsdonotrelyongenericityoftheEDCS.Someofourproofsemployanunusualinterplayofclassicalanalyticandalgebraicresultsrelatedtopositive 12

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Westudytheclassof1-dofHenneberg-Igraphsinordertotakethenextstepinasystematicandgradedprogramofcombinatorialcharacterizationsofecientcongurationspaces.Weproveanalgebraictheoremthatmakescombinatorialclassicationmeaningful.Wegivethegraphcharacterizationaccordingtotheclassication.Weproveourresultsaretightandourdenitionsarerobust.OurresultshaveimmediateCADapplications. Wegivepreliminaryresultsandconjecturesfortwonaturalextensions:which2DEDCShavecongurationspacewithatmosttwoconnectedcomponentsandwhich3DEDCShaveconnectedcongurationspace. Finally,wediscusstwoapplicationproblems:characterizingcongurationspaceofpackingZeoliteandHelix.WegiveasurprisingcongurationspacedescriptiontheoremforZeoliteproblem.WeshowthatournovelsimulationofHelixpackingviageometricconstraintsolvingprovidesqualityandeciencyguaranteesthatothermethodsdonot. 13

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WeseekecientrepresentationsoftherealizationspaceofanEDCS.Wedenearepresentationtobe(i)achoiceofparameterset,specicallyachoiceofasetFofnon-edgesofG,and(ii)asetdF(G;)ofpossibledistancevalues(f)thatthenon-edgesf2F 6 ; 25 ; 7 ).ThisisalsothesetofddimensionaljVjjVjEuclideandistancematrixcompletionsofthepartialdistancematrixspeciedby(G;)( 1 ). WerefertotherepresentationdF(G;)(resp.dF(G;[l;r]))asthecongurationspaceoftheEDCS(G;)(resp.(G;[l;r]))ontheparametersetFofnon-edgesofG. 14

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ThisdissertationformulatestheconceptofecientcongurationspacedescriptionforEDCSbyemphasizingtheexactchoiceofparametersusedtorepresenttherealizationspace.Thissetsthestageforamostlycombinatorial,andcomplexity-gradedprogramofinvestigation.Aninitialsketchofthisprogramwaspresentedin( 12 ). ExistingmethodsforsamplingEDCSrealizationspacesoftenuseCartesianrepresentations,factoringouttheEuclideangroupbyarbitrarily\pinning"or\grounding"someofthepoints'coordinatevalues.Evenwhenthemethodsuse\internal"representationparameterssuchasCayleyparameters(non-edges)oranglesbetweenunconstrainedobjects,thechoiceoftheseparametersisadhoc.WhileEuclideanmotionsmaybeautomaticallyfactoredoutintheresultingparametrizedcongurationspace,formostsuchparameterchoices,thecongurationspaceisstillatopologicallycomplexsemi-algebraicset,sometimesofreducedmeasureinhighdimensions. Aftertherepresentationparametersarechosen,themethodofsamplingthecongurationspaceoftenreducesto\takeauniformgridsamplingandthrowawaysamplecongurationsthatdonotsatisfygivenconstraints."Sinceevencongurationspacesoffullmeasure(representationusinglowestpossiblenumberofparametersordimensions)oftenhavecomplexboundaries,potentiallywithcuspsandlargeholes,thistypeofsampling 15

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WhenparametersfortheEDCSin(a)arechosentobethetwodashednon-edges,wegetaconvex2Dcongurationspaceshownin(b).Thehorizontalaxisdenotes(v2;v4)andtheverticalaxisdenotes(v1;v4). Figure1-2. WhenparameterfortheEDCSin(a)ischosentobethedashednon-edge,wegetadisconnected2Dcongurationspaceshownin(b):therealizationp(v1)canlieineitherofthetwosolidarcsegmentsofthecirclelabeledp(v1),yieldingtwodisconnectedintervalsforthecongurationspaceonthenon-edge(v1;v3)asshownin(c).Theaxisin(c)denotes(v1;v3). islikelytomissextremeandboundarycongurationsandismoreovercomputationallyinecient.Todealwiththis,numerical,iterativemethodsaregenerallyusedwhentheconstraintsareequalities,andinthecaseofinequalities,probabilistic\roadmaps"andothergeneralcollisionavoidancemethodsareused.Theyareapproximatemethods.Ifthecongurationspaceisrelativelylowdimensional,theninitialsamplingisusedtoprovideanapproximateandrenablerepresentationofthecongurationspace,usingtraditional 16

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TworelatedproblemsadditionallyoccurinNMRmolecularstructuredeterminationandwirelesssensornetworklocalization:completingapartiallyspeciedEuclideanDistanceMatrixinagivendimension;andndingaEuclideanDistanceMatrixinagivendimensionthatcloselyapproximatesagivenMetricMatrix(representingpairwisedistancesinametricspace)( 5 ; 9 ).ThelatterproblemalsoarisesinthestudyofalgorithmsforlowdistortionembeddingofmetricspacesintoEuclideanspacesofxeddimension( 2 ).BothofthesecanbeviewedassearchingoveracongurationspaceofanEDCS.Butthecommonmethodsfortheseproblemsaredierentfromthoseusedforexploringcongurationspaces.Onereasonforthisisthatusuallyonlyonerealizationisusuallysought,whichoptimizessomeappropriatelychosenfunction;thegoalisnotsamplingordescriptionoftheentirecongurationspace.Commonmethodsfortheseproblemsare:(i)eitherusesemi-deniteprogramming,sinceEuclideanDistanceMatricesinaspecieddimensionaredirectlyrelatedtoGrammatriceswhicharepositivesemidenitematricesofaspeciedrank;(ii)oriterativelyenforcetheCayley-MengerdeterminantalconditionsthatcharacterizeEuclideanDistanceMatricesinaspecieddimension. Motivatedbytheseapplications,ouremphasisisonexact,ecientdescriptionofthecongurationspaceofunderconstrainedorindependentandnotrigidEDCS.(i)Anexactalgebraicdescriptionguaranteesthatboundaryandextremecongurationsarenotmissedduringsampling,whichisimportantformanyapplications.(ii)Anecientdescription(i.e,lowdimensional,fullmeasure,convex,usingfewpolynomialorevenlinearinequalities,whosecoecientsareobtainedecientlyfromthegivenEDCS)isimportantfortractabilityofthesamplingalgorithm. 17

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Concerning(ia),itisimportanttonotethatmostchoicesofCayleyparameters(non-edges)torepresenttherealizationspaceof(G;)giveinecientdescriptionsoftheresultingparametrizedcongurationspace(seeforexampleFigures 1-1 1-2 ).Henceweplaceastrongemphasisisonasystematic,combinatorialchoiceoftheCayleyparametersthatguaranteeacongurationspacewithalltheeciencyrequirementslistedhere.Further,weareinterestedincombinatoriallycharacterizingforwhichgraphsGsuchachoiceevenexists. Thesecondeciencyfactoristherealizationcomplexity.NotethatthepricewepayforinsistingonexactandecientcongurationspacesisthatthemapfromthetraditionalCartesianrealizationspacetotheparametrizedcongurationspaceismany-one.I.e,eachparametrizedcongurationcouldcorrespondtomany(butatleastone)Cartesianrealizations. However,wecircumventthisdicultybydeningandstudyingrealizationcomplexityasoneoftherequirementsonecientcongurationspacesi.e.,wetakeintoaccountthattherealizationsteptypicallyfollowsthesamplingstep,andensurethatoneorallofthecorrespondingCartesianrealizationscanbeobtainedecientlyfromaparametrizedsampleconguration. Beforeourworkin( 12 )andthisdissertation,thedierentiationbetweenthesamplingcomplexityandtherealizationcomplexityisnotaddressed.RefertoFigure 1-3 ,distance 18

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Figure(left)isa1-DofHenneberg-Igraphwhosecongurationspaceonthebasenon-edgehasintervalendpointsthatarenotalwaysquadraticallysolvable.Figure(right)ontheotherhandhasquadraticallysolvableendpoints.Fortheedgedistancesshow,theintervalsare[1 8p 8p 5p 5p betweenv1andv2ischosenastheparameterinboth(a)and(b).Ineithercase,therealizationcomplexityisquadraticallysolvable;however,in(a)thesamplingcomplexityisnotquadraticallysolvablewhilein(b)thesamplingcomplexityisquadraticallysolvable. Athirdeciencyfactorisgenericcompleteness,i.e,wewouldlike(a)eachcongurationinourparametrizedcongurationspacetogenericallycorrespondtoatmostnitelymanyCartesianrealizationsand(b)wewouldlikethecongurationspacetobeoffullmeasure,andinparticular,theyuseatmostasmanyparametersordimensionsastheinternaldegreesoffreedomofG.Specically(a)meansG[Fisrigidand(b)meansG[Fisnotoverconstrained,i.e,itisindependent.ThisgenericcompletenessmeansthatthegraphG[Fiswellconstrainedi.e.,minimallyrigid. Afourthandfthimportanteciencyfactorsaretopologicalandgeometriccom-plexityforexample,numberofconnectedcomponents,andconvexity.Convexityandconnectednessarenaturalpropertiestostudysincetheyfacilitateconvexprogrammingandotherecientmethodsforsampling.Anothercrucialreasonforstudyingconvexityisthatresults(suchasthosepresentedhere)aboutconvexcongurationspacesreadily 19

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2 ,wecharacterizegraphandnon-edgesetpair(G;F)suchthatforalldistanceassignment(E)(distanceintervalassignment[l;r]),the2Dcongurationspace2f(G;(E))(2f(G;[l;r]))isalwaysconnected(aconvexoralinearpolytope). 3 ,wecharacterizegraphHsuchthatforallpartitionG[FofH(GisasubgraphofHandFisasetofedgesofH;thusFisasetofnon-edgesofG),foralldistanceassignment(E)(ordistanceintervalassignment[l;r]),the3Dcongurationspace3F(G;)(3F(G;[l;r]))isalwaysconnected.Wealsocharacterizethegraphswithsuchapropertyin2D.Inaddition,weprovethataknownclassofgraphshavethisproperty. 4 ,westudythecongurationspacesofanimportantclassofgraphs,1-DofHenneberg-Igraphs.Wenaturallychoosenon-edgefthatguaranteesthattherealizationcomplexityisquadraticallysolvable.Weproveanalgebraictheoremthatdescribeshowthecongurationspacesonthespeciednon-edgesareexactlydetermined.Thistheoremalsomakesthecombinatorialclassicationonsamplingcomplexitymeaningful.Weshowthesamplingcomplexityvaries(quadraticallysolvableornot)althoughtherealizationisalwaysquadraticallysolvable.Basedontheseresults,weformalizetheproblemtocharacterize1-DofHenneberg-Igraphswithecientcongurationspaceswhere\ecient"ispreciselydened.Wegiveagraphminorcharacterizationtheoremforasubclassof1-DofHenneberg-Igraphsandprovethatthetheoremistight.Wegiveanalgorithmiccharacterizationforgeneral1-DofHenneberg-Igraphs.Weproveaquantierexchangetheoremthatshowsthechoiceoffdoesnotmakedierencewithrespecttoourclassication. 5 ,wediscusstwoopenproblemsextendedfromtheresultsinChapter 2 .Therstproblemaimstocharacterizepair(G;f)suchthatforalldistanceassignment(E),the2Dcongurationspace2f(G;(E))alwayshasatmosttwoconnectedcomponents;thesecondproblemaimstocharacterizepair(G;f)suchthatforalldistanceassignment(E),the3Dcongurationspace3f(G;(E))isalwaysconnected.Webuildframworksforbothproblemsandreducebothofthemtospecicsubproblems. 6 ,twoapplicationproblems(ZeoliteandHelixpacking)arestudied.FortheZeoliteproblem,weproveatheoremthatshowsthecongurationspacesoncertainsetsofnon-edgesareextremelysimple.Wealsocharacterizethenon-edgesetsuchthatthecongurationspaceshassuchasimpleproperty.FortheHelixpacking 20

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21

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Inthischapterwegive4naturalquestionsconcerningecientcongurationspacesandthecontributionofthismanuscripttowardansweringthem. (1) InTheorem 3 ,wegiveanexactcharacterizationoftheclassofgraphsGallofwhosecorrespondingEDCS(G;)admita2D(genericallycomplete),linearpolytopecongurationspace.Thetheoremalsoshowsthatthecharacterizationremainsunchangedifthecongurationspaceismerelyrequiredtobeconvex,andfurtherifitismerelyrequiredtobeconnected. (2) Foragraphintheaboveclass,inTheorem 4 wegiveanexactcombinatorialcharacterizationofthechoicesofCayleyparameters(non-edges)thatensurea(genericallycomplete),linearpolytopecongurationspace. (3) BothaboveresultsrelyonkeyTheorem 1 (inturnbasedonTheorem 2 )thatcharacterizesagraphGalongwithanon-edgefsuchthatforalldistanceassignments(E),the2Dcongurationspace2f(G;),isasingleinterval.WeadditionallyshowinObservation 10 (deferredtoChapter 5 )thatthisresultistightinthattheobviousanalogofthisresultfailsin3D. (4) Observation 1 showsthatwhiletheforwarddirectionofTheorem 3 ,Theorem 4 ,andTheorem 1 forpuredistanceconstraintsholdsdirectlyforintervalconstraints(G;[l;r]),thereversedirectionfails.However,inTheorem 5 ,wegiveanex-actcharacterizationoftheclassofgraphsGallofwhosecorrespondingEDCS(G;[l;r])admita2D(genericallycomplete),linearpolytope,convexorconnectedcongurationspace. Forwarddirectionoftheabovetheorems(thatthegraph-theoreticpropertyalwaysadmitsaconvex,connected,linearpolytopecongurationspace)isstraightforward.It 22

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2.2 ). Conceptually,theresultsonglobalrigidity( 17 ; 18 )(resp.globallylinkedpairs( 19 ))arerelated,sincetheycombinatoriallycharacterizewhenthecongurationspaceonall(resp.specic)Cayleyparameters(non-edges)isasinglepoint.However,thesecharacterizationsholdonlygenerically( 8 ; 15 )asiscustomaryformanycombinatorialpropertiesrelatedtorigidity.Incontrast,ourcharacterizationsapplytoallEDCS'(frameworks)andnotjustgenericframeworks.Thisisacrucialdistinctionthatisneededtoreconcileapparentdiscrepanciesofthetwotypesofresults,asweelaborateinSection 2.4 Ourresultsadditionallyyieldacombinatorialcharacterizationofsamplingcomplexity.Thisincorporatesthecomplexityof(i)obtainingthechosensetofparametersand(ii)thealgebraiccomplexityofobtainingthedescriptionofthecongurationspacefromthegivengraph.Thisinturnincludesthedescriptivecomplexityofthecongurationspaceasasemi-algebraicset,suchasthenumberanddegreeofthepolynomials.Thecharacterizationsmoreoverincorporaterealizationcomplexity,i.e,thecomplexityofobtainingarealizationfromaparametrizedconguration. Tothebestofourknowledge,theonlyresultsofasimilaravorare:theresultof( 27 )thatshowstheequivalenceofTree-orTriangle-decomposability( 26 )andQuadraticorRadicalrealizabilityforplanargraphs. ConcerningtheuseofCayleyparametersornon-edgesforparametrizingagenericallycompletecongurationspace:( 30 )aswellas( 21 ; 32 )studyhowtoobtain\completions"ofunderconstrainedgraphsG,i.e,asetofnon-edgesFwhoseadditionmakesG

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21 )alsoguaranteesthatthecompletionensuresTree-orTriangle-decomposability,therebyensuringlowrealizationcomplexity.However,theydonotaddressthegeometric,topologicaloralgebraiccomplexityofthesetofdistancevaluesthatthesecompletionnon-edgescantake,northecomplexityofobtainingadescriptionofthiscongurationspace,giventheEDCS(G;)andthenon-edgesF.Thelatterishoweverthemainfactorinuencingthetractabilityofsamplingthesecongurationspacesbeforeobtainingthecorrespondingrealizations.Theproblemhasgenerallybeenconsideredtoomessy,andbarringeectiveheuristicsforcertaincases,forexamplein( 24 ),therehasbeennosystematic,formalprogramtostudythisproblem. Someoftheproofs,e.g.Theorem 2 ,useanoveltypeofrestrictededge-contractionreductiontoagraphminorwhichdisallowsedgeremovalsaspeciedpairofverticestoremaindistinctandtoremainanon-edge.Weanticipatethatthisnewstrategycouldbeusefulinothersituations. 2.3.1Basics 3 ).Agraphisak-treeifitisak-sumofKk+1's.Givenagraph,wecanruntheinverseoperationsofk-sumtogetasetofk-sumcomponents.Ifwecannotruntheinverseoperationsofk-sumforacomponent,wesaythatcomponentisaminimal(k-sum)component.Givenanon-edgef,aminimalk-sumcomponentcontainingfisaminimalsubgraphthatisbothak-sumcomponent 24

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Agraphisapartialk-treeifitisasubgraphofak-tree.PleaserefertoFigure 2-1 for2-treesandpartial2-treesandFigure 2-2 for2-Sumand2-Sumcomponent.Itisnothardtoseethatpartial2-treesareexactlythe2-realizablegraphs.Whilepartial3-treesareinfact3-realizable,theclassof3-realizablegraphsincludenonpartial3-treesaswell. In( 3 ; 4 )ausefulforbidden-minorcharacterizationofsuchgraphsisgiven.AgraphGhasagraphKasaminorifthereisavertexinducedsubgraphofGthatcanbereducedtoKviaedgeremovalsandedgecontractions(coalescingoridentifyingthe2verticesofanedge).Itisnothardtoseethatpartial2-treesareexactlythegraphsthatavoidK4minors. Nextwegivebasic2DcombinatorialrigiditydenitionsbasedonLaman'stheorem( 22 ).For3D,nocombinatorialdenitionsexist.Forthecorrespondingalgebraicdenitions,pleaseseeforexample( 16 )(combinatorialrigidityterminology)( 13 )(geometricconstraintsolvingterminology). In2D,agraphG=(V;E)iswellconstrainedorminimallyrigidifitsatisestheLamanconditions( 22 );i.e.,jEj=2jVj3andjEsj2jVsj3forallsubgraphsGs=(Vs;Es)ofG;GisunderconstrainedorindependentandnotrigidifwehavejEj<2jVj3andjEsj2jVsj3forallsubgraphsGs.AgraphGisoverconstrainedordependentifthereisasubgraphGs=(Vs;Es)withjEsj>2jVsj3.GiswelloverconstrainedorrigidifthereexistsasubsetofitsedgesE0suchthatthegraphG0=(V;E0)iswellconstrainedorminimallyrigid.Agraphisexibleifitisnotrigid. 25

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Thegraphofonlysolidedgesisanunderconstrainedpartial2-treewhilethegraphofbothsolidanddashededgesiswellconstrainedandisa2-tree. Figure2-2. A2-Sumofveminimal2-Sumcomponents(markedbydashedcircles).The2-Sumcomponentinthemiddleisapartial2-treebuttheentiregraphisnotapartial2-tree.Theunionofthemiddlecomponentwithanyothercomponentisalsoa2-Sumcomponentbutnotminimal. Furthemore,givenGinthischaracterizedclass,wewouldliketocharacterizethecorrespondingsetsFofnon-edges.Theseareexactlytheparameterchoicesthatyieldwell-behavedcongurationspaces. Figure 2-1 givesanexamplegraphthatadmitsaconnected,convexandlinearpolytope2Dcongurationspaceonthespeciednon-edges;andviceversa,inFigure 2-3 ,weprovideanexampleinwhichthegraphdoesnotadmitsuchacongurationspaceonanynon-edge.ThegraphcharacterizationofTheorem 1 canbeeasilyveriedforbothexamples. 26

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Nonon-edgefexistssuchthat2f(G;)isalwaysconnected. Figure2-4. BaseCase1ofTheorem 2 1 .Toproveone(harder)directionofTheorem 1 ,weneedthefollowingpurelygraph-theoretictheoremandthefollowingLemma 1 .Theother(easy)directionfollowsfromLemma 4 whichisinturnprovengraduallyusingLemma 2 andLemma 3 2-4 orFigure 2-5 byasequenceofedgecontractions(noedgeremovals)ifandonlyifthereexistsaminimal2-SumcomponentofG[fcontainingfthatisnotapartial2-tree.

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BaseCase2ofTheorem 2 :Theverticesui:i=1;;mwherem1aretheonlyverticesotherthanv1andv2withdegreetwoandtheyareadjacenttobothv1andv2;f=(v1;v2)isnotanedgeofthegraph. 2-4 andFigure 2-5 arebasedonK4.Basedonthefactthatpartial2-treesdonothaveK4minorsandpropertiesof2-Sum,wecanproveonedirectionofTheorem 2 .FortheotherdirectiontheexistenceofaK4minoraloneisinsucient.Werequireaspecialtypeofpureedge-contractionreductionwithoutedgeremovals,whichadditionallypreserveselectednon-edges:i.e,preventthemfrombecomingedgesorfromcollapsingtoasinglevertex. Proof. 2 .WerstprovethatGcannotbereducedtoFigure 2-4 orFigure 2-5 byedgecontractionsifalltheminimal2-SumcomponentsofG[fcontainingfarepartial2-trees.Becausepartial2-treesdonothaveK4minors,andsinceK4existsasaminorinbothFigure 2-4 andFigure 2-5 ,wecannotreduceGtoeitherofthetwobasecasesbyedgecontractions(noedgeremovals).Infact,incasethereexistsa2-SumcomponentG[fthatdoesnotcontainf,ourproofwillnotchangesinceedgecontractionseitherpreserve2-Sumrelationshiportransforma2-Sumtoa1-sum. WeprovetheotherdirectionbyinductiononthenumbernofverticesofG.Thestatementistrueforthe2basecases.AssumethestatementistrueforjVjn1;weproveitforjVj=n.First,weremovev1andv2togetasetofconnectedcomponentsH1;;Hk(Figure 2-6 ).WeuseGitodenotethesubgraphofGwhichisinducedbyverticesofHitogetherwithv1andv2,wheref=(v1;v2).NotethateachGi[fisa 28

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GraphGhasaconnected2Dcongurationspaceonf=(v1;v2)ifandonlyifforall1ik,thegraphGihasaconnected2Dcongurationspaceonf. Figure2-7. Vertexv3isanarticulationvertexforv1andv2. 2-SumcomponentofG[f.Withoutlossofgenerality,weassumeG1[fisoneofthese2-SumcomponentsofG[fbutnotapartial2-tree. 2-5 2-7 for 29

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Casek=1,Subcasel2inproofofTheorem 2 .Thereareatleasttwodisjointpathsfromv1tov2;G[fisaminimal2-Sumcomponentcontainingbothv1andv2;andG[fisnotapartial2-tree. thiscase).SinceG[fisaminimal2-Sumcomponentcontainingbothv1andv2,both(v1;v3)and(v2;v3)havetobenon-edges.Inaddition,atleastoneofG1[(v1;v3)andG2[(v2;v3)isnotapartial2-tree,otherwiseG[fwillalsobeapartial2-tree.Withoutlossofgenerality,supposeG1[(v1;v3)isnotapartial2-tree.Bytheinductionhypothesis,wecanreduceG1tooneofthetwobasecases.BycontractingalltheedgesinG2wecanalsoreduceGtooneofthetwobasecases(v3isidentiedwithv2). 2-8 ).Ifwefurtherremovev1,v2,t1andz1,wegetnewconnectedcomponents.Thenwecontractalltheedgesinsidethesenewconnectedcomponentssuchthateachofthembecomesasinglevertexthatwedenotebyq1;;qm.Beforewecontractpaths(t1;;ts)and(z1;;zn),ifweremovev1andv2,theremaininggraphisstillconnected(k=1),soatleastoneofq1;;qmisconnectedtobotht1andz1. Fortheremainingproof,weonlyneedtocontractedgesinacertainwaysuchthatwedonotoverlapthenon-edgesbetweenv1andv2.Nowwecontractedgesasfollows(refertoFigure 2-8 ). 30

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Ifqiconnectstobotht1andz1,wecanidentifyqiwitht1byedgecontraction; (2) Ifqiconnectstoonlyv1andv2(nottot1orz1),weleaveitunchanged; (3) Ifqiconnectstoonlyoneofv1,v2,t1andz1,weidentifyitwiththecorrespondingvertexinv1,v2,t1andz1; (4) Ifqiconnectstov1,v2,t1,wecanidentifyqiwitht1; (5) Ifqiconnectstov1,v2,z1,wecanidentifyqiwithz1. Weenumerateallthepossiblecasesforqi'sconnectivitytotheothervertices.Thatcoversallthecasesandcompletestheproofoftheinductionstep. Theorem 2 givesusthefollowingindependentlyinterestingcorollary.Thiscorollarydirectlygivesusanalgorithmtocheckwhetheragraphandspeciednon-edgearesatisedwiththeconditionsprescribedinTheorem 2 2-4 orFigure 2-5 byasequenceofedgecontractionsprovidedthefollowinghold. (1) (2) Foranyvertexviotherthanv1andv2,eitherdeg(vi)is2andviisadjacenttobothv1andv2,ordeg(vi)isatleastthree. (3) Atleastonevertexviotherthanv1andv2hasdegreeofthreeormore. Proof. 2 ,alltheminimal2-SumcomponentsofthegraphG[fcontainingv1andv2arepartial2-trees.Notethatatleastonevertexviotherthanv1andv2hasdegreeofthreeormore.Weconsiderthe2-SumcomponentCcontainingvi.NotealsothatChasmorethan3vertices,andwithinC,since(2)holds,therecanbenoverticesotherthanv1andv2thathavedegreeoftwoorless.By(2),anyothervertexofdegreetwoorlesswouldbeadjacenttobothv1andv2andwouldformitsown2-Sumcomponents. 31

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UsingCorollary 1 ,weseethatthegraphinFigure 2-3 canbereducedtooneofthetwobasecasesnomatterwhichnon-edgewechoose. 2-4 andFigure 2-5 ,thereexistsadistanceassignments.t.2f(G;)isnotconnected. Proof. Asnotedearlier,Theorem 2 andLemma 1 haveprovedthedicultdirectionforTheorem 1 .Thefollowinglemmasprovetheeasydirection. Proof. 2 ]Simplyhingeallthe2-Sumcomponents'realizationsalongthe2-Sumedgestogetarealizationof(G;),withanyoneoftworeectionchoicesacrossthe2-Sumedge.Theotherdirectionisimmediate. Proof. 2

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IfaGhasa2-SumcomponentG0=(V0;E0)thatisanunderconstrainedpartial2-tree,thenthereexistsanonemptynon-edgesetFentirelyinG0suchthatforany,2F(G;)isalinearpolytope.Moreover,thereissuchasetFsuchthat2F(G0;)isgenericallycompleteforG0. (b) IfaGisanunderconstrainedpartial2-tree,thenforanynonemptynon-edgesetF0thatpreserves(V;E[F0)asapartial2-tree,andforall,2F0(G;)isalinearpolytope. Proof. NotethatG0isanunderconstrainedpartial2-tree,sowecanndanonemptysubsetofnon-edgesofG0byaddingwhichwegeta2-tree.WeletFbethisnonemptyset.NotethatsuchanFisacompletionforG0,i.e.,makesG0minimallyrigid.Henceweknowthat2F(G0;)isoffull-measureandgenericallycomplete,provingthelastsentenceofthetheorem. Togetthelinearpolytope,notethata2-treecanbewrittenasthe2-Sumoftriangles.Forexample,let(vi;vj),(vj;vk)and(vk;vi)denotethelengthofthethreeedgesofthetriangle4vivjvk,thenthetriangleinequalitiesare(vi;vj)(vi;vk)+(vj;vk),(vi;vk)(vj;vk)+(vi;vj)and(vj;vk)(vi;vk)+(vi;vj). Thus,forall,2F(G0;)isalinearpolytope.NowsinceFisentirelyinG0,Lemma 3 appliesandforall,2F(G;)=2F(G0;)or2F(G0;)isempty.Thus,forall,2F(G;)isalsoalinearpolytope. Forproving(b):foranyunderconstrainedpartial2-treeG=(V;E),wecanndanonemptynon-edgesetFthatmakes(V;E[F)a2-tree;andweshowedthatforany,2F(G;)isagenericallycompletelinearpolytope2Dcongurationspace.TakeanynonemptysubsetofF0ofsuchaF-theseareexactlythesubsetsofnon-edgeswhoseadditionwouldpreservethepartial2-treepropertyofG.Then2F0(G;)istheprojectionof2F(G;)onF0andsincethelatterisalinearpolytope,theformerisalinearpolytopeaswell. 33

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ForObservation 10 ,G's3Dcongurationspaceonnon-edgefisoneinterval,althoughG[ffghasaK5minor. Figure2-10. ForObservation 10 .GraphGhasconnected3Dcongurationspaceonnon-edgef;GdoesnothaveK5minororK2;2;2minor;G[fhasaK2;2;2minorbutdoesnothaveaK5minor;inparticular,withoutcontractingedgefinG[fwecannotgetaK2;2;2minor. 1 .Theproofofone(harder)directionfollowsdirectlyfromTheorem 2 andtheLemma 1 .Specically,topickadistanceassignmentforGthatyieldsadisconnectedcongurationspaceonf,wesetallthecontractededgesduringtheprocedureofTheorem 2 to0.Theuncontractededgesarenowmappedbythereductiontoedgesofoneofthebasecases.Lemma 1 tellsushowtochoosethosedistancevaluestoensuredisconnectednessofthecongurationspaceonf.Theother(easy)directionisimmediatefromLemma 4 NextweshowthatTheorem 1 istightinthatneitherofthetwostraightforwardextensionsto3Dhold. 34

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2-9 andpartial3-tree's(resp.3-realizablegraphs)G=(V;E)andnon-edgefsuchthatG[fisnotapartial3-tree(resp.3-realizablegraph)andyet3f(G;)isalwaysconnected.WedefertheprooftoChapter 5 Observation 10 3 ,wegiveanexactcharacterizationoftheclassofgraphsG,allofwhosecorrespondingEDCS(G;)admita2D(genericallycomplete),linearpolytopecongurationspace.Thetheoremalsoshowsthatthecharacterizationremainsunchangedifthecongurationspaceismerelyrequiredtobeconvex,andfurtherifitismerelyrequiredtobeconnected. (a) thefollowingfourstatementsareequivalent: (a.1) thereexistsanon-emptysetofnon-edgesFsuchthatforall2F(G;E)isconnected; (a.2) thereexistsanon-emptysetofnon-edgesFsuchthatforall,2F(G;E)isconvex; (a.3) thereexistsanon-emptysetofnon-edgesFsuchthatforall2F(G;E)isalinearpolytope. (a.4) (b) AnunderconstrainedgraphGalwaysadmitsagenericallycompletelinearpolytope,connectedorconvexcongurationspaceifandonlyifeveryunderconstrained2-SumcomponentofGisapartial2-tree. Proof. 4 .Alinearpolytopeisconvex,so(3))(2)follows.Convexityimpliesconnectedness,so(2))(1)follows.Theorem 1 andtheproofofLemma 4 proves (4)) Foronedirectionof(b):ifeveryunderconstrained2-SumcomponentofGisanunderconstrainedpartial2-tree,thenbyLemma 4 ,Galwaysadmitsagenericallycompletelinearpolytope,connectedorconvexcongurationspace.Thereversedirectionof(b)followsfrom(a)(1,2,3)4):ifGalwaysadmitsagenericallycompletelinearpolytope, 35

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Theorem4. Proof. 1 andtheproofofLemma 4 Proof. 2-11 However,givenapathoflength2withdistanceconstraintoneachedge,saypath(v1;v2;v3)anddistanceconstraints(v1;v2)and(v2;v3),wecangetanequivalentdistanceintervalconstraint[l(v1;v3);r(v1;v3)].Thissimpleobservationgivesusacharacterizationtheoremfordistanceintervalconstraints. 36

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ForObservation 1 .Alltheminimal2-SumcomponentsofG[fcontainingfarepartial2-treesbut2f(G;[l(E);r(E)])isnotalwaysconnected. 2suchthatthedistanceconstraintsonthesubdivision,namely(u1;u)and(u;u2)togetherimposetheoriginaldistanceintervalconstraintontheedge(u1;u2).ThenbyTheorem 4 ,2F(G0;)isalinearpolytope,connectedorconvexforallifandonlyifalltheminimal2-SumcomponentsofG0FcontainingasubsetofFarepartial2-Trees. 1 )includingcomplexityofsamplingandrealization.Thisistherststepinasystematicandgradedprogram-forthecombinatorialcharacterizationsofecientcongurationspaces.Inparticular,theresultspresentedherecharacterizegraphsandtheirCayleyparameters 37

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Genericallygloballyrigidgraphin2D. thatyield2Dcongurationspacesthatareconnected,convex,linearpolytopes,andecientalgorithmsforsamplingrealizations. Ourresultscharacterize(forthecaseofdistanceequalitiesandframeworks)congurationspacepropertiesthatholdforalldistanceassignments,andarehenceincorrectifwerequirethepropertiestoholdonlyingenericsituations.Forexample,Figure 2-12 showsagenericallygloballyrigidgraphG:thegeneric2Dcongurationspaceofthisgraphonthenon-edgefisasinglepoint.However,byourTheorem 1 ,sinceaminimal2-Sumcomponentcontainingfisnot2-realizable,thecongurationspaceonfisdisconnected.TheapparentdiscrepancyarisesbecausetheproofoftheLemma 2 usesnon-genericspecializationsoftheedgedistancesintheprocessofreductiontoaminorwhichthenshowsthatthecongurationspaceonfisdisconnectedforthat,whichissucienttoprovethestatementofLemma 2 andTheorem 1 .Webelieve,howeverthatboththeseresultsstillholdunderagenericityassumption,providedthegraphGisnotoverconstrained(independent,inrigidityterminology)andthenon-edgefisnotglobally-linked.

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8 ; 15 ; 17 ; 18 )and( 19 ). 39

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Inthischapter,wedeneagraphpropertynameduniversallyinherentsquareconvexcongurationspaces.Weprovethatifagraphisd-realizablethenitadmitsuniversallyinherentsquareconvexcongurationspacesind-Dimension.Wealsoprovethatin2Dand3Dagraphis3-realizable(2-realizable)ifandonlyifitadmitsuniversallyinherentsquareconvexcongurationspacesin3D(2D). 2 ,wecharacterizepairs(G;F)suchthatGalwaysadmitsaconnectedorconvexcongurationspaceonF.Sometimes,itismoreconvenienttoinsteadcharacterizethegraphsH=G[F:Inparticular,wesaythatagraphHalwaysadmitsanin-herentconnectedorconvexcongurationspace,ifthereexistsapartitionoftheedgesofHintoE[FsothatthegraphG=(V;E)alwaysadmitsaconnectedorconvexcongurationspaceonF.Inotherwords,foralldistanceassignments(E)(resp.intervals[l(E);r(E)])forthethegraphG=(V;E),thed-Dimensionalcongurationspacedf(G;)(resp.df(G;[l;r])),isconnectedorconvex.Weadditionallyconsiderthefollowingstrongproperty.WesaythatagraphHalwaysadmitsuniversallyinherentconnectedorconvexcongurationspaces,ifforeverypartitionoftheedgesofHintoE[F,thegraphG=(V;E)alwaysadmitsaconnectedorconvexcongurationspaceonF.WeareinterestedincombinatoriallycharacterizinggraphsHthatalwaysadmituniversallyinherentconnectedorconvexcongurationspaces. (1) Agraphisd-realizableifforeveryforwhichtheEDCS(G;)hasaEuclideanrealizationinanydimension,italsohasarealizationinRd.Thisusefulnotionofd-realizabilitywasintroducedby( 3 ; 4 ),whichalsoshowedaforbiddenminorcharacterizationofsuchgraphsford3.Foranydimensiond,weshowinTheorem 6 thattheclassofd-realizablegraphsalwaysadmituniversallyinherentconnectedcongurationspaces,thatareinfactconvexoversquaredCayleyparameters.Werefertothoseassquareconvexcongurationspaces.ThisresultholdsalsowhenthecorrespondingEDCSusedistanceintervals. 40

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Theorem 7 showsthereversedirectionof(5)for3D,andthusshowsthat3-realizablegraphsHareexactlytheonesthatalwaysadmituniversallyinherentconnectedandsquareconvex3Dcongurationspaces,alsowhenthecorrespondingEDCSusedistanceintervals.Thus,by( 3 ; 4 ),thisclassalsohasforbiddenminorcharacterization.InObservation 10 (deferredtoChapter 5 ),weobservethatbothTheorem 6 andTheorems 7 areweakstatementsfor2D{muchstrongerstatementsfollowdirectlyfrom(2)above.Forexample,itfollowsfrom(2)thatifagraphisnot2-realizable,thenthereisanaturalcomponentofthegraphthatdoesnotevenadmitaninherentconnectedcongurationspacedescription,leavealoneauniversallyinherentone. Conceptually,theresultsonglobalrigidity( 17 ; 18 )(resp.globallylinkedpairs( 19 ))arerelated,sincetheycombinatoriallycharacterizewhenthecongurationspaceonall(resp.specic)Cayleyparameters(non-edges)isasinglepoint.However,thesecharacterizationsholdonlygenerically( 8 ; 15 )asiscustomaryformanycombinatorialpropertiesrelatedtorigidity.Incontrast,ourcharacterizationsapplytoallEDCS'(frameworks)andnotjustgenericframeworks.Thisisacrucialdistinctionthatisneededtoreconcileapparentdiscrepanciesofthetwotypesofresults,asweelaborateinSection 2.4 Ourresultsadditionallyyieldacombinatorialcharacterizationofsamplingcomplexity.Thisincorporatesthecomplexityof(i)obtainingthechosensetofparametersand(ii)thealgebraiccomplexityofobtainingthedescriptionofthecongurationspacefromthegivengraph.Thisinturnincludesthedescriptivecomplexityofthecongurationspaceasasemi-algebraicset,suchasthenumberanddegreeofthepolynomials.Thecharacterizationsmoreoverincorporaterealizationcomplexity,i.e,thecomplexityofobtainingarealizationfromaparametrizedconguration. 41

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27 )thatshowstheequivalenceofTree-orTriangle-decomposability( 26 )andQuadraticorRadicalrealizabilityforplanargraphs. ConcerningtheuseofCayleyparametersornon-edgesforparametrizingagenericallycompletecongurationspace:( 30 )aswellas( 21 ; 32 )studyhowtoobtain\completions"ofunderconstrainedgraphsG,i.e,asetofnon-edgesFwhoseadditionmakesGminimallyrigidorwell-constrained.BotharemotivatedbyrealizationcomplexityofunderconstrainedEDCS:i.e,ecientlyobtainingarealizationgiventheparametersvaluesofaconguration,i.e,oncethedistancevaluesofthecompletionedgesinFaregiven.Inparticular( 21 )alsoguaranteesthatthecompletionensuresTree-orTriangle-decomposability,therebyensuringlowrealizationcomplexity.However,theydonotaddressthegeometric,topologicaloralgebraiccomplexityofthesetofdistancevaluesthatthesecompletionnon-edgescantake,northecomplexityofobtainingadescriptionofthiscongurationspace,giventheEDCS(G;)andthenon-edgesF.Thelatterishoweverthemainfactorinuencingthetractabilityofsamplingthesecongurationspacesbeforeobtainingthecorrespondingrealizations.Theproblemhasgenerallybeenconsideredtoomessy,andbarringeectiveheuristicsforcertaincases,forexamplein( 24 ),therehasbeennosystematic,formalprogramtostudythisproblem. Theproofs(e.g.Theorem 6 )relatedtouniversallyinherentCCSemployanunusualinterplayof(i)classicalanalyticresultsrelatedto(squared)Euclideandistancematrices,suchaspositivesemi-denitenessthatdatebackto( 29 ),with(ii)recentgraph-theroreticcharacterizations( 3 ; 4 )relatedtod-realizability.Thisfurtherpermitsustodirectlyapplyarecentresultaboutecientrealizationof3-realizableEDCS( 31 ). 6 thatd-realizablegraphsadmituniversallyinherentconnectedandsquareconvexd-Dimensionalcongurationspaces.Beforethat,werstshowinLemma 5 thatsquareconvexityimpliesconnectedness. 42

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Proof. Nowwearereadytogivethetheoremthatd-realizablegraphsadmituniversallyinherentconnectedandsquareconvexd-Dimensionalcongurationspaces. Proof. 5 ,weonlyneedtoprovead-realizablegraphadmitsuniversallyinherentsquareconvexcongurationspaces. AnnmatrixMisaEuclideansquaredistancematrix(EDM)if9p1;:::;pn2Rdforsomedsuchthatjjpipjjj2=M(i;j).Aclassicalresult( 29 )thatfollowsfrompositivesemidenitenessofGrammatricesisthatthesetofallEDM'sisaconvexcone(notethatd,andhencetherankofthesematricesisnotxed).TheprojectionofthisconeonanysetE[Fofpairs(i;j)isalsoconvex.Bythedenitionofd-realizabilityofagraphH=(V;E[F),withjVj=n,thisprojectionisexactlythesetofallsquareddistanceassignments()2tothepairsinE[Fforwhich(H;)hasarealizationinRd.Wedenotethis(dE[F)2((V;);).Sinceconvexityispreservedbybothsectionsandprojections,the 43

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Theorem 6 givesonedirectionforalldimensions.WeconjectureinSection 3.5 thereversedirectionisalsotrue.Inthenextsection,wewillprovethatthereversedirectionistrueford3. Theorem7. (1) (2) (3) 4 showsthatifagraphHisnot2-realizable,thenithasaminimal2-Sumcomponentthatisnot2-realizable(notapartial2-tree).Andthiscomponentdoesnotadmitanyinherentcongurationspace,leavealoneuniversallyinherentones.Inotherwords,inthisnon2-realizableminimal2-SumcomponentHC,onavertexsetVCforeverypartitionofedgesintoEC[FC,thecongurationspace2FC(GC;)ofgraphGC=(VC;EC)isdisconnected.Ford=3,ontheotherhandnosuchstrongstatementholdsasshowninthecounterexampleofObservation 10 .Toshowtheabovetheorem,wemerelyshowthatifagraphHisnot3-realizable,thenthereexistsapartitionoftheedgesofHintoE[F,suchthatthecongurationspace3F(G;)ofthegraphG=(V;E)isdisconnected.

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5 proves(3))(2),andTheorem 6 proves(1))(3)foralldimensionsd.Basedontheaboveremark,werestrictourselvestod=3,andjustprove(2))(1). Foranynon-3-realizablegraphH=(V;E0),wendapartitionofE0asE[FwhereGisthegraph(V;E)andndadistanceassignmentsuchthatthecongurationspace3F(G;)isdisconnected.HereG:=(V;E)andfisasinglenon-edgeofG,sothisisa1-parametercongurationspaceandweshowthatithas2isolatedpoints.Todothis,westartfromthefollowingtheorem. 3 ; 4 )). 2 ,weshowhowtopickfromHthegraphGanditsnon-edgesFsuchthatbyarestrictedreductionthatusesonlyedgecontractions(noedgeremovals)andpreservingthenon-edgesF,wecanreducethegraphGtoaK5orK2;2;2thatismissingexactlyoneedge,namelyf,ontowhichallthenon-edgesinFhavebeenmappedbythereduction.Finally,weobtainusingLemma 6 thedistanceassignmentforGbysettingthedistancesforthecontractededgesto0;and,similartoLemma 1 wepickdistanceassignmentsfortheun-contractededgesinsuchawaythatthetwobasecasesdonothaveconnected3Dcongurationspacesonf.Werstproveasimplefact. 8

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DistanceassignmentforK2;2;3withoneedgeremoved:(a)K2;2;2withoneedgefremoved;(b)seeproofofLemma 6 :adistanceassignmenttoK222nfsuchthatthe3Dcongurationspaceonfisnotconnected. 8 .IfagraphhasaK5minor,useFact 1 ,itcanbereducedtoK5byedgecontractionsalone,soweonlyneedtoprovethecasethatGhasaK2;2;2asminor.IfGhasaK2;2;2asminor,bydenitionofminorwecangetaK2;2;2byrstcontractingsomeedges,thenremovingsomeedges,andnallyremovingsomeisolatedvertices.Followthisreductionpathbutstopafteredgecontractions,anddenotethenewgraphbyG0.DenotetheK2;2;2subgraphofG0byM. NowwewillshowwecaneithergetMoraK5byedge-contractionsandremovingsomeisolatedverticesifnecessary.Thestrategyisstraightforward:successivelycontractoneedgeatatime,whosetwoverticesarenotbothinMuntilwecannotcontinue.Afterremovinganypossibleisolatedvertices,theremaininggraphhasexactly6vertices(thesameasM)andwedenotedthisgraphbyM0.WeknowthatK2;2;2isasubgraphofM0.NowweuseasimpleobservationthatbyaddingoneormoreedgestoK2;2;2,wecangetaK5byedgecontractionsalone,thusGcanbereducedtoK5orK2;2;2byedgecontractionsalone. Forprovingthenextlemma,wegiveanappropriatedistanceassignmenttoK5andK2;2;2toshowtheydonotadmituniversallyinherent,connectedcongurationspaces. 46

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Proof. ForK5case,wetakeftobe(v1;v2)andassignthesamedistancestoalltheedgesinK5nf.Ifwextetrahedron(v1;v3;v4;v5),thenv4caneitherbecoincidentwithv1oristhereectionofv1aboutplane(v3;v4;v5).Sincev1isnotintheplane(v3;v4;v5),sointhelattercase,(f)isnotzero.Intheformercase,(f)iszero.Thesetwovaluesareallthepossiblevaluesof(f),sowehaveprovedthat3f(K5nf;)isnotconnected. ForK2;2;2case(Figure 3-1 ),wechooseedge(v5;v6)asf.Wechooseadistanceassignmentsothatthefollowingconditionsaresatised:(v1;v2)=(v2;v3)=(v1;v3)=(v1;v4)=(v3;v4)=(v2;v5)=(v3;v6)=(v4;v5),(v1;v4)+(v4;v6)=(v1;v6),(v4;v6)>0,and(v2;v6)willlet\(v2;v1;v6)= Because(v1;v4)+(v4;v6)=(v1;v6),v1,v4andv6arecollinearandthefourverticesv1,v2,v4andv6arecoplanar.Because\(v2;v1;v6)= 47

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7 7 .ByTheorem 6 ,a3-realizablegraphHadmitsuniversallyinherentconnectedandsquareconvex3Dcongurationspaces,soweonlyneedtoprovethereversedirection.IfHisnot3-realizable,byTheorem 9 ,wecangetaK5orK2;2;2byedgecontractionsalone. Asmentionedbefore,ndapartitionoftheedgesetofHintoF[EthatdenesasubgraphG=(V;E)andanon-edgesetFforG.ThenndadistanceassignmentforEsuchthat3F(G;)isdisconnected. SinceHcanbereducedtoK5orK2;2;2byedgecontractionsalone,wepickanedgefromthecorrespondingminoranddenoteitf.WechooseFtobealltheedgesthatwereidentiedwithfbythereduction.ForthedistanceassignmenttotheedgesofG,wewilluseasimilarmethodasintheproofofTheorem 1 .Lemma 6 givesadistanceassignmentforK2;2;2andK5thatensuresthatthecongurationspaceontheedgefisdisconnected.SetthedistancesofeachuncontractededgeeduringthereductionofGtothedistanceassignmentoftheedgeinK5orK2;2;2thatewasidentiedwith.SetthedistancesofallthecontractededgesofGtobe0.Thisensuresthat3F(G;)isdisconnected. 48

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5 .OurrstconjectureisthereversedirectionofTheorem 6 8 ; 15 )aboutconnectedcomponentsofthed-Dimensionalrealizationspacesofagraphbasedonitshigher,d0-Dimensionalrealizationspaces,whered0>d. NotethatwhileweknowfromTheorem 7 that3-realizablegraphsalwaysadmituniversallyinherentsquare-convex3Dcongurationspaces,wedonotyetknowanecientalgorithmtodeterminetheirdescription.Thisisnecessarytodeterminethesamplingcomplexity.Thisisinstarkcontrasttothelineartimealgorithmforobtainingsuchdescriptionsinthecaseof2D(Theorem 4 ). OnestraightforwardalgorithmforobtainingthedescriptionofthecongurationspacedF(G;)asasemi-algebraicsetistostartwiththeCayley-Menger( 7 ; 6 ; 25 )determinantalequalitiesandinequalitiesforEuclideandistancematricesind-Dimensions.ThesearepolynomialrelationshipsbetweenallthejVj2Cayleyparameters,includingthoseinE,Fandthosein Viewedinthismanner,itappearsremarkablethatin2D,forthegraphsGandnon-edgesetsFsatisfyingtheconditionsofTheorem 4 ,thisabove-describedeliminationleavesonlythetriangleinequalitiesrelatingtheCayleyparametersinEandF(notethatthesewerepartoftheoriginalCayley-Mengersetofinequalities)andhencewegetalinearpolytopedescriptionof2F(G;).Notehoweverthatwedidnotusesuchaneliminationforourproof!.Ourproofthatthesetriangleinequalitiesgiveadescriptionof2F(G;)wasthroughamoredirectroute:wedeterminedforwhatconguration(F)forthe 49

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Wewouldliketoshowasimilarresultin3DeitherbyusingeliminationorusingEuclideandistancematrixcompletionforxedrank( 1 ; 5 )byamoredirectrouteofdeterminingwhen3Drealizationscanbeconstructed. 4 .Forthispurpose,itmakessensetostudycomplete2-treesHonasetofverticesV.Inparticular,afterpartitioningtheedgesofHintoE[FandeliminatingallexceptthesetofCayleyparametersinEweobtainasystemoflinearinequalitiesinthoseparameters,whichwecallthe2DadmissibledistancepolytopeofG=(V;E).I.e,itspeciesthedistanceassignmentsforwhich(G;)hasa2Drealization.Notethatthisistheprojectionofthe2DadmissibledistancepolytopeforH,ontotheparametersinE.Furthermore,thecongurationspace2F(G;)isasectionorabreofthispolytope.Itisfutureworktostudythedetailedstructureofthesepolytopes. 50

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InChapter 2 ,wehavecharacterizedthegraphthatalwaysadmits2Dconnectedcongurationspaces.Thatclassofgraphsistightlyrelatedtopartial2-Tree.Inthischapter,wegobeyond2Dconnectedcongurationspacesintopologycomplexity.Thegraphofinterestis1-DofHenneberg-Igraph,whichisaccordinglybeyondpartial2-Tree. Thisquestionreferstofoureciencyfactors(samplingcomplexity,realizationcomplexity,genericcompletenessandtopologicalcomplexity)aswellasanadditionalimportanteciencyfactor:geometriccomplexity,e.g.,convexity.InChapter 2 andChapter 3 ,aseriesofexactcombinatorialcharacterizationsaregivenforconnected,convexandcompletecongurationspacesoflowsamplingandrealizationcomplexityforgeneral2Dand3DEDCSs(includingdistanceinequalities),andasomewhatweakercharacterizationisgivenforarbitrarydimensionalEDCSs. Inthischapter,wetaketherststepinoneoftwonaturaldirectionstomovebeyondChapter 2 andChapter 3 whichcharacterizesgraphswhoseEDCSalwaysadmitconvexand/orconnected2Dcongurationspaces.Onepossibleextensiondirectionistoaskwhichgraphsalwaysadmit2Dcongurationspaceswithatmost2connectedcomponents.Asecondpossibledirection,istotakethesimplestnaturalclassofgraphswith1-Dof(genericmechanismswith1-degree-of-freedom)thatdonothaveconnectedcongurationspaces,andcombinatoriallyclassifythembasedontheirsamplingcomplexity.Thisisthedirectionwetakehere. 51

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22 ).WecallthemSimple1-DofHenneberg-Igraphs(formallydenedinSection 4.3.1 ). ForthisclassofgraphsG,werstobservethatthenaturalchoiceofcongurationspaceparameteristhebaseedgefthathasbeendeleted,i.e,thebasenon-edge.Thischoiceyieldstheoptimalrealizationcomplexityforobtainingarealization,givenaparameterizedcongurationin2f(G;).FixingthisparameterchoicereducestheabovequestiontowhensuchacongurationspacehaslowsamplingcomplexitysincetherealizationproblemforHenneberg-Igraphsisknowntobeeasy.Specically,thisisa1-parametercongurationspace,andhenceitconsistsofaunionofintervals.Thesamplingcomplexityisthusthecomplexityofcomputingtheendpoints(orboundaries)oftheseintervals,startingwiththeEDCS(G;)asinput.OurmaincontributionstoQuestion1atthebeginningofthesectionarethefollowing. (1) InordertoquantifysamplingcomplexityweprovethecrucialTheorem 10 thatassociatesacombinatorialmeaningtotheendpointsoftheintervalsinthecongurationspace2f(G;)foraSimple1-DofHenneberg-IgraphGwithbasenon-edgef.Specically,weshowthattheendpointsareassociatedwithcongurationsofextremegraphsobtainedfrom(G;f).TheseareformallydenedinSection 4.3.1 areareusedprominentlyinTheorem 10 .TheproofofthislemmarequiresbasicalgebraandrealanalysisandalsoyieldsObservation 2 thatensureslinearrealizationcomplexityfor2f(G;). Basedonthecombinatorialdescriptionofthecongurationspaces2f(G;)giveninTheorem 10 ,anaturaldenitionoflowsamplingcomplexityisobtainedbyrequiringalltheextremegraphsobtainedfrom(G;f)tobelonginaclassofgraphscalledTree-orTriangle-decomposablegraphswhoseEDCS(G;)areknown( 27 )tohaverelativelylowalgebraiccomplexityofrealization.Hencethecorrespondingcongurationscanberelativelyeasilyobtained.ByTheorem 10 ,thisimpliesalowcomplexityforcomputingtheendpointsoftheintervalsofthe1-parametercongurationspace2f(G;). (2) Basedontheabovedenitionoflowsamplingcomplexity,Theorem 11 givesasurprisingandexactforbidden-minorcharacterizationoflowsamplingcomplexity 52

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3 notesthattheintervalendpointsofthe1parametercongurationspacesofsuchgraphsGcanbeobtaineddirectly,withoutrealizingtheTriangle-decomposableextremegraphscorrespondingtoG.Thisissomewhatcounterintuitiveforthefollowingreasons:asmentionedabove,thecharacterizationinTheorem 11 reliesontheTriangle-decomposablilityofalltheextremegraphsofG;additionally,byTheorem 10 ,theintervalendpointsofthecongurationspacecorrespondtocongurationsofextremegraphs;andthisinturnwasourmotivationforequatinglowsamplingcomplexitywithpurelyTriangle-decomposableextremegraphs,sincetheircongurationscanbeeasilyobtained. (3) Next,thetightnessofTheorem 11 isillustratedbyObservation 4 ,Observation 5 andObservation 6 ,whichgivesseveralexamplesofgraphsthatindicateobstructionstoobtainingforbiddenminorcharacterizationswhenanyoftherestrictionsofTheorem 11 areremoved.Whensomeofthemareremoved,however,Theorem 12 givesacharacterizationofmoregeneralSimple1-DofHenneberggraphswithlowsamplingcomplexity,withTriangle-decomposableextremegraphs.Observation 7 howevergivesexamplesfromthismoregeneralclassforwhichitisnotposibletousethedirectmethodofObservation 3 forobtainingtheintervalendpointsinthecongurationspace. (4) WeshowinTheorem 13 thatforHenneberg-IgraphsH,allpossiblebaseedgesfyieldequallyecient(orinecient)congurationspacesforthecorrespondingSimple1-DofHenneberg-IgraphG,i.e,whereH=G[f.Thisisaninterestingquantierexchangetheorem.Besidesprovidingacharacterizationofallpossibleparametersthatyieldecientcongurationspaces,thetheoremillustratestherobustnessofourdenitionoflowsamplingcomplexity. 53

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Tothebestofourknowledge,theonlyknownresultinthisareathathasasimilaravorofcombinatoriallycapturingalgebraiccomplexityistheresultof( 27 )thatrelatesquadraticsolvabilityandTree-orTriangle-decomposabilityforplanargraphs. ConcerningtheuseofCayleyparametersornon-edgesforparametrizingthecongurationspace:thepapers( 21 ),( 30 )and( 32 )studyhowtoobtain\completions"ofunderconstrainedgraphsG,i.e,asetofnon-edgesFwhoseadditionmakesGwell-constrainedorminimallyrigid.AllaremotivatedbytheneedtoecientlyobtainrealizationsofunderconstrainedEDCS.Inparticular( 21 )alsoguaranteesthatthecompletionensuresTree-orTriangle-decomposability,therebyensuringlowrealizationcomplexity.However,theydonotevenattempttoaddressthequestionofhowtondrealizabledistancevaluesforthecompletionedges.Nordotheyconcernthemselveswiththegeometric,topologicaloralgebraiccomplexityofthesetofdistancevaluesthatthesecompletionnon-edgescantake,northecomplexityofobtainingadescriptionofthiscongurationspace,giventheEDCS(G;)andthenon-edgesF,noracombinatorialcharacterizationofgraphsforwhichthissamplingcomplexityislow.Thelatterfactorshoweverarecrucialfortractablyanalyzinganddecomposingunderconstrainedsystemsandforsamplingtheircongurationspacesinordertoobtainthecorrespondingrealizations.Theproblemhasgenerallybeenconsideredtoomessy,andtherehasbeennosystematic,formalprogramtostudythisproblem.Ontheotherhand,( 24 )givesacollectionofusefulobservationsandheuristicsforcomputingtheintervalendpointsinthecongurationspacedescriptionsofcertaingraphsthatariseinrealCADapplications. 54

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4.3.1DenitionandBasicPropertiesofSimple1-DofHenneberg-IGraphs 4-2 .Weconsiderthisclassbecauseitisthesmallestnaturalclassthatcontains2-trees(sometimescalledgraphsoftree-width2)whichgureprominentlyinthecombinatorialcharacterizationsofconvexandconnectedcongurationspacesfor2DEDCSinChapter 2 Inotherwords,Henneberg-Igraphsarethesimplestgeneralizationof2-treeswhichdonothaveconvexorconnectedcongurationspaces.Henneberg-IgraphsareanaturalsubclassofLamanorminimallyrigidgraphs,andalsoofanothercommonclassofgraphscalledTree-orTriangle-decomposablegraphs( 10 ),thatareconjecturedtobeexactlyequivalenttoquadraticallysolvablegraphs,aconjecturethathasbeenprovenforplanar( 27 ). Figure4-1. Tree-DecomposableGraph:agraphGisTree-DecomposableifitcanbedividedintothreeTree-DecomposablesubgraphsG1,G2andG3suchthatG=G1[G2[G3,G1\G2=(fv3g;;),G2\G3=(fv2g;;)andG1\G3=(fv1g;;)wherev1,v2andv3arethreedierentvertices;asbasecases,apureedgeandatrianglearedenedtobeTree-Decomposable. AgraphGisTriangle-DecomposableorTree-Decomposable,if: 55

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4-1 )( 10 ). WealsosayG1,G2andG3areclustersandv1,v2andv3aresharedvertices. Figure4-2. Henneberg-Igraph,Simple1-DofHenneberg-Igraphandextremegraph.(a)Henneberg-Igraph:(v1;v2)isthebaseedge;(b)Simple1-DofHenneberg-Igraph:(v1;v2)isthebasenon-edge;(c)Theextremegraphof(b)thatcorrespondstov7/(v5;v6);itisalsoaK3;3graph.Forboth(a)and(b),theHenneberg-Iconstructionscontain(v3/(v1;v2);v4/(v1;v2);v5/(v1;v3);v6/(v2;v4);v7/(v5;v6)). ASimple1-DofHenneberg-IgraphGisobtainedbyremovingabaseedgeffromaHenneberg-Igraph(notethattherecanbemorethan1possiblebaseedgeforagivenHenneberg-Igraph,refertoFigure 4-14 ).Suchanedgefiscalledabasenon-edgeofG.TheEDCSs(G;)basedonsuchgraphsgenericallyhaveoneinternaldegreeoffreedomandhenceacomplete,1-parametercongurationspace. Figure4-3. SimpleHenneberg-Igraph:(a)Henneberg-Igraphwith(v1;v2)asbaseedge;(b)Simple1-DofHenneberg-Igraphwith(v1;v2)asbasenon-edge;(c)extremegraphof(b)thatcorrespondstov7/(v5;v6);itisalsoaC3C2graph.Forboth(a)and(b),theHenneberg-Iconstructionscontain(v3/(v1;v2);v4/(v1;v2);v5/(v3;v4);v6/(v1;v2);v7/(v5;v6)). 56

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22 );i.e.,jEj=2jVj3andjEsj2jVsj3forallsubgraphsGs=(Vs;Es)ofG;GisunderconstrainedorindependentandnotrigidifwehavejEj<2jVj3andjEsj2jVsj3forallsubgraphsGs.AgraphGisoverconstrainedordependentifthereisasubgraphGs=(Vs;Es)withjEsj>2jVsj3.GiswelloverconstrainedorrigidifthereexistsasubsetofitsedgesE0suchthatthegraphG0=(V;E0)iswellconstrainedorminimallyrigid.Agraphisexibleifitisnotrigid. ThenotionofanextremegraphofaSimple1-DofHenneberg-IgraphGwithbasenon-edgefwillbeusedprominentlyinourresults.ThekthextremegraphXkbasedonGandfisobtainedfromGbyaddinganewedge(u;w)betweenthebasepairofverticesuandwofthekthHennebergconstructionstepvk/u;w,providedu;wdonotbelongtoanywell-constrainedsubgraphofG(otherwise,thekthextremegraphisoverconstrainedandirrelevant-dependingonthecontextitcouldbeleftundened).FortheEDCS(G;)andthenon-edgef,thekthextremeEDCSXk;j;j=1;2is(Xk;j),wherethej=1;2representstwopossibleextensionsoftothenewedge(u;w):1(u;w):=(u;vk)+(vk;w),and2(u;w):=j(u;vk)(vk;w)j. Nextweproveaseriesoffactsgivingbasicpropertiesof1-DofHenneberg-Igraphsthatwillbeusedinourmainresultsandareadditionallyofindependentinterestsincethesegraphsarecommonlyoccuring. Proof. 22 ):i.e,thenumberofedgesofanysubgraphisatmosttwicethenumberofverticesminus3.First,weconsiderthelistofverticesofGobtainedfromaHenneberg-IconstructionsequencesforGwithbasenon-edgef.Thatis,sisaHenneberg-I 57

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ThenweprovethatasubgraphG0ofaSimple1-DofHenneberg-Igraphiswellconstrained(minimallyrigid)ifandonlyifG0isaHenneberg-Igraph.OnedirectionisclearsinceanyHenneberg-Igraphiswellconstrained. Fortheotherdirection,byLaman'stheorem( 22 ),thenumberofedgeofG0hastobe2n3ifG0iswellconstrained.WehavejustprovedthatthenumberofedgeinG0doesnotexceed2n3.Fortheequalitytobetrue,theremustbeoneedgebetweenthersttwoverticesinthesublistandanyvertexinthethirdorhigherslotinthesublistmustbeadjacenttoexactlytwoverticesbeforeitinthesublist.BythedenitionofHenneberg-Igraph,thisimpliesG0hastobeaHenneberg-Igraph. Proof. 2 anditsproof.ThatlemmastatesthatG0mustbeaHenneberg-IgraphifitiswellconstrainedanditsproofpointsoutthatifG0containsv1andv2,theremustbeanedgethersttwoverticesinthesublistwhicharev1andv2here.Thiscontradicts(v1;v2)beingthebasenon-edgeofG. 58

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(1) foranym,theextremegraphcorrespondingtovm/(um;wm),i.e.,thegraphobtainedbyaddingtheedge(um;wm)iswellconstrainedifandonlyifthereisnowellconstrainedsubgraphinGthatcontainsbothumandwm. (2) IfthereexistsasubgraphG0containingumandwmthatiswellconstrained,thenwecansaythefollowing.TakingGm1tobethegraphconstructedbeforevmandletGm=Gm1[vmNowforanydistanceassignmentwehave2f(Gm;)=2f(Gm1;)or2f(Gm;)=;. Proof. 22 )).Thisprovestheotherdirection. For(2),byFact 2 ,G0isaHenneberg-Igraphwithabaseedge,say(vi;vj).IfweremovealltheverticesofG0otherthanviandvj.wecangetasubgraphG.NowGisa2-sumofG0andG,i.eG0andGhingedtogetheratanedge,soforany(G;)hasarealizationifandonlyif(G;)hasarealizationand(G0;)hasrealization.Furthermore,either2f(G;)=2f(G;)or2f(G;)=;.NotethispropertyholdsifweaddmoreverticestoG0byHenneberg-Isteps.Thus,wehave2f(Gm;)=2f(Gm1;)or2f(Gm;)=;. 59

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Notethatthisrealizationprocesscouldleadtoanexponentialcombinatorialexplosionbecausethereare2possibleorientationsforeachpointp(v)andonlyoneofthemmaysuccessfullyleadtoarealizationoftheentireEDCS.However,wewillshowinObservation 2 thatwecancircumventthisproblembyencodingalongwitheachparametrizedconguration(f),one(orall)oftheorientations(denedbelow)ofitscorrespondingrealizations.ThustherealizationcomplexityisessentiallylinearinjVj. Withthisinmind,weonlyneedtocharacterizewhichSimple1-DofHenneberg-IgraphsGhavelowsamplingcomplexityfortheircongurationspaceonthebasenon-edgef.Specically,thisisa1-parametercongurationspace,andhenceitconsistsofaunionofintervals.Thesamplingcomplexityisthusthecomplexityofdeterminingtheendpointsoftheseintervals,startingwith(G;)asinput. InordertoquantifyanddenelowsamplingcomplexityweproveacrucialresultTheorem 10 thatgivesacombinatorialmeaningtotheendpointsoftheintervalsinthecongurationspace2f(G;).ThetheoremreliesonatechnicalLemma 7 thatgivescombinatorialdescriptionofthecongurationspace.Theproofrequiresbasicalgebraandrealanalysis. 60

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Infact,observethatthisisa1-1correspondenceprovidedassignsdistinctdistancestotheedgesofH.I.e,foranysuch,thereexistsatmostone2Drealizationpof(H;),whenanorientation(;f)isspecied.Thecoordinatesofp(vk)arenotuniqueonlyifatthekthstepoftheconstructionsequencecthevertexvkisconstructedfromverticesuKandwkforwhichp(uk)andp(wk)arecoincidentand(vk;uk)isequalto(vk;wk).SeeFigure 4-4 NowconsideranEDCS(G;)whereGisaSimple1-DofHenneberg-Igraphwithbasenon-edgef;andassumeassignsdistinctvalues.Foranysuchanddistanceassignment(f)distinctfromthevaluesassignedby,anorientation(;f)(andrealization)for(G[f;;)givesacorrespondingorientation(andrealization)for(G;).Atanyconstructionstep,wecanregard(u;w)forthebasepairofverticesasafunctionof(f).Thenextlemmaanalyzesthisfunctiontogiveacombinatorialdescriptionfor2f(G;). Figure4-4. Whenp(v7)andp(v8)arecoincident,distance(v5;v9)isnotafunctionof(v1;v2). 61

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(1) (2) Foranyintervalendpoint(f)in2f(G;),thereisauniquerealizationfor(G[f;;(f))withtheorientationandthereexistsaHenneberg-Istepv/(u;w)suchthatthethreeverticesv,uandwarecollinearinthisuniquerealization; (3) Foranypairofvertices(u;w)andanyrealizationpof(G[f);;(f))thedistancep(u;w)isacontinuousfunctionof(f)oneachclosedintervalof2f(G;).Furthermore,foranyvertex,v,thecoordinatesofthepointp(v)arecontinuousfunctionsof(f)oneachclosedintervalof2f,ifwepinthecoordinatesofp(v1)tobe(0;0)andthey-coordinateofp(v2)tobe0. 7 .WeprovebyinductiononthelengthofthegivenHenneberg-Iconstructionsequencestartingfromf. Inthebasecase,thelengthofthegivenHenneberg-Iconstructionsequenceis1.Supposev3istheonlyothervertex.Bythetriangleinequality,weknow2f(G;)is[j(v3;v1)(v3;v2)j;j(v3;v1)+(v3;v2)j],so(1)and(2)aresatised.For(3),weonlyneedtoconsiderwhetherthecoordinatesofp(v3)whichwedenoteas(xv3;yv3)areacontinuousfunctionof(f).DenoteR1=(v1;v3),R2=(v3;v2)andR3=(v1;v2)=(f).Wecancompute 62

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2R3:(4{2) NotethatsinceR3isnot0,bothxv3andyv3arecontinuousfuctionsofR3,whichisour(f)now. Byinductionhypothesis,weassumethat(1),(2)and(3)holdforaSimple1-DofHenneberg-IgraphGk1=(V;E)withbasenon-edgefwithlessthankHennebergsteps.SupposewegetanewgraphGkbyonemoreHenneberg-Istepvk/(uk;wk)withbaseverticesuk;wkinGk1.I.e.,Gk=(V[vk;E[(vk;uk)[(vk;wk)).Wewillprove(1),(2)and(3)holdforGk. Figure4-5. ForLemma 7 .Newconstrainton(uk;wk)changestheintervalendpointsin2f(Gk;). AccordingtotheStatement(3)oftheinductionhypothesis,intherealizationpofGk1withaxedorientation,foranypairofvertices(u;w)ofGk1,thedistancevaluep(u;w)isacontinuousfunction,saypu;w,of(f).Weextendtherealizationptothenewlyaddedvertexvk.Nowtheedges(vk;uk)and(vk;wk)willrestrictp(uk;wk)tobein[min;max]wheremin=j(vk;uk)(vk;wk)jandmax=(vk;uk)+(vk;wk).Thisrestrictionwillcreatenewcandidateintervalendpointsin2f(Gk;),namelyp1uk;wk(p(u;v));y2[j(vk;uk)(vk;wk)j;j(vk;uk)+(vk;wk)],asisshowninFigure 4-5 .Sincethesenewcandidateintervalendpointsinf(Gk;)correspondtotherealizationinwhichp(uk),p(vk)andp(uk)arecollinear,(1)and(2)arealsotrueforgraphGk. Toshowtheinductionstepfor(3),takeanynon-edge(u;w).Wehave: 63

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Ifu6=vkandw6=vk,p(u;w)isclearlyacontinuousfunctionof(f),soweonlyneedconsiderthecasethateitheru=vkorw=vk. Forconvenience,rstrotateandtranslatethecoordinatesystemsothatinthetriangle4(uk;wk;vk),ukisattheoriginand~uk;wkisthex-axis.Withoutlossofgenerality,letp(vk)belocatedabovethelinejoiningp(uk)andp(wk),bythegivenorientationinthestatementoftheLemma.DenoteR1=(vk;uk),R2=(vk;wk)andR3=p(uk;wk).Then, and 2R3:(4{5) SincewehaverestrictedR16=R2,wehaveR3>0.Considertherotationandtranslationthatnowputthepointp(v1)attheoriginandp(v2)onthex-axisasinthestatementoftheLemma.Denotetherotationangleas.Thenwehave: cos=xwkxuk sin==ywkyuk Sowecangetthetransformedcoordinatesofp(vk): 64

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4{3 isacontinuousfunctionof(f)evenifu=vkorw=vkandthisprovestheinductionstepofStatement(3)oftheLemma 7 forgraphGk. 7 ,werequirethatthetwodistances(vk;uk)and(vk;wk)arenotequalforthekthHenneberg-Istepvk/(uk;wk).Thisrequirementguaranteesthatthetwopointsp(uk)andp(wk)inarealizationpfor(Gk1;)arenotcoincident,wherebythequantityR3>0andthuswecanuseacontinuityargument. NowwecanstatethetheoremthatinterestsacombinatorialmeaningtothecongurationspaceofaSimple1-DofHenneberg-Igraphusingthenotionofextremegraphsdenedearlier. whereXm;k;j(f)denotesthelengthordistancevalueoffinthemthrealizationpmwithorientationsequenceofthekthextremeEDCSXk;jdeterminedbythepair(G;f). 4 (1)guaranteesthatthegraphcorrespondingtothisextremeEDCSiswellconstrainedprovidedthetwoverticesincidentonthenewedgewerenotpreviouslyinawellconstrainedsubgraph.Iftheywereinawellconstrainedsubgraph,thenthecorrespondingtwoEDCSsXk;1andXk;2canbeleftundened,andthecorrespondingintervalendpointsdonotappearinE(G;)byFact 4 (2). 7 (2). 10 implieslinearrealizationcomplexityofthecongurationspace2f(G;):foreachcandidateorientationsequence,wecanreadoasetofin-tervalsIfromthedescriptionE(G;)asinTheorem 10 ,suchthataconguration

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ForObservation 2 .(Left)showsextremeEDCScongurationsinE(G;)thatareinsomeproperintervalofI,butnotendpoints;(Middle)showsextremeEDCScongurationsthatareendpointsofintervalsinofI;and(Right)showsextremeEDCScongurationsthatareisolatedpointsinI.Thehorizontalaxisdenotes(f),theverticalaxisdenotesfunctionpuk;vk((f))=puk;vk((f)). Proof. 10 theendpointsof2f(G;)formasubsetofthecandidatesetE(G;),whichweviewasaunionovercandidatesetsforeachorientation:SE(G;).Whileeverysuchcandidateconguration(f)isacongurationofanextremeEDCSofG,noteverycandidatecongurationisactuallyanintervalendpointfor2f(G;),norevenanendpointofthesetofintervalsIrequiredinthestatementoftheObservation.Toseethis,recalltheproofforLemma 7 (Figure 4-5 );letvkbethevertexconstructedinthekthstepoftheHennebergconstructionofG[fstartingfromf,andletukandwkbethebaseverticesofthisstep.Considerthecontinuousfunctionpuk;wkinthevariable(f)whichgivesthedistancebetweenukandwkinaparticularrealizationpwithorientation;i.e,thevalueofthiscontinuousfunctionpuk;vkevaluatedat(f)isthedistancep(uk;wk).Figure 4-6 showsthatbasedonthiscontinuousfunction,the2distancevaluesmin=j(v(k);uk)(v(k);wk)jandmax=j(v(k);uk)(vk;wk)j,allthefollowingfourcasesarepossibleforacandidatecongurationXm;k;j(f):neithertheleftnortherightneighborhoodfallsinto2f(G;);boththeleftandtherightneighborhoodfallinto 66

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Basedonsuchadescriptionofthecongurationspace2f(G;),wesayithaslowsamplingcomplexityifalloftheextremeEDCSareTree-orTriangle-decomposable,whichensuresthattheintervalendpointsXki;j(f)intheabovetheoremcanbecomputedessentiallyusingasequenceofsolvingonequadraticequationatatime.ThisensurescomplexitylinearinjVj.IthasadditionallybeenconjecturedthesegraphsexactlycaptureQuadraticSolvabilityandtheconjecturehasbeenprovenforplanargraphs( 27 ). 67

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Allthe1-PathTriangle-FreeSimple1-DofHenneberg-Igraphswithlessthen8vertices;neither(d)nor(h)haslowsamplingcomplexityonbasenon-edge(v1;v2)whilealltheotherhave;both(d)and(h)haveaK3;3minorwhilealltheothersdonothave. ASimple1-DofHenneberg-Igraphwithbasenon-edgefhasthe1-Pathpropertyifexactlyonevertexotherthantheendpointsoffhasdegree2.WesayagraphGisTriangle-FreeifGhasnosubgraphthatisatriangle(seeFigure 4-7 ). (1) (2)

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LetGbea1-PathSimple1-DofHenneberg-Igraphwithbasenon-edgef=(v1;v2).Then (1.a) ifthenumberofverticesdirectlyconstructedwithv1andv2asbaseverticesis3ormore,thenGhasaK3;3minor. (1.b) ifthenumberofverticesdirectlyconstructedwithv1andv2asbaseverticesisexactly2andbothdeg(v1)anddeg(v2)areatleast3,thenGhasaK3;3orC3C2minor. (2) LetGbea1-PathSimple1-DofHenneberg-Igraphwithbasenon-edgef.ThenGdoesnothavelowsamplingcomplexityonfifeitherofthefollowingholds (2.a) thenumberofverticesdirectlyconstructedwithv1andv2asbaseverticesis3ormore,thenGdoesnothavelowsamplingcomplexityonf. (2.b) thenumberofverticesdirectlyconstructedwithv1andv2asbaseverticesisexactly2andbothdeg(v1)anddeg(v2)areatleast3,thenGdoesnothavelowsamplingcomplexityonf. Proof. 4-8 (a)).Sincevnwillbeadjacenttoallui(i=1;;m)inthecontractedgraph,thecontractedgraphhasaK3;3minorwhichisinducedbyv1,v2,vn,u1,u2andu3(v1,v2andvnareasonepartitionandu1,u2andu3astheother. 4-8 (c))orusingabaseedgewhoseverticesareamongv1,v2,u1andu2.Forthelattercase,withoutlossofgenerality,weassumev5isconstructedwithv1andu1(seeFigure 4-8 (b)).Forbothcases,wecontractalltheedgeswhichhaveatleastonevertexotherthanv1,v2,u1,u2,v5andvn.FortheformercaseshowninFigure 4-8 (b), 69

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4-8 (c),thereisaK3;3minorinthecontractedgraphwherev1,v2andv5areinonepartitionandu1,u2andvnareintheother. Figure4-8. EdgecontractionsandgraphminorsforLemma 8 3 thatv1andv2cannotbothbelongtoanywellconstrainedsubgraph,sov1andv2cannotbothbelongtoanyofC1,C2andC3.Vertexu1isadjacenttobothv1andv2,whicharenotbothinacluster,sou1mustbeavertexsharedbytwodierentclustersofC1,C2andC3.Similarly,u2andu3aresharedvertices.Nowu1,u2andu3arethethreesharedvertices(RefertoFigure 4-1 )butv1andv2areadjacenttoallthesethreesharedverticeswhichisimpossible(seeFigure 4-1 ). 3 ,ifC1doesnotcontainedge(u;w),C1willnotbewellconstrained,soC1mustcontainedge(u;w).BecauseGis1-Path,verticesuandwarethetwobaseverticesofthelastconstructedvertexvn,andC1containsv1andv2which 70

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4-9 Figure4-9. ProofofLemma 8 (2b). NextitisnothardtoshowthatGisalsoaSimple1-DofHenneberg-Igraphwithbasenon-edge(u1;u2).ByFact 3 ,nosubgraphofGcontainingbothverticesofabasenon-edgeiswell-constrained.NowC1containsbothu1andu2whicharethetwoverticesofabasenon-edgeforG,soforC1tobewellconstrained,theedge(u;w)hastobelonginC1.ThisimpliesthatbothC2andC3aresubgraphsofG.ByFact 2 ,awell-constrainedsubgraphofa1-DofHenneberg-IgraphhastobeaHenneberg-Igraph,soC2isaHenneberg-Igraph.Further,accordingtotheorderofverticesintheHenneberg-IconstructionsequenceofGstartingfrom(v1;v2)andtheconclusionsofthepreviousparagraph,theedges(u1;v2)and(u2;v2)havetobethebaseedgesforHenneberg-IgraphsC2andC3respectively.ThisrestrictsC2andC3tobepureedges,otherwise,avertexin 71

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(1) ifu1andu2aretheonlyverticesconstructedwithv1andv2asbaseverticesanddeg(v1)is2,then (1.a) (1.b) (2) ifu1andu2aretheonlyverticesconstructedwithv1andv2asbaseverticesandbothdeg(v1)anddeg(v2)are2,then (2.a) ForproofofLemma 9 72

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For(2.a)and(2.b),wecandirectlyextendtheargumentfor(1.a)and(1.b)exceptthatweneedadd/removebothv1andv2thathavedegreeof2. 11 .Onedirectionof(2)inTheorem 11 istrivial:ifthe(extreme)graphG[(uk;wk)isaHenneberg-Igraphwithbaseedge(uk;wk)foranyHenneberg-Iconstructionvk/uk;wkassociatedtoGandthebasenon-edge(v1;v2),thenbythedentionoflowsamplingcomplexity,graphGhaslowsamplingcomplexityon(v1;v2). Weprovethereversedirectionof(1).Considerthenumberofverticeswhicharedirectlyconstructedon(v1;v2).Denoteitbym. 8 (1.a)andLemma 8 (2.a),GhasK3;3orC3C2minorandGdoesnothavelowcomplexityon(v1;v2). 8 (1.b)andLemma 8 (2.b),Gdoesnothavelowsamplingcomplexityon(v1;v2)andhasK3;3orC3C2minor. 73

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8 (2.b),eitherdeg(v1)ordeg(v2)is2.Withoutlossofgenerality,supposedeg(v1)is2andu1,u2arethetwoverticesconstructedwithv1andv2asbasevertices.DenoteGnfv1gbyG0. SinceGhaslowsamplingcomplexityon(v1;v2),byLemma 9 (1.a)and(1.b)soG0haslowsamplingcomplexityon(u1;u2). WecanprovenowbycontradictionthatifGdoesnothavelowsamplingcomplexityon(v1;v2),thenGhasaK3;3orC3C2minor.Assumenot,thenwecanndaGwithminimumnumberofverticessuchthatGdoesnothavelowsamplingcomplexityon(v1;v2)andGdoesnothaveaK3;3orC3C2minor.Considerthenumberofverticesdirectlyconstructedonv1andv2,GcannotbeinCase1,Case2orCase3.So,GcanonlybeinCase4.SinceGdoesnothavelowsamplingcomplexityon(v1;v2),G0doesnothavelowsamplingcomplexityon(u1;u2).GraphGhasnoK3;3orC3C2minorsoG0doesnothaveK3;3orC3C2minoreither.GraphG0haslessnumberofverticesthanGanddoesnothavelowsamplingcomplexityon(u1;u2)anditdoesnothaveK3;3orC3C2minor,sowehaveacontradiction. For(2)andthereversedirectionof(1),wewillproveastrongerargument:ifHenneberg-IgraphGhaslowsamplingcomplexityonbasenon-edge(v1;v2),thenallextremegraph~Gk=G[(uk;wk)isalsoaHenneberg-Igraphwhereukandwkarethetwobaseverticesforthek'thHenneberg-Istepvk/(uk;wk).WeprovethisbyinductiononthenumberofverticesofG. Basecase:ifthenumberofverticesofGis3,Ghaslowsamplingcomplexityon(v1;v2)and~G3isanedge,atrivialHenneberg-Igraph. Assumethat~GkisHenneberg-Iifkn.Fortheinductionstep,wewillprove~Gk+1isalsoHenneberg-I.Recalltheabovefourcases.SinceGhaslowsamplingcomplexityon(v1;v2)andCase1istrivial,soweonlyneedtoconsiderCase4.NowGhaslowsamplingcomplexityon(v1;v2)impliesthatG0haslowsamplingcomplexityonbasenon-edge 74

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NextweshowthatalthoughthelowsamplingcomplexityofthegraphscharacterizedinTheorem 11 havelowsamplingcomplexityresultsfromTriangle-decomposableextremegraphs,theircongurationspacedescription(i.e.,intervalendpoints)canbeobtainedusingadirectmethod,withoutrealizingtheextremegraphs. SeeFigure 4-11 whichillustratesthevariouscasesthatmustbedistinguishedindeterminingthedistanceintervalforf,giventhedistanceinterval[l(e);r(e)]fore.The 75

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Casesthatmustbedistinguishedindeterminingthedistanceintervalforf,giventhedistanceinterval[l(e);r(e)]fore.Thehorizontalaxisdenotes(f);theverticalaxisdenotes(e).Themeaningofthesymbolsare:min:l(e);max:r(e);1:rl(f);2:rr(f);3:ll(f);4:lr(f);5:min(f);6:max(f);e:(e);f:(f). variousquantitiesthatcomeintoplayare:(i)l(e),r(e);(ii)thecorrespondinglengths(iftheyexist)forfll(f),lr(f),andrl(f),rr(f);andmoreover(iii)theoverallmaximumandminimumvaluesforthelengthsofeandfthatarepermittedbythecurve:min(e),max(e),min(f),andmax(f)-asmentionedearlierthesearedeterminedeasilybytriangleinequalitiesusingthe2trianglesbasedontheedgese1;:::;e4ande(resp.f). Inparticular,forFigure 1-3 (right),quadrilateral(v1;v2;v3;v4)has4distanceequalityconstraints:(v1;v3)=7,(v2;v3)=7,(v1;v4)=6and(v2;v4)=8.Boundedbytriangleinequalitiesin4(v3;v4;v5),diagonal(v3;v4)hasanintervalconstraintas[4;5].By 76

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8p 8p 5p 5p Forexample,usingthefollowingstepswecanget2f(G;)(f=(v1;v2))inFigure 4-7 (e): AsimilaralgorithmappliesforFigure 4-7 (f). IntheFigure 4-7 (e)thetwoquadrilateralsforStep(i)andstep(i+1)donotshareanyedgeswhileforFigure 4-7 (f)thetwoquadrilateralsforStep(i)andstep(i+1)maysharetwoedges.GenerallythenumberofthequadrilateraldiagonalintervalmappingstepsisbetweenjVj=2andjVj.NowwegivetheproofforObservation 3 3 .InfacttheobservationissubsumedintheproofofTheorem 11 .Ifthe1-PathTriangle-FreegraphG=(V;E)haslowsamplingcomplexityonbasenon-edgef=(v1;v2),weonlyhavethreepossiblecases.Case1:jVjis3. 77

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4-10 (b)).ForCase3,wecanseethequadrilateralstructurebyanalyzingGnfv1g(seeFigure 4-10 (b)),whichalreadyhasonevertexv2whichisconstructedonbasenon-edge(v3;v4)andweknowdeg(v2)isnot2.Withoutlossofgeneralityweusev5todenotetheothervertexconstructedon(v3;v4)byaHenneberg-Istep.InGnfv1g,ifbothdeg(v3)anddeg(v4)are2(correspondingtoCase2),thenwehavetwoquadrilaterals(v1;v2;v3;v4)and(v2;v3;v4;v5)whichsharetwoedges(v2;v3)and(v2;v4)(refertoFigure 4-7 (c)).InGnfv1g,ifonlydeg(v4)is2(correspondingtoCase1),thenwealsohavetwoquadrilaterals(v1;v2;v3;v4)and(v2;v3;v4;v5)whichalsosharetwoedges(v2;v3)and(v2;v4)(refertoFigure 4-7 (g)).Sincewecanrecursivelyrepeatthisanalysis,ifGhaslowsamplingcomplexityonf,2f(G;)canbecomputedbyanO(jVj)sequenceofquadrilateraldiagonalintervalmappings. 11 istightbyillustratingobstaclestoobtainingaforbidden-minorcharacterizationafterremovingeitheroftherestrictionsofTriangle-Free(Figures 4-12 and 4-13 )and1-Path(Figure 4-14 )usedinthetheorem. Proof. 4-12 .WecanverifythatGisa1-PathSimple1-DofHenneberg-Igraphwithbasenon-edgef=(v1;v2).Amongalltheextreme 78

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Figure4-12. ForObservation 4 .ASimple1-DofHenneberg-IgraphGonbasenon-edge(v1;v2)whichisnotTriangle-FreebuthasasingleHenneberg-Iconstructionpathforv14onbasenon-edge(v1;v2);Ghascongurationspaceoflowsamplingcomplexityon(v1;v2);butG1hasaK3;3minorandG2hasaC3C2minor. ByminormodicationofFigure 4-12 ,wehavethefollowingstrongerobservation.

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4-12 suchthatKmisaminorofG1.Ifwecandothis,theprooffollowssincebyusingthesameargumentasproofforObservation 4 ,weadditionallyhave:bothG1andG2areHenneberg-Igraphswithbaseedge(v1;v3)and(v2;v3)respectively;thereareonlytwoextremegraphswhicharebothwellconstrainedandTriangle-decomposablesoGhaslowsamplingcomplexityon(v1;v2). Weprovebyinductionthatwecanconstructa1-PathHenneberg-IgraphG1withbaseedge(v1;v3)suchthatG1canbereducedtoKmbyedgecontractions.Thebasecases(m=1;2;3)havebeenshowninFigure 4-12 .Astheinductionhypothesis,weassumethatwecanconstructa1-PathHenneberg-IgraphGm1withbaseedge(v1;v3)suchthatGm1canbereducedtoKmbyedgecontractions.NowweprovetheinductionstepforKm+1.WestartfromGm1toconstructGm+11.Wepickmverticesu1;;umfromGm1containingthelastconstructedvertexofGm1andadditionallysuchthattheymaptodistinctverticesinthecontractedgraphKm.Weaddavertexw1byaHenneberg-Istepwithu1andu2asbasevertices.Thenweaddavertexw2byaHenneberg-Istepwithw1andu2asbaseverticesandsoon.Finallyweaddwm1byHenneberg-Istepwithwm2andumasbaseverticestogetGm+11(PleaserefertoFigure 4-13 foraK5example).Clearly,Gm+11isaHenneberg-Igraphwithbaseedge(v1;v3).Thenbycontractingalltheedgesthathaveatleastonevertexotherthanu1,,umandwm1,wegetaKm+1.Thus,wehaveprovedthatGm+11isaHenneberg-IgraphandcanbecontractedtoKm+1. Proof. 4-14 .TheSimple1-DofHenneberg-Igraphisconstructedwithbasenon-edge(v1;v2)anditisnota1-Path.WecanverifythatalltheextremegraphsareinfactHenneberg-IgraphssotheyareTriangle-decomposable.ThisshowsthatGhaslowsamplingcomplexityonf.Ifwecontractalltheedgesthathaveat 80

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ForObservation 5 .ASimple1-DofHenneberg-Igraphonbasenon-edge(v1;v2)thathasoneHenneberg-Iconstructionpathonbasenon-edge(v1;v2);ithasacongurationspaceoflowsamplingcomplexityon(v1;v2)butithasaK5minorshownintheleftcircledsubgraph;ingeneral,itcanhaveaarbitrarycliqueasaminor. leastonevertexotherthanv1,v2,v3,v4,v5andv6,wecangetacliqueK6,soGhasbothK3;3andC3C2minors.Ghasalltherequiredpropertiesoftheobservation. Figure4-14. ForObservation 6 .ASimple1-DofHenneberg-Igraphonbasenon-edge(v1;v2)thathasmorethanoneHenneberg-Iconstructionpathsonbasenon-edge(v1;v2);ithasacongurationspaceoflowsamplingcomplexityon(v1;v2);butithasbothK3;3andC3C2minors.Aside:(v3;v4),(v5;v6),(v1;v5)and(v2;v6)arealsobasenon-edgesandallofthemyieldcongurationspacesoflowsamplingcomplexity. 81

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Figure4-15. A1-Path1-DofHenneberg-Igraphthathaslowsamplingcomplexityonbasenon-edge(v1;v2);exactly1vertexnamelyv3isconstructedonv1andv2.SeeproofofTheorem 12 andObservation 7

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Proof. 8 (1.a),m<3.Whenm=2,assumev3andv4areconstructedwithv1andv2asbasevertices.Sincev1andv2areadjacenttobothv3andv4,sobothdeg(v1)anddeg(v2)areatleast2.ByLemma 8 1.b,deg(v1)anddeg(v2)cannotbebothgreaterthan2,soeitherv1orv2hasdegreeoftwo. ByLemma 9 (1.a)and(1.b)wehave(2)andbyLemma 9 (2.a)and(2.b)wehave(3). 12 leavesthecasewherethenumberofverticesdirectlyconstructedusingv1andv2asbaseverticesisexactly1.SuchgraphsoflowsamplingcomplexityarecapturedinFigure 4-15 TheabovetheoremcharacterizestheSimple,1-Dof,1-PathHenneberg-IgraphsthathaveTriangle-decomposableextremegraphsandlowsamplingcomplexity.ItisnaturaltoaskifthecongurationspacedescriptionforthesegraphscanalsobeobtaineddirectlyasinObservation 3 ,withoutactuallyrealizingtheextremegraphs.Thenextobservationgivesanegativeanswer. 4-15 showsanexampleofaSimple,1-Dof,1-PathHenneberg-Igraphwithlowsamplingcomplexity,forwhichtheintervalendpointsinitscongurationspacecannotdirectlybeobtainedbythemethodofquadrilateraldiagonalintervalmapping(inObservation 3 ). Proof. 4-15 ,thegraphisa1-Path1-DofHenneberg-Igraphthathaslowsamplingcomplexityonbasenon-edge(v1;v2).However,wecannotndasequence 83

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4-14 Proof. TheproofrestsonseveralclaimsontheaboveG,f1andf2whichexcludethepossibilityofminimality. Proof. 84

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2 thesubgraphisaHenneberg-Igraphandalsoa2-sumcomponentofG.SoG0willbealsoaHenneberg-Igraphwithtwobasenon-edgesf1andf2andjustlikeG,G0haslowsamplingcomplexityonf1butdoesnothavelowsamplingcomplexityonf2.NowweconsiderthecasethatGdoesnothaveawellconstrainedsubgraphwhichincludebothunandwn.Inthiscase,sinceGhaslowsamplingcomplexityonf1,itfollowsthatG[(un;wn)isTriangle-decomposable.ComparetheextremegraphsassociatedwithGand(v3;v4)andtheextremegraphsassociatedwithG0and(v3;v4),theformerhasonemoreextremegraphG[(v1;v2).Gdoesnothavelowsamplingcomplexityonf2=(v3;v4),sooneextremegraphassociatedwith(G;f2)isnotTriangle-Decomposable.NowG[(vi;vj)isTriangle-Decomposable,sooneextremegraphassociatedwith(G0;f2)mustnotbeTriangle-Decomposable,thus,G0musthaveoneextremegraphwhichisnotTriangle-DecomposablesuchthatG0doesnothavelowsamplingcomplexityonf2.Inbothcases,G0isaHenneberg-Igraphwithbasenon-edgesf1andf2andjustlikeG,G0haslowsamplingcomplexityonf1butdoesnothavelowsamplingcomplexityonf2.ThisviolatestheminimalityofG,soourassumptionisincorrectandnovertexofGotherthanv1,v2,v3andv4canhavedegree2. Proof.

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2 ,weprovedthatatleastoneofdeg(v1)anddeg(v2)istwoweassumethatdeg(v1)is2sothenumberofverticesconstructedwithv1andv2asbaseverticesisatmosttwo.Wecanshowbycontradictionthatthisnumberisnotexactly2.Supposev5andv6arethetwoverticesconstructedwithv1andv2asbasevertices.ByLemma 9 (2.a),(v5;v6)isalsoabasenon-edgeforG.Ghaslowsamplingcomplexityon(v1;v2),sobyLemma 9 (1.bor2.b),Galsohaslowsamplingcomplexityon(v5;v6).Ifv1isdierentfrombothv3andv4,wehave:Gdoesnothavelowsamplingcomplexityon(v3;v4),Ghaslowsamplingcomplexityon(v5;v6),v1hasdegree2,andv1isdierentfromv3,v4,v5andv6.ThiscontradictstoCliam 1 ,soweonlyneedtoconsiderthecasethatv1isthesameasv3orv4.Withoutlossofgenerality,weassumethatv1isthesameasv3.Gisa1-DofHenneberg-Igraphwithbasenon-edge(v3;v4)andGhasatleast3vertices,soatleastonevertexofGotherthanv3andv4hasdegree2.ByCliam 1 ,onlyv1,v2,v3andv4canhavedegree2,sodeg(v2)hastobe2.Now,Ghaslowsamplingcomplexityon(v5;v6),Gdoesnothavelowsamplingcomplexityon(v3;v4),vertexv2hasdegreeof2andv2isdierentfromv3,v4,v5andv6.ThisagaincontradictstoCliam 1 thusprovestheclaim. 13 Continued.ByClaim 3 thereisonlyonevertexconstructedwithv1andv2asbasevertices,withoutlossofgenerality,supposev9issuchavertex.ConsidertheHenneberg-Istepthatimmediatelyfollowsv9/(v1;v2).Sincev9isthevertexconstructedwithv1andv2asbasevertices,thebaseverticesforthenextHenneberg-Istepareeitherv1andv9orv2andv9.Sincewehavelabeledv1asthevertexwhichhasdegreeof2,wehavetodierentiatethesetwocases.InClaim 4 wediscussthecaseinwhichthenextHenneberg-Istepisv10/(v1;v9)andinClaim 5 wediscussthecaseinwhichthenextHenneberg-Istepisv10/(v2;v9).

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ProofsofClaim 4 andinClaim 5 .Simple1-DofHenneberg-IgraphGhaslowsamplingcomplexityonbasenon-edge(v1;v2)whileGdoesnothavelowsamplingcomplexityonbasenon-edge(v3;v4);vertexv9istheonlyvertexdirectlyconstructedonv1andv2;triangle4(v9;v10;v1)correspondstothesecondHenneberg-Iconstructionfrom(v1;v2);in(a),(v1;v2)and(v3;v4)donotshareanyvertex;in(b)and(c),(v1;v2)and(v3;v4)shareavertex. 4-16 ,(v2;v10)isalsoabasenon-edgeforSimple1-DofHenneberg-IgraphG.FurtherGhaslowsamplingcomplexityon(v2;v10)sinceGhaslowsamplingcomplexityon(v1;v2).ThisisaresultwhichissimilartoLemma 9 (1.b)andcanbeprovedbycomparingallthepossibleextremegraphs.Foranyextremegraphcorrespondingto(v2;v10),thereisanextremegraphcorrespondingto(v1;v2)whichhasoneextraHenneberg-Istepv1/(v9;v10).Ghaslowsamplingcomplexityon(v1;v2),soalltheextremegraphscorrespondingto(v1;v2)aretriangledecomposable.Thus,alltheextremegraphscorrespondingto(v2;v10)arealsotriangledecomposablesinceremovingverticesfromatriangledecomposablegraphbyinverseHenneberg-Istepskeepsthegraphstilltriangledecomposable.So,Ghaslowsamplingcomplexityon(v2;v10). NotethatGhasdoesnothavelowsamplingcomplexityon(v3;v4)anddeg(v1)is2.ByClaim 1 ,v1cannotbedierentfrombothv3andv4(Figure 4-16 (a)).Soweonlyneedtoconsiderthecasev1iscoincidentwithv3orv4.Althoughwelabeledv3asthevertexwithdegreeof2butwedonotusethispropertyhere,sowesupposev1iscoincidentwithv4. Since(v3;v4)isabasenon-edgeforGandv4(justlikev1)isonlyadjacenttov9andv10,sov3mustbeadjacenttoeitherv9(Figure 4-16 (b))orv10(Figure 4-16 (c))inordertoguaranteetheHenneberg-Istepwithv3andv4asbaseverticesispossible.ForFigure 4-16 (b),(v3;v10)isabasenon-edgeforGsince(v3;v4)isabasenon-edgeforG. 87

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1 ,sothecaseshowninFigure 4-16 (b)isimpossible.WecanusethesameargumentforthecaseshowninFigure 4-16 (c)andget:Gdoesnothavelowsamplingcomplexityon(v3;v9),Ghaslowsamplingcomplexityon(v1;v2),v1hasdegreeof2andv1isdierentfromv2,v3,v9andv10.ThisagaincontradictstoClaim 1 ,sothecaseshowninFigure 4-16 (c)isalsoimpossible.NowwehaveshownthatwecannothaveaHenneberg-Istepv10/(v1;v9). Proof. 4-17 andFigure 4-18 ). Letv12denotetheothervertexthatv1isadjacent(weknowthatv1isadjacenttov9).Observethat(v1;v2)isabasenon-edgeforG,sov1mustbeoneofthetwobaseverticesforv12'sconstruction.Denotetheothervertexbyv11.Clearly,beforev12isconstructed,wemusthaveconstructeda1-PathHenneberg-Igraphwith(v2;v9)asbaseedgeandv11asthelastvertex.Wedenotethis1-PathHenneberg-IgraphbyG1andmarkitbyadashedcircleinFigure 4-17 Wecanshowthatv1hastobedierentfromv3andv4.SinceGissimple1-DofHenneberg-Igraphwithbasenon-edge(v3;v4),atleastonevertexotherthanv3andv4shouldhavedegree2.ByClaim 1 ,v1andv2aretheonlypossibleverticeswithdegreeof 88

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ProofofClaim 5 :v1iscoincidentwithv3. 2.Ifv1isv3orv4,deg(v2)is2.Butdeg(v2)cannotbe2byClaim 4 ,sowehaveprovedthatv1hastobedierentfromv3andv4anddeg(v2)isnot2(seeFigure 4-17 ). Fortheremainingcases,wewilluseClaim 1 totargettheimpossbilitywhichisstatedinthelaimwewanttoprove.Todothat,wechangetheedgesofGtogetanewgraphGsuchthat:G0haslowsamplingcomplexityonbasenon-edgef3;G0doesnothavelowsamplingcomplexityonbasenon-edge(v3;v4);v1isdierentfromv3,v4andthetwoverticesoff3. IfweconsidertheHenneberg-Isequencestartingfrom(v1;v2),G1isaHenneberg-Igraphwith(v2;v9)asbaseverticeandthelastvertexisv11.ThismeansthatanyvertexinG1otherthanv2,v9andv11donothavedegree2.ConsiderhowwecanconstructG1intheHenneberg-Isequencestartingfrom(v3;v4).RecalleachHenneberg-Istepinvolves1vertexand2edges.ByusingthesamedofcountingmethodthatusedforFact 2 ,theremustbeanedgebetweenthersttwoverticesinG1,withoutlossofgeneralityweassumethattherstvertexisv13andthesecondisv14.So,G1hastobeaHenneberg-Igraph(maynotbe1-Path)withbaseedge(v13;v14). NowwemodifyG1togetG01suchthatwegettheG0thatweexpect.WekeepalltheverticesinG1butremovealltheedges.OurobjectiveistoaddedgestogetanewgraphG01suchthatG01isaHenneberg-Igraphwithboth(v2;v9)and(v13;v14)asbaseedgesandG01containsedges(v2;v11)and(v9;v11).Toachievethis,werstaddedges(v2;v9),(v2;v11)and(v9;v11).Thenweconsideraddingedgesforv13andv14:ifbothv13andv14

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ForproofofClaim 5 :v1isdierentfromv3andv4. areamongv2,v9orv11,wedonotaddanyedge;ifexactlyoneofv13andv14isoneofv2,v9orv11,weaddedges(v13;v14)andanotheredgebetweenv11andwhicheverofv13andv14isnotoneofv2,v9orv11;ifneitherofv13andv14isoneofv2,v9orv11,weaddedges(v13;v2),(v13;v11)(v14;v13)and(v14;v11).FinallyforeachvertexuinG1otherthanv2,v9,v11,v13andv14,weaddoneedgebetweenuandv2andanotheronebetweenuandv9.WeuseG01todenotethisnewsubgraphthatreplacesG1andG0fortheentiregraph.Bythemannerinwhichweaddedges,ourobjectiveisachieved:G01isHenneberg-Igraphwithboth(v2;v9)and(v13;v14)asbaseedgesandalsocontainsedges(v2;v11)and(v9;v11). Nowobservethatboth(v1;v2)and(v3;v4)arestillbasenon-edgesforG0.Further,(v9;v12)isalsoabasenon-edgeforG.NowweconsiderwhetherG0haslowsamplingcomplexityon(v9;v12)and(v3;v4).Todothat,werefertoTheorem1provedin( 11 ):ifagraphisTriangle-decomposable,wecanperformtheclustermerging(inverseoperationoftriangle-decomposition)inanyorder(aChurch-RosserProperty)butnallywegetoneclusterwhichisthesameasthewholegraph.Soforanygivengraph,ifwereplaceoneofitsTriangle-decomposablesubgraphsbyanothertriangledecomposablesubgraphwhilekeepingtheverticesunchanged,thegraphpreservesTriangle-decomposability.Hereinourtransform,bothG1andG01areHenneberg-IgraphsandthusbothareTriangle-decomposable.ComparetheextremegraphscorrespondingtoGandG0forwhichbasenon-edgeischosenas(v1;v2). 90

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4 ,anextremegraphcorrespondingtotheHenneberg-Istepv/(u;w)iswellconstrainedifandonlyifuandwarenotinanywellconstrainedsubgraph.ObservethatthedierencebetweenGandG0isexactlythedierencebetweenG1andG01.BothG1andG01arewellconstrained,sointhecomparisonofextremegraphswedonotneedtoconsiderextremegraphscorrespondingtotheHenneberg-IstepsinsideG1andG01. ForalltheotherHenneberg-IstepsoutsideG1andG01,thedierencebetweentheextremegraphsforGandG0isexactlythedierencebetweenG1andG0.ThisprovesG0haslowsamplingcomplexityon(v1;v2)sinceGhaslowsamplingcomplexityon(v1;v2).Similarly,wecanshowthatG0doesnothavelowsamplingcomplexityon(v3;v4)sinceGdoesnothavelowsamplingcomplexityon(v3;v4).NowverifyingFigure 4-18 again,(v9;v12)isalsoabasenon-edgeforG0.BycomparisonofextremegraphsaswedidinClaim 4 ,G0haslowsamplingcomplexityon(v1;v2)sinceG0haslowsamplingcomplexityon(v9;v12).ThiscontradictstoClaim 1 ,sowehaveprovedwhenv9istheonlyvertexconstructedwithv1andv2asbaseverticesanddeg(v1)is2,thennovertexcanbeconstructedwithv2andv9asbaseverticeseither. 13 Continued.Nowwecanputallthe5claimstogether.WeassumethatGhaslowsamplingcomplexityonbasenon-edge(v1;v2)butdoesnothavelowsamplingcomplexityonbasenon-edge(v3;v4).WealsoassumethatthenumberofverticesinGisminimumamongallthegraphswiththisproperty.InClaim 1 toClaim 5 ,wediscusswhatpropertiessuchaGshouldhaveinordertokeeptheminimalityofthenumberofvertices.InClaim 1 weshowthatanyvertexotherthanv1,v2,v3andv4cannothavedegree2;inClaim 2 ,weshowatleastoneofdeg(v1)anddeg(v2)(resp.atleastoneofdeg(v3)anddegv4)is2andwithoutlossofgeneralityweassumethatdeg(v1)anddeg(v3)are2;inClaim 3 ,weshowthatthereisonlyvertexthatisconstructedwithv1andv2asbaseverticesandwedenotethevertexbyv9;theresultinClaim 3 narrowstheHenneberg-Istepthatfollowsv9/(v1;v2)toeitherv10/(v2;v9)orv10/(v1;v9),so 91

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4 weshowthatv10/(v1;v9)isinfeasible;nallyClaim 5 showsthattheonlyremainingpossibilitynamelyv10/(v2;v9)resultsinaconsequencethatcontradictstoClaim 1 .ThisimpliesnominimalgraphGcanexistthatcontradictstheconditionsofthetheorem,thusprovingTheorem 13 2 andChapter 3 Asimmediatefuturework,itwouldbedesirabletogiveacleanercombinatorialcharacterizationoflowsamplingcomplexityforcongurationspacesof1-PathSimple1-DofHenneberg-Igraphs.I.e,itwouldbedesirabletoimprovethecharacterizationofTheorem 12 .Thenextnaturalcontinuationistostudycongurationspacesofgraphswithkdofs(k>1)obtainedbydeletingkedgesfromHenneberg-IorTree-orTriangle-decomposablegraphs. 92

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InChapter 2 wehavecharacterizedtheclassofgraphswhosecorrespondingEDCSadmit2Dconnectedcongurationspace;inChapter 3 ,wehavecharacterizedgraphsthatadmituniversallyinherentsquareconvexcongurationspacesandconnectedcongurationspacein2Dand3D.Asfuturework,weupgradethealgebraiccomplexityandaskthesamecombinatorialgraphcharacterizationquestions.Particularly,inSection 5.1 ,wediscusswhichgraphsadmit2Dcongurationspaceonnon-edgesthatalwayshaveatmost2connectedcomponents;inSection 5.2 ,wediscusswhichgraphsadmit3Dconnectedcongurationspaceonnon-edges. 5.1.1QuestionsandContributions (1) InObservation 8 andObservation 9 ,wegivethemajordierencebetween2Dconnectedcongurationspaceand2D2CCS.Lemma 10 givesagraphcharacterizationthatisbasedonObservation 8 .BasedonObservation 9 ,werestrictthediscussionheretothecasesthatthegraphisconnectedafterthetwoendverticesofthenon-edgeareremoved.LaterinConjecture 4 ,werevisitObservation 9 (2) Givenagraphandnon-edgepair(G;f)whereG=(V;E)andf=(v1;v2),wediscusstheminimumnumberofdisjointpathsbetweenv1andv2inG.Wefurthergetsimplepathsfromthedisjointpaths(simplepathsmeansavertexinducedpath 93

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14 thatisacomplete2D2CCSgraphcharacterizationforthecasethattheminimumnumberofdisjointpathsbetweenv1andv2isexactly2. (3) InTheorem 15 ,wegiveasetofbasecasesinwhichgraphsdonothold2D2CCS.Thissetcancovermostcasesinwhichtheminimumnumberofdisjointpathsbetweenv1andv2is3ormore.Weshowtheexceptionisrelatedtooverconstraints. 2-7 ). Proof. 2-6 ). Proof. 94

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ProofforLemma 10 is[1:5;2],[3;3:5]and[3:8;4].Itisclearwecanndappropriategraphsanddistanceassignmenttoachievetheseintervals. ForObservation 8 ,thefollowinglemmagivesagraphcharacterization.ToovercomethehurdledescribedinObservation 9 ,werstassumeGnfv1;v2gisconnected. 5-1 )andGistheminimumsum-componentthatcontainsbothv1andv2,thenGholds2D2CCSonfifandonlyifthereexistsa1-sumcomponentGisuchthatGiadmits2D2CCSonfiandallGj[fj(j6=i)arepartial2-tree. Proof. 2 ,byedgecontractionsbothGiandGjcanbereducedtothebasecasesshowninFigure 2-4 orFigure 2-5 .Ineithercase,fiandfjarenon-edgesofa2-Sumoftwotriangles,thus,wecanndanappropriatedistanceassignmentiandjforGiandGjsuchthatfi(Gi;i)andfj(Gj;j)aretwodistinctvalues.ByObservation 8 ,Gdoesnothold2D2CCSonf. Fortheotherdirection,refertoFigure 5-1 ,ifG1admits2D2CCSonf1andG1admits2Dconnectedcongurationspaceonf2,G1[G2admits2D2CCSon(v1;v4).Byrepeatingthis,wecanshowthatGadmits2D2CCSon(v1;v2). 95

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GraphGhas2D2CCSonnon-edgef=(v1;v2):thecurverepresentsapartial2-treewhichhasthetwoendpointsasvertices;thetwographsareequivalentwithrespecttowhetherGhas2D2CCSonf. Proof. Proof. Nowweonlyneedtoprove:givenquadrilateralG=(v1;v2;v3;v4)thatcontainsnon-edge(v1;v2)andedges(v1;v3),(v1;v4),(v2;v3),(v2;v4)and(v3;v4)andinterval 96

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Case4:Gdoesnothave2D2CCSonnon-edgef. constrainsonallthe5edges,thecongurationspace(v1;v2)(G)alwayshasatmosttwointervals. Letusxpv3andtheorientationfrompv3topv4.Wehavetwocasesthateitherpv1andpv2areonthesamesideofpv3andpv4ordierentsides.Werstprovethatforbothcasesthespaceofpv1isconnectedandthespaceofpv2isconnected.Becauseofthesymmetry,weonlyneedtoproveonecaseandwithoutlossofgeneralitywewillprovethecasethatpv3andpv4areondierentsideofpv1andpv2.Oncethisisproved,clearlyforeithercase,thepossiblevaluesfor(v1;v2)isconnectedso(v1;v2)hasatmosttwointervals. Giventwodistinctvalues1(v1;v2)and2(v1;v2)for(v1;v2),weprovewecancontinuouslychangethedistancevaluesfor(v1;v3),(v1;v4),(v2;v3),(v2;v4)and(v3;v4)such(v1;v2)iscontinuouslychangedfrom1(v1;v2)to2(v1;v2).Thisisclearsincethecongurationspaceon(v1;v3),(v1;v4),(v2;v3),(v2;v4)and(v3;v4)isalwaysaconvexbyTheorem 4 .Thisargumentalsostandsforthecasepv1andpv2areondierentsideofpv3andpv4. 5-3 ,thenGdoesnotadmit2D2CCSonnon-edgef. Proof. 5-3 (a)byG1.Firstwecontractallthe 97

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Weexcludesuchapossibilitybyenumeratingallthepossiblecases: (a) (b) (c) (c.1) ifziisnotadjacenttov1andv5,wecontract(v4;zi); (c.2) ifziisnotadjacenttov1andv3,wecontract(v4;zi); (c.3) ifziisnotadjacenttov5andv3,welet(v5;v1)=(zi;v1),(v5;v4)=(zi;v4)and(v5;v2)=(zi;v2); (c.4) ifziisnotadjacenttov3andv4,wecontract(v5;zi); (c.5) ifziisnotadjacenttov5andv4,we(v5;v1)=(zi;v1),(v5;v3)=(zi;v3)and(v5;v2)=(zi;v2); (c.6) ifziisnotadjacenttov2andv4,wecontract(v5;zi); (c.7) ifziisnotadjacenttov2andv5,wecontract(v3;zi); (c.8) ifziisnotadjacenttov1andv4,wecontract(v5;zi); (c.9) ifziisnotadjacenttov2andv3,wecontract(v5;zi); (c.10) ifziisnotadjacenttov1andv2,wecontract(v3;zi). (d) (d.1) Ifziisnotadjacenttov1,wecontract(v4;zi); (d.2) ifziisnotconnectedtov2,wecontract(v3;zi); (d.3) ifziisnotadjacenttov5,welet(v5;v1)=(zi;v1),(v5;v3)=(zi;v3),(v5;v4)=(zi;v4)and(v5;v2)=(zi;v2); 98

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ifziisnotadjacenttov3,wecontract(v5;zi); (d.5) ifziisnotadjacenttov4,wecontract(v5;zi). (e) Wewillassigndistancevaluestotheedgesthatremain:weassign(v1;v5)=(v4;v5)=(v3;v5)=(v2;v5)=1;weassignappropriatevaluesto(v1;v3),(v3;v4)and(v2;v4)suchthat\(v1;v5;v3)= Fortheremainingedgesbetweencertainzi(1ik)andverticesofG1,weassignallofthemtobe2suchthatthe4valuesabovefor(v1;v2)allarepreserved. Nowwegivethegraphcharacterizationtheoremforclassofgraphsinwhichthenumberofdisjointpathsbetweenthetwoendverticesofthenon-edgeisexactly2. 5-4 (h)). Proof. 5-4 (h)),byLemma 12 Fortheotherdirection,weprovebycases.Firstwetakethetwodisjointpathsbetweenv1andv2inG,whichwedenoteby(v1;t1;;ts;v2)and(v1;z1;;zs;v2).WetakethesubgraphofGwhichisinducedbytheverticesofpath(v1;t1;;ts;v2)andndtheshortestpathbetweenv1andv2inthissubgraph.Clearly,onepropertyofthisshortestpathisthatthereisnoedgebetweenanytwonon-adjacentverticesinthisshortestpath.Inthesameway,wegetashortestpathbetweenv1andv2inthesubgraph 99

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ProofforTheorem 14 inducedbytheverticesofpath(v1;z1;;zs;v2).Withoutlossofgeneralityweassume(v1;t1;;ts;v2)and(v1;z1;;zs;v2)aresuchshortestpaths. WeremovealltheverticesinthesetwoshortestpathsfromGandgetasetofconnectedcomponents.SinceGnfv1;v2gisconnected,atleastonecomponentsisadjacenttoatleastonevertextiandatleastonevertexzj.WetakesuchacomponentH.WeproveHisadjacenttoexactlyonevertexin(t1;;ts)andexactlyonevertexof(z1;;zs)butneitherv1norv2.Otherwise,therearetwocases: 5-4 (a)and(b)).Then,theunionofHandthetwoshortestpathscanbereducedtoFigure 5-4 (c)byedgecontractions.BythepropertyoftheshortestpathsandthewaywecangetH,thisunionisalsoavertexinducedsubgraph.ByLemma 13 ,Gdoesnotadmit2D2CCSonf.So,thecaseshowninFigure 5-4 (a)and(b)cannothappenwhenGdoesnotadmit2D2CCSonf. 5-4 (d))andeitherv1orv2(withoutlossofgeneralityweasumev2).NotethecaseHisadjacenttobothv1andv2cannothappen,otherwise,there 100

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5-4 (e)byedgecontractions.BythepropertyoftheshortestpathsandthewaywecangetH,thisunionisalsoavertexinducedsubgraph.AlsobyLemma 13 ,Gdoesnotadmit2D2CCSonf.So,thecaseshowninFigure 5-4 (d)cannothappenwhenGdoesnotadmit2D2CCSonf. Next,fortwosuchgraphsHiandHj,bothofthemhavetobeadjacenttothesametiandzj.Otherwise,theunionofHi,HjandthetwoshortestpathscanbereducedtoFigure 5-4 (c).AlsobyLemma 13 ,insuchacaseGdoesnothold2D2CCSonf. ThesubgraphinducedbytheunionofverticesofHandfti;zjghastobeapartial2-Tree.Otherwise,wecanreduceGtoFigure 5-4 (f)thatclearlydoesnothold2D2CCSonf. Finally,refertoFigure 5-4 (g),theunionofv1,v3andthesubgraphofGthatisadjacenttov1andv3isnotapartial2-Tree,wecancontractthegraphFigure 5-4 (g)thatclearlydoesadmit2D2CCSonf.Wehavethesameargumentforfv1;v4g,fv2;v3gandfv2;v4g. ThisprovestheonlypossiblecaseisFigure 5-4 (h)whenGadmits2D2CCSonfundertheconditiongiven. Theorem 14 givesthefollowingcorollarythatisveryusefultoarguethatagraphdoesnothold2D2CCSonnon-edges. Proof. 14 101

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GraphGdoesnotadmit2D2CCSonnon-edgef=(v1;v2). 14 tothecasesinwhichthenumberofdisjointpathsbetweenv1andv2is3ormore.Theorem 15 provesthatinmostcasessuchgraphsdonothold2D2CCSonspeciednon-edges. 5-5 (b)-(f)doesadmit2D2CCSonnon-edgef=(v1;v2). Proof. (1) ProofforcaseinFigure 5-5 (b).ThisdirectlyfollowsLemma 13 case(b). Forthecases(c)-(f),wedirectlyconstructexamples.WecandividethegraphintotwosubgraphsG1thatisinducedbyverticesfv1;v2;v4;v5;v6gandG2thatisinducedbyverticesfv1;v2;v3;v4;v7g.Clearly,foranydistanceassignmentforG(thenthedistanceassignmenttoG1andG2isdenedaccordinglythatwedenoteby1and2),thecongurationspace2f(G;)is2f(G1;1)^2f(G1;2).Foreachcasefrom(c)to(f),wechoosesuchthatboth2f(G1;1)and2f(G1;2)have2intervalsand2f(G1;1)^2f(G1;2)has3intervals. (2) ProofforcaseinFigure 5-5 (c). 102

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Considertheangleconstraintvaluefor\(v1;v4;v5),and\(v2;v4;v3);angleconstraintintervalfor\(v1;v4;v3)and\(v2;v4;v5).Wecarefullypicktheseangleconstraintssuchthat\(v1;v4;v2)hasthreeintervalsand(v1;v2)hasthreeintervalsaccordingly. Wechoosethefollowingangleconstraintvalues:\(v1;v4;v5)=11 18,\(v2;v4;v3)=14 18,2 18\(v2;v4;v5)4 18,1 18\(v1;v4;v3)5 18.Consideringthereectionof4(v2;v3;v4)about(v2;v4),thepossiblevaluesfor\(v1;v4;v2)are[7 18;9 18],13 18and15 18.Thesethreedierentangles(angleintervals)correspondtothreedierentvalues(intervals)for(v1;v2). (3) ProofforcaseinFigure 5-5 (d). Clearly,G1in(d)isequivalenttoG1in(c)andG2in(d)isequivalenttoG1in(c).Sowecanassignthesameangleandangleintervalconstraintsto(d)asthatfor(c).Thentheintersection2f(G1;1)^2f(G1;2)in(d)isthatsameas(c). (4) ProofforcaseinFigure 5-5 (e)follows(c)and(d). (5) ProofforcaseinFigure 5-5 (f). Wechoose(v1;v5)=8,(v2;v5)=10,(v1;v4)=4,(v2;v4)=8,(v1;v3)=8,(v2;v3)=12.WeuseMapletodrawthetwoquadrilateraldiagonalintervalmappingcurveinFigure 5-6 .Theverticalaxisdenotes(v3;v4)and(v4;v5).Wecanadjust(v5;v6)=11:725,(v4;v6)=0:225,(v3;v7)=10:9995and(v4;v7)=0:9995tocontroltheminimumpossibleandthemaximumpossiblevaluefor(v3;v4)and(v4;v5).The 103

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Proofforcase(f)inTheorem 15 fourlineshow(v3;v4)2[10;11:999](v4;v5)2[11:5;11:95].Thenumericalcomputationveriesthattheresultingvaluesfor(v1;v2)fallsin3intervals. OnecasethatisnotcoveredinTheorem 15 is:theunionofthegraphandthenon-edgehaveaK4subgraphthatcontainsbothofthetwoendverticesofthenon-edge.Note2D2CCSisdenedoverallthepossibledistanceassignmentanddistance0isalsoallowed.Weemphasizethatgraphsmaynothave2D2CCSonnon-edgesalthoughthenon-edgeisamissingedgeinaK4.Lemma 14 followingDenition 2 givesanequivalenceresultsforsuchcases.

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Overconstraintsresultincongurationspaceof2connectedcomponents. Proof. Theorem 15 ,Lemma 14 andTheorem 14 coversmostcasesbutnottheexampleshowninFigure 5-7 .Becausethesubgraphinducedbyvertexfv1;v4;v6;v7gisoverconstrained,therealizationforthissubgraphisalwaysuniqueifwefactorouttheEuclideantransformandrotation.Thisessentiallymakesthegraphhold2D2CSSonnon-edge(v1;v2).AgeneralizationofthisexamplewillmakethecharacterizationmoreprecisethantheclassofgraphsprescribedbyTheorem 14 ,Lemma 14 .Inthesameway,dierentiatingthe 105

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5-7 fromwhatisprescribedinTheorem 15 willalsoadvancethegraphcharacterizationfor2D2CCS. NowwecomebacktoObservation 9 foranotherhurdletothegraphcharacterizationto2D2CCS.Thedicultyis:ifweintersectiononetwo-intervalwithanothertwo-interval,thenumberoftheintervalsintheresultdependsontheexactvaluesofalltheinputintervals.Forexample,ifeachtwo-intervalonlycontainstwodistinctvalues,theintersectionwillbealsoatwo-interval.Wedenethisandgiveacharacterizationconjecture. GivengraphGandnon-edgef=(v1;v2),wesayGadmits2Dcongurationspaceoftwovalues(2D2CCSoftwovalues)ifforalldistanceassignment,thecongurationspacef(G;)eitherisconnectedorhaveexacttwovalues. 9 ,supposewegetgraphGbyidentifyingG1andG2onverticesv1andv2.Westudythecongurationspaceonf=(v1;v2).IfeitherG1orG2holds2D2CCSoftwovaluesonf,Gholds2D2CCSoftwovalues;otherwise,ifneitherG1norG2holds2D2CCSoftwovaluesonf,weconjectureGdoesnothold2D2CCSoftwovaluesandtheproofisdirectlyimpliedbythegraphcharacterizationfor2D2CCSoftwovalues. 5.2.1QuestionsandContributions 2 ,wehavegivenanexactcharacterizationoftheclassofgraphsGallofwhosecorrespondingEDCS(G;)admita2Dconnectedcongurationspace.WeareinterestedintheclassofgraphsallofwhosecorrespondingEDCS(G;)admita3Dconnectedcongurationspace.Astherststeptowardansweringthisquestion,werestrictourselvestothecharacterizationofgraphandnon-edgepair(G;f)wherefisan 106

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1 InChapter 3 ,wehaveprovedinTheorem 6 thatifG[fis3-Realizable(havenoK5orK2;2;2minor)thenGadmits3Dconnectedcongurationspace3f(G;).AlsoinChapter 3 ,wehaveprovedinLemma 6 thatGdoesnothold3Dconnectedcongurationspaceonnon-edgefwhenG[fisaK5orK2;2;2.Theseresultsmotivateourworkincharacterizing3Dconnectedcongurationspace.Particularly,wearecuriousabouttherelationshipbetweenK5/K2;2;2and3Dconnectedcongurationspace. 2 Wehavegivenacompletegraphcharacterizationfor2DconnectedcongurationspaceinChapter 2 Theorem 1 .InObservation 10 andObservation 11 weshowadirectextensionofTheorem 1 from2Dto3Ddoesnotstand.WegenerizeObservation 10 asLemma 18 andLemma 19 .InLemma 20 ,weprovethatamaximally3-Realizablegraphdoesnothold3Dconnectedcongurationspaceonanynon-edge.WesummerizethediscussionasConjecture 5 thatmaypotentiallyexplaintheextensionfailurefrom2Dto3DforTheorem 1 3 Weconjecturethestrategyusedfor2Dconnectedcongurationspace(intheproofofTheorem 1 ),edgecontractionspreservingthespeciednon-edge,canbeusedforcharacterizing3Dconnectedcongurationspace.Underthisassumption,wegiveasetofbasecaseswhicharenecessarytoargue-byusingthestrategy-thatagraphGdoesnotadmit3Dconnectedcongurationspaceonnon-edgef.Wegetthisnecessarysetbydecomposinggraphs.ThecompletenessofthissetisconjecturedinConjecture 6 .Theproofforthelistedbasecasesisbasedonconstruction. 5.2.2.1Lemmasstandforboth2Dand3D Proof. 107

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2-6 )bethesetofconnectedcomponentsafterv1andv2areremovedfromGandGibesubgraphofGwhichisinducedbyverticesofHitogetherwithv1andv2,thenGadmits3DconnectedcongurationspaceonfifandonlyifallGiadmit3Dconnectedcongurationspaceonf. Proof. Fortheotherdirection,weprovethecontrapositive:assume(withoutlossofgenerality)thatG1doesnotadmit3DconnectedcongurationspaceonfandproveGdoesnotadmit3Dconnectedcongurationspaceonf.SinceG1doesnotadmit3Dconnectedcongurationspaceonf,sowecanndadistanceassignment1toedgesofG1suchthatf(G1;1)hasmorethanoneinterval(notconnected).Wetaketheminimumandmaximumvalueoff(G1;1)anddenotethembyminandmax.NowweonlyneedtoassigndistancevaluestotheedgesthatarenotinG1whilepreservingthepossiblevaluesfor(f)isthesameasf(G1;1).Todothat,weassigndistancevalue0totheedgesofHi(i>1),orequivalentlycontractalledgesofHi(i>1).Intheresultinggraph,alltheverticesthatarenotinG1havedegreeexactly2andareadjacenttobothv1andv2.WeassigndistancemaxtoalltheedgesthatarenotinG1.Bythetriangleinequality,thedistancesofedgesnotinG1restrict(f)tobe[0;2max].Bytakingtheintersection,f(G)willbethesameasf(G1;1)sowendadistanceassignmentforGsuchthatf(G)isnotconnected. 2-7 ),thenGadmits3DconnectedcongurationspaceonfifandonlyifG1admits

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Proof. Fortheotherdirection,supposeG1admits3Dconnectedcongurationspaceon(v1;v3)(or(v1;v3)isanedgeofG)andG2admits3Dconnectedcongurationspaceon(v2;v3)(or(v2;v3)isanedgeofG).WeconsidertherealizationsofGwitharbitrarygivendistanceconstraints.Ifwexpv3in3D,thepossiblelocationsforbothpv1andpv2areconnected.Meanwhile,pv1andpv2areindependentofeachother,sothepossiblevaluesfor(v1;v2)isconnectedandwehaveprovedGadmits3Dconnectedcongurationspaceonf=(v1;v2). ByLemma 15 ,Lemma 17 andLemma 16 ,tofurtherlydiscusswhetherGadmits3Dconnectedcongurationspaceonf=(v1;v2),wecanassumethatGitselfistheminimumsumcomponentofGcontainingbothv1andv2,Gnfv1;v2gisconnectedandthereisnoarticulationvertexinGthatseparatesv1andv2. 1 ,wehaveprovedthatforagraphGandanon-edgef,thecongurationspace2f(G;)isconnectedifandonlyifalltheminimal2-SumcomponentsofG[fthatcontainsfarepartial2-trees.AlsoinTheorem 6 wehaveprovedthat3f(G;)isconnectedifG[fis3-Realizable.However,thefollowingtwoobservationsshowthatadirectextensionofTheorem 1 to3Ddoesnotstand.

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Proof. 2-9 .Thenon-edgefis(v1;v2).Obviously,GdoesnothaveK5orK2;2;2minorbutG[fhasaK5minor.SinceGnfv1;v2gisacliqueK4,therealizationforGnfv1;v2gisuniqueifwefactoroutthe3Dtranslateandrotation.Thenthepossiblelocationforbothpv1andpv2arecircles.So,thepossiblevaluesfor(v1;v2)isconnected. RefertoFigure 2-10 .Thenon-edgefisalso(v1;v2).Clearly,GdoesnothaveK5orK2;2;2minorbutG[fhasaK2;2;2minor.Firstconsiderthecongurationspaceonf2=(v4;v5)andf3=(v3;v6).NotebothG[f2andG[f3hasnoK5orK2;2;2minor,byTheorem 6 thecongurationspace3f2(G;)and3f3(G;)arebothconnected.Actually3f2;f3(G;)isalsoconnected. Note4(v3;v5;v7)and4(v5;v6;v7)restrict(f3)tobeoneintervalwhosebothendpointscorrespondtothecasesinwhichtetrahedron(v3;v5;v6;v7)isin2Dplane.Inthesameway,4(v3;v4;v7)and4(v4;v6;v7)restrict(f3)tobeoneintervalwhosebothendpointscorrespondtothecasesinwhichtetrahedron(v3;v4;v6;v7)isin2Dplane.So,thebothendpointsof3f3(G;)correspondtothecasesinwhicheithertetrahedron(v3;v4;v6;v7)ortetrahedron(v3;v5;v6;v7)isin2Dplane.Notetheonlypossibilitythat3f2;f3(G;)isnotcontinuousis:givenavaluefor(f3),(f4)hastwopossiblevaluesthatcorrespondtowhetherpv4andpv5areinthesamesideof4(v3;v6;v7).Now,giventwovaluesin3f2;f3(G;),wecanalwaysrstlycontinuouslychange(f2)and(f3)tilleithertetrahedron(v3;v4;v6;v7)ortetrahedron(v3;v5;v6;v7)isin2Dplane,andthencontinuously(f2)and(f3)toanothergivenvalue. Becausepv1canbecontinuouslyrotatedaroundedge(v3;v5)andpv6canbecontinuouslyrotatedaroundedge(v4;v7).So,(f1)isoneinterval. 110

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ForproofofObservation 11 Proof. 5-8 (a)and(b).TheproofisdirectlyimpliedbyObservation 10 andLemma 16 ForthetwoexamplesusedintheproofofObservation 10 ,wegeneralizetheexampleinFigure 2-10 asLemma 18 andFigure 2-9 asLemma 19 Proof. 111

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Proof. 18 .SupposeFisanon-edgesetsuchthatG[Fisa2-SumofK4's.Supposev1andv2arethetwoverticesoff.WeprovethatforalldistanceassignmenttotheedgeofGandanytwovalues1f(G;)and2f(G;)forf,wecanbychangingFsuchthatfischangedfrom1f(G;)to2f(G;).SinceG[Fis2-Sum'sofK4,G[Fispartial3-Treeand3-Realizable,byTheorem 6 ,F(G;)isconnected.SinceforeachK4,acontinuousmovinginthedistancespaceofthe6edgesresultsinthecontinuousmovingofthe6pointsin3DspaceifwefactorouttheEuclideantransformandrotation.Also,forthe2-Sumof2K4,wecancontinuouslyrotatethetwoK4'salongtheedgethatthe2-Sumoperationison.Thisprovesthatfcanbechangedfrom1f(G;)to2f(G;). WedeneagraphGismaximally3-Realizableifforanynon-edgefofG,G[fisnot3-Realizable. 5-9 )where3-sumisonlyallowedbetweenK4'sand2-sumisnotallowedbetweenK4's. Proof. 8 ),ithasbeenprovedthatagraphGis3-Realizableifandonlyifitisasubgraphof2-Sumor3-SumofK4,C5C2andV8(seeFigure 5-9 )where3-SumisonlyallowedbetweenK4's.So,ifGismaximally3-Realizable,Gmustbe2-Sumor3-SumofK4,C5C2andV8where3-sumisonlyallowedbetweenK4's.Toprovethisdirection,weonlyneedtoexcludethepossibilitythat2-Sumoccursbetween 112

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Threebasicsum-componentsofmaximally3-Realizablegraphs:(a)K4;(b)C5C2;(c)V8. Nowweproveitissucient.WetaketheminimumsumcomponentofG,sayG,thatcontainsboththetwoverticesofnon-edgef,sayv1andv2.Notev1andv2areattheendsoftheG.IfGcontainsanyC5C2asasum-component,wecancontracttheedgesthatarenotinthisC5C2suchthat(v1;v2)isannon-edgeofthisC5C2.ByLemma2in( 3 ),theunionofanynon-edgeandC5C2willhaveeitherK5orK2;2;2asminor.IfGcontainsanyV8asasum-component,wecancontracttheedgesthatarenotinthisV8suchthat(v1;v2)isannon-edgeofthisV8.ByLemma1in( 3 ),theunionofanynon-edgeandV8willhaveeitherK5orK2;2;2minor.WhenGdoesnotcontainanyC5C2orV8,ithastobethe3-SumofK4.Bycontractingedges,G[fcanbecomeaK5.Thus,addinganynon-edgewillresultinaK5orK2;2;2minorandthisprovesthesuciency. Proof. 20 ,G[fhaveK5orK2;2;2minor.BycheckingtheedgecontractionsintheproofofLemma 20 (suciencydirection)andthegraphreductionofbothLemma1andLemma2in( 3 ),we 113

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6 ,the3Dcongurationspace3f(G)isnotconnected. 5-10 ),ifwerestrictthedistancevalueofonediagonal(v3;v4)betwoarbitrarydistinctvalues,thenwecanndadistanceassignmenttotheedgesofthequadrilateralsuchthatthepossibledistancevalueoftheotherdiagonal(v1;v2)isalsotwodistinctvalues. Proof. 5-10 .Intheleftsolution,(v1;v2)willbe 4(max+min 4(max+min Itcanbesimpliedtobeq Inthesameway,(v1;v2)willbe 4(max+min 4(max+min Itcanbesimpliedtoq 114

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Giventwodierentvaluesfor(v3;v4),wecanndappropriatedistanceassignmenttothefouredgesofquadrilateral(v1;v2;v3;v4)suchthat(v1;v2)hastwodierentvalues. Figure5-11. Basecasesthatdonothold3Dconnectedcongurationspaces. 115

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ProofforLemma 23 case(c)(Figure 5-11 (c)). 5-11 ,graphGdoesnotadmit3Dconnectedcongurationspaceonnon-edgef=(v1;v2). Proof. 6 wehaveprovedthecases(a)and(b).ByusingLemma 22 ,(e)and(f)havebeenproved.Inthesameway,weonlyneedtoprovecases(c)and(d)since(g)and(h)areimpliedbyandcase(c)and(d)respectivelyalsoaccordingtoLemma 6 Werstprovethecase(c).RefertoFigure 5-11 (c),weassigndistances(v1;v7)=(v1;v4)=(v1;v5)=(v6;v4)=(v6;v5)=(v6;v7)=(v1;v5)=(v3;v5)=(v4;v5)=(v4;v7)=1,(v1;v3)=2,(v3;v7)=p 3,(v2;v3)=1+p 3.RefertoFigure 5-12 .Since(v1;v5)+(v3;v5)=(v1;v3)=2,sopv1,pv5andpv3arecollinear.Accordingtothevaluesof(v1;v7),(v1;v3)and(v3;v7),cos(\(v7;v1;v3))=1 2and\(v7;v1;v3)is 3.(v6;v3)isthedoubleof(v8;v5),so(v6;v3)=2p 3.Because(v2;v6)=1p 3and(v2;v3)=1+p 3,bytriangleinequality,pv2,p3andpv6arecollinear.Wedenotethembypv6andpv6

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Nowweprovecase(d)(seeFigure 5-11 (d)).Werstlyxtriangle4(v1;v6;v5),becauseofthereectionabout4(v1;v6;v5),vertexv3andv4maybeondierentsidesof4(v1;v6;v5)oronthesamesideof4(v1;v6;v5).Weusethiskindofreectiontoconstructourexampletoshowthat(v1;v2)canhavemorethanoneinterval.Weset(v6;v3)?4(v1;v6;v5),(v5;v4)?4(v1;v6;v5),(v6;v3)=(v5;v4)=xand(v6;v5)=y.Withthissetting,v6,v3,v5andv4arealwaysinaplane.Ifv3andv4areonthesamesideof(v6;v5),(v3;v4)willbey;ifv3andv4areonthedierentsidesof(v6;v5),(v3;v4)willbep 2(v6;v3)(v3;v2)=p 5-11 byedgecontractionswhilepreserving(v1;v2)asannon-edge.

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ZeoliteandHelixpackingquestionsarediscussedinthischapter.BothquestionswereintroducedtousbypeersincomputationalChemistryormolecularbiology.Intherstpart,weabstractzeolitesasZ-Graphswhichareunderconstrained.Wegiveacompletionandcongurationspaceonthecompletionaswell.Thecongurationspaceturnsouttobeextremelysimpleandquitesurprising.Inthesecondpart,geometricconstraintproblemsareformalizedforHelixpacking.Theextremecongurationweintroduceisimportantinbothsamplingsolutionspaceandwalkinginsolutionspace. InChapter 2 wehavecharacterizedgraphsthatadmit2Dconnectedcongurationspaces.Thecharacterizationsapplytoalldistanceassignmentsordistanceintervals.Thisplacesstrongrestrictionsonthecharacterizedclassesofgraphswithecientcongurationspaces.However,ifoneassumesspecialdistanceassignments,well-behavedcongurationspacesmayexistformuchlargerclassesofgraphs.Forexample,considerthe2D2-directiongrid(seeFigure 6-1 ),whichisnotapartial2-tree.However,undertherestrictionofunit-distanceedges,wecanshowthatsuchanEDCShasacongurationspacethatisaconvexpolytope,infactarectilinearbox.Wediscussthe2Dversionoftheproblem.Itwouldbedesirabletoobtainsimilarresultsfor3D. 118

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ZeolitegraphG6;7andone2Dcompletion. 6-1 showsZ-GraphG6;7. Aswehaveintroducedearlier,thecorrespondinggeometricconstraintsystemsforZ-Graphsrequiresthatalltheedgelengthsshouldbeequal.ThispropertyfollowstherealphysicalpropertiesofZeolites.Welimitthediscussionhereto2D.Sincedistanceassignment(E)isgiven,azeolitegraphGactuallyrepresentsageometricconstraintsystemandwedonotexplicitlydierentiatethem. Proof. AlsoinFigure 6-1 ,acompletionisgivenasthesetofnon-edges,whichwedenotebyF.Accordingly,wedenotethedistancesonthenon-edgesby(F).WedenethecongurationspaceonF-denotedbyF-as: F(G)=f(F)jthereisa2Drealizationfor(G[F;(E)[(F))g. 119

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FirstrowandrstcolumnsubgraphofG6;7;itisapartial2-tree. Theorem16. 6-1 ,thecongurationspaceonFis[0;2]jFj.(notewepickasetofcellsandpickexactlyonediagonalfromeachcelltoformthecompletion;thereisnodirectionrequirementforthediagonals). Proof. LetustaketherstrowandrstcolumntogetthesubgraphG1ofG.RefertoFigure 6-2 ,obviously,F(G1)is[0;2]jFj. Thenweprovethatgivena2DrealizationforG1wecanalwaysndarealizationforG.Letususepi;jtodenotethecoordinateofvi;jintherealization.Foranyvertexvi;jinGnG1(equivalenttoi>2Vj>2),weplacepi;jto:pij+2;2ifiisgreaterthanj;otherwisep2;ji+2.Anintuitivewaytolookatthisprocedureis:werecursivelyplacepi;j

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Alsonotethat,wecantakeanarbitraryrowandanarbitrarycolumntogetacompletionandthecongurationspaceonsuchacompletionisalways[0;2]jFj.Actually,theone-row-one-columnconditioncanbefurtherlyloosened(seeFigure 6-3 ).Thefollowingtheoremgaveusageneralwaytondsuchcompletion. Proof. 16 .Actually,theprocedureremovingavertex(sayv1)withdegree2canbeviewedasidentifyingv1toanothervertexv4wherev1andv4areinthesamecellandv1andv4arenotadjacent.Sincewecanreducethegraphto2-SumsofthesecellswhichwedenotebyG1,wecanidentifyalltheverticesthatarenotinG1toonevertexinG1bytheremovingprocedure.DistanceconstraintsareclearlysatisedasthatintheproofofTheorem 16 Itwouldbeinterestingtocharacterizethatwhatm+n3cellswouldbethetypeasdescribedinTheorem 17 .Thecompletionwithsuchapropertyisnotunique.Figure 6-3 givesanothercompletion.Thefollowinglemmasgivesanecessarycondition. Proof. 121

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AnothercompletionforG6;7. 6.2.1IntroductionandMotivation 6-4 6-5 ).Thespheresaresometimescalledatomsandrepresentaminoacidsequences.Thespheresofonehelixcanintersectoneanother,butthespheresofdierenthelicesmustnotintersect.Thiscollisionavoidanceiscalledpackingconstraint.

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Thealphahelixisatightlycoiled,rodlikestructurewhichhasanaverageof3.6aminoacidsperturn.ThehelixisstabilizedbyhydrogenbondingbetweenthebackbonecarbonylofoneaminoacidandthebackboneNHoftheaminoacidfourresiduesaway.Allmainchainaminoandcarboxylgroupsarehydrogenbonded,andtheRgroupsstickoutfromthestructureinaspiralarrangement.[Fromwiz2.pharm.wayne.edu/biochem/prot.html] Figure6-5. Twohelices 123

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Bi-incidence 124

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6-7 .Tagsrepresentthecontactsthatmakethecongurationwell-constrained.Thelistofcontactscanbeasupergraphofanytagbutitmustcontainatleastonetagasasubgraph.Thecontactbipartitegraphofaboundarycongurationhasatagandanunder-constrainedtag.OneshouldalsocheckwhethereveryoneofthesetypesisrealizableforSOMEsetofradiiofthespheres(whichtheverticesrepresent).Notethatthesphereswithinasinglehelixcouldinfactintersect. Allcongurationsontheinteriorofthesameboundaryfacethavethesametag.ThepackingcongurationspaceisnonemptyifandonlyifithasanextremecongurationinwhicheachBi-incidence(ai1;ai2),(bj1;bj2)inthespanningsetoccurswithai1,ai2beinginfa1;a2;a3;a4;a5gandbj1,bj2beinginfb1;b2;b3;b4;b5;b6ginoneoftheabovetags. 125

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Eachtagrepresentsabipartitegraphwith(left,right)partshaving(i;j)verticesrespectively,wherei+1j6;i3:Thereshouldbe6edgesinthebipartitegraphandEachvertexcanhaveatmost3edgesincidentuponit. 126

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Weinitializeasystematicandgradedprogramofcombinatoriallycharacterizinggraphswithcongurationspacesofdierentgeometryandalgebraiccomplexity.Ourresultsinclude: 127

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[1] A.Y.Alfakih,A.KhandaniandH.Wolkowicz.SolvingEuclideanDistanceMatrixCompletionProblemsviaSemideniteProgramming.Comput.Optim.Appl.,12:13{30,1999. [2] M.Badoiu,K.Dhamdhere,A.Gupta,Y.Rabinovich,H.Racke,R.RaviandA.Sidiropoulos.Approximationalgorithmsforlow-distortionembeddingsintolow-dimensionalspaces.SODA'05:ProceedingsofthesixteenthannualACM-SIAMsymposiumonDiscretealgorithms,page119{128,2005. [3] MariaBelk(Sloughter)andRobertConnelly.RealizabilityofGraphs.Discrete&ComputationalGeometry,37(2):125{137,2007 [4] MariaBelk.RealizabilityofGraphsinThreeDimensions.Discrete&ComputationalGeometry,37(2):139{162,2007 [5] PratikBiswas,Tzu-ChenLian,Ta-ChungWangandYinyuYe.Semideniteprogrammingbasedalgorithmsforsensornetworklocalization.ACMTrans.Sen.Netw.,2(2):188{220,2006. [6] L.M.Blumenthal.TheoryandApplicationsofDistanceGeometry.OxfordUniver-sity,1953. [7] A.Cayley.Atheoreminthegeometryofposition.CambridgemathematicalJournal.,II:267{271,1841. [8] RobertConnelly.Genericglobalrigidity.Discrete&ComputationalGeometry,33(4):549{563,2005 [9] G.M.CrippenandT.F.Havel.DistanceGeometryandMolecularConformation.Chemometricsseries,15,Taunton,Somerset,England:ResearchStudiesPress,1998. [10] I.FudosandC.M.Homann.Agraph-constructiveapproachtosolvingsystemsofgeometricconstraints.ACMTransactionsonGraphics,16(2):179{216,1997. [11] I.FudosandC.M.Homann.Correctnessproofofageometricconstraintsolver.Int.J.Comput.Geom.Appl.,6:405{420,1996. [12] H.Gao,andM.Sitharam.CombinatorialClassicationof2DUnderconstrainedSytems.ProceedingsoftheSeventhAsianSymposiumonComputerMathematics(ASCM2005),Sung-il.Pae.andHyungju.Park.,Eds.,2005,pp.118{127. [13] ChristophM.Homann,AndrewLomonosovandMeeraSitharam.Decompositionofgeometricconstraintssystems,partI:performancemeasures.JournalofSymbolicComputation,31(4),2001. 128

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HepingGaowasborninTongzhong,Hanchuan,Hubei,China.HegrewupmostlyinTongzhong,Hanchuan.HereceivedhisB.S.degreeintelecommunicationengineeringin1998fromChongqingUniversityofPostsandTelecommunications(CQUPT)ofChinaandhisM.S.degreeinelectricalengineeringin2003fromShanghaiJiaoTongUniversity(SJTU)ofChina.HewasawardedthePh.D.incomputerandinformationscienceandengineeringfromtheUniversityofFloridain2008. 131