Walks, Transitions and Geometric Distances in Graphs

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Thomas Bellitto

To cite this version:

Thomas Bellitto. Walks, Transitions and Geometric Distances in Graphs. Other [cs.OH]. Université de Bordeaux, 2018. English. �NNT : 2018BORD0124�. �tel-01926589�

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parThomas Be

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POUROBTENIRLEGRADEDE

DOCTEUR

SP´ECIALIT´E:INFORMATIQUE

Wa

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ionsand Geometr

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istancesin Graphs

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Soutenuepubliquementle27aoˆut2018 apr`esavisdesrapporteurs:

JørgenBang-Jensen,Professeur,UniversityofSouthernDenmark SylvainGravier,Directeurderecherche,Universit´eJosephFourier devantlacommissiond’examencompos´eede:

ChristineBachoc,professeure,Universit´edeBordeaux,directricedeth`ese JørgenBang-Jensen,professeur,UniversityofSouthernDenmark,rapporteur SylvainGravier,directeurderecherche,Universit´eJosephFourier,rapporteur Mickael Montassier,professeur,Universit´ede Montpellier,examinateur ArnaudPˆecher,professeur,Universit´edeBordeaux,directeurdeth`ese ´

EricSopena,professeur,Universit´edeBordeaux,pr´esidentdujury

Abstract Thisthesisstudiescombinatorial,algorithmicandcomplexityaspects ofgraphtheoryproblems,andespeciallyofproblemsrelatedtothenotionsofwalks, transitionsanddistancesingraphs.

Wefirststudytheproblemoftraffic monitoring,in which wehavetoplace asfewcensorsaspossibleonthearcsofagraphtobeabletoretrace walksof objects. Thecharacterizationofinstancesofpracticalinterestsbringsustothe notionofforbiddentransitions,whichstrengthensthe modelofgraphs. Ourwork onforbidden-transitiongraphsalsoincludesthestudyofconnectingtransitionsets, whichcanbeseenasatranslationtoforbidden-transitiongraphsofthenotionof spanningtrees.

Alargepartofthisthesisfocusesongeometricgraphs,whicharegraphswhose verticesarepointsoftherealspaceandwhoseedgesaredeterminedbygeometric distancebetweenthevertices. ThisgraphsareatthecoreofthefamousHadw iger-Nelsonproblemandareofgreathelpinourstudyofthedensityofsetsavoiding distance1invariousnormedspaces. Wedevelopnewtoolstostudytheseproblems andusethemtoprovetheBachoc-Robinsconjectureonseveralparallelohedra. We alsoinvestigatethecaseofthe Euclideanplaneandimprovetheboundsonthe densityofsetsavoidingdistance1andonitsfractionalchromaticnumber.

Finally,westudythecomplexityofgraphhomomorphismprob lemsandestab-lishdichotomytheoremsforthecomplexityoflocally-injectivehomomorphismsto reflexivetournaments.

Keywords Graphs,walks,forbiddentransitions,graphhomomorphisms ,indepen-dencenumber,geometricdistances,setsavoidingdistance1,NP-completeness Affiliation UniversityofBordeaux,CNRS,LaBRI,UMR5800,F-33400Talence, France

R´esum´e Cetteth`ese´etudielesaspectscombinatoires,algorithmiquesetla complexit´edeprobl`emesdeth´eoriedesgraphes,ettoutsp´ecialementdeprobl`emes li´esauxnotionsde marches,detransitionsetdedistancedanslesgraphes.

Nousnousint´eressonsd’abordauprobl`emedetraffic monitoring,quiconsiste`a placeraussipeudecapteursquepossiblesurlesarcsd’ungraphedefa¸con`apouvoir reconstituerdes marchesd’objets.Lacaract´erisationd’instancesint´eressantesdans lapratiquenousam`ene`alanotiondetransitionsinterdites,quirenforcele mod`ele degraphe. Notretravailsurlesgraphes`atransitionsinterditescomprendaussi l’´etudedelanotiond’ensembledetransitionsconnectant,quel’onpeutvoircomme l’analogueentermedetransitionsdelanotiond’arbrecouvrant.

Unepartieimportantedecetteth`eseportesurlesgraphesg´eom´etriques,qui sontdesgraphesdontlessommetssontdespointsdel’espacer´eeletdontlesarˆetes sontd´etermin´eesparlesdistancesg´eom´etriquesentrelessommets. Cesgraphes sontaucœurduc´el`ebreprobl`emede Hadwiger-Nelsonetnoussontd’unegrande aidedansnotre´etudedeladensit´edesensemblesqui´evitentladistance1dans plusieurstypesd’espacesnorm´es. Nousd´evelopponsdesoutilspour´etudierces probl`emesetlesutilisonspourprouverlaconjecturedeBachoc-Robinssurplusieurs parall´elo`edres. Nousnouspenchonsaussisurlecasduplaneuclidienetam´eliorons lesbornessurladensit´edesensembles´evitantladistance1etsursonnombre chromatiquefractionnaire.

Enfin,nous´etudionslacomplexit´edeprobl`emesd’homomorphismesdegraphes et´etablissonsdesth´eor`emesdedichotomiesurlacomplexit´edeshomomorphismes localementinjectifsverslestournoisr´eflexifs.

Mots-cl´es Graphes,marches,transitionsinterdites,homomorphismesdegraphes, nombredestabilit´e,distancesg´eometriques,ensemble´evitantladistance1, NP-compl´etude

Affiliation Universit´edeBordeaux,CNRS,LaBRI,UMR5800,F-33400Talence, France

Acknow

ledgements

Jecommencebiensˆurcettesectionparuntr`esgrand merci`a Arnaud,qui m’encadredepuis monstagede masteretqui m’a´enorm´ementaid´e`atrouverce quejevoulaisfaireet`aarriverl`ao`uj’ensuis. Mercipourtonsoutien,tonaideet tesconseilsdanstellementdedomaines:larecherche,l’enseignement,lar´edaction, larecherchedestagespuisdepostdocs,lescandidatures... Merciauss ipourtaconfi-anceetlalibert´equetum’aslaiss´eependantmath`esepourtrouverlesprobl`emessur lesquelsjevoulaistravailleretlesdirectionsquejevoulaisexplorer. Mercisurtout pourtoutletempsquetuastoujoursr´eussi`atrouverdanstonemploidutemps charg´epour´ecouter,relireetessayerdecomprendre mesid´eestordues. Ungrand merciaussi`aChristine,quim’abeaucoupaid´e`aenrichirmondomainederecherche, quim’afaitd´ecouvrirdesprobl`emespassionnantsetquim’a´enorm´ementaid´e`ales relier`acequejesavaisfaireet`apartirdanslabonnedirection.

IalsothankSylvain GravierandJørgenBang-Jensenforacceptingtoreadthe manuscriptand ´EricSopenaand Mickael Montassierforbeingpartofthejury.

IwouldalsoliketothankJørgenBang-JensenagainandAndersYeoforchoosing meforanopenpostdocpositionintheirteam.Iamreallyexcitedforthisnext importantstepof myjourney.

Ofcourse,Iwouldliketothank myco-authorsandallthepeopleIhaveworked withandwithoutwhoseideasthisthesiswouldprobablyneverhaveexisted. Arnaud etChristine,unefoisencore. Philippe, monfr`eredeth`ese,avecqui¸caatoujours ´

et´eungrandplaisirdetravailleroudediscuter. Gary,ofcourse;Ihadanamazing timein Victoriathankstoyouandyouhavedonealottohelp mesince. Thank youto myotherco-authorsfrom Victoriatoo: Chris,StefanandFeiran. Merci`a Benjaminavecquij’aibeaucoupaim´etravailleret´echanger. Merciaussi`aAntoine avecqui¸caa´et´eunplaisirdetravaillercesderniers mois.

Thankyouto mypreviousadvisors: GunnarKlau,Tobias Marschall,Alexander Sch¨onhuth,DavidCoudert,NicolasNisseandEndreBoros.

Merci´evidemment`atoutel’´equipedeth´eoriedesgraphesduLaBRIpourde nombreux´echangespassionnantsetpouruncadredetravailetuneambiancetr`es agr´eables. Merciaussi`amon´equipeINRIA,R´ealopt,et`amesco-bureaux,Henriet Th´eo.

Merci`aHerv´eet`atoutel’´equipede Maths`a Modelerpourcetteexp´eriencetr`es enrichissante.

Merciaussi`atousceuxquej’aioubli´esetqui m’ontaid´e`aarriverl`ao`ujesuis aujourd’hui.

Merci`atous mesamis,dulaboetd’ailleurs. Un´enorme mercitoutpart ic-uli`erement`a Th´eoet Antoninpourleursoutienetpourtouslesbons moments pass´esensemblecesderni`eresann´ees.

Mercienfin`a mafamilleetsurtout`a ma m`ereet monfr`erepourleursoutien inconditionneldepuistoutescesann´ees.

Contents

Introduction 9

1 Preliminaries 17

1.1 Fundamentalsofgraphtheory... 18

1.1.1 Coredefinitions... 18

1.1.2 Homomorphismsandcolouring ... 23

1.1.3 Specialvertexsets ... 26

1.1.4 Hypergraphs ... 28

1.2 Elementsofcomplexity ... 30

1.2.1 P,NPandpolynomialreductions... 30

1.2.2 3-SATandNP-completeness ... 31

1.2.3 Approximations... 33

1.3 Walks,connectivityandtransitions... 33

1.3.1 Walksandconnectivityinusualgraphs ... 33

1.3.2 Forbidden-transitiongraphs... 36

1.4 Polytopesandlattices ... 38

1.4.1 Normsanddistances... 38

1.4.2 Lattices ... 40

1.4.3 Polytopes ... 41

1.4.4 Classificationoftheparallelohedraindimension2and3... 43

1.5 Rationallanguagesandautomata... 45

1.5.1 Rationallanguages... 46

1.5.2 Automataandrecognition... 47

1.6 Linearprogramming ... 50

1.6.1 Definitions ... 50

1.6.2 Integerlinearprogramming ... 52

2 Separatingcodesandtraffic monitoring 55 2.1 Introduction... 55

2.2 Thetraffic monitoringproblem ... 56

2.2.1 Definition... 56

2.2.2 Limitationsoftheexistingseparation models ... 57

2.3 Anew modelofseparation:separationonalanguage... 59

2.3.1 Presentationoftheproblem... 59

2.4 Separationofafinitesetofwalks... 60

2.5 Separationofwalkswithgivenendpoints ... 63

2.5.1 Studyofthereachablelanguages... 64

2.5.2 Reductiontheoremandresolution ... 65

2.6 Separationofwalkswithforbiddentransitions... 69

2.6.1 Motivationoftheproblem... 69

2.6.2 StudyoftheFTG-reachablelanguages... 70

2.6.3 Reductiontheoremandresolution ... 72

2.7 Conclusion ... 75

3 Minimumconnectingtransitionsetsingraphs 77 3.1 Introduction... 77

3.2 Polynomialalgorithmsandstructuralresults ... 79

3.2.1 Generalbounds... 79

3.2.2 Connectinghypergraphs... 81

3.2.3 Polynomialapproximation... 87

3.3 NP-completeness ... 89

3.3.1 MCTSinFTGs... 89

3.3.2 MCTSinusualgraphs... 91

3.3.3 Intuitionoftheproof... 97

3.4 Conclusion ... 102

4 Densityofsetsavoidingparallelohedrondistance1 103 4.1 Introduction... 103

4.1.1 Unit-distancegraphsandtheHadwiger-Nelsonproblem ... 103

4.1.2 Densityofsetsavoidingdistance1 ... 105

4.2 Preliminaryresultsand method... 108

4.2.1 Independenceratioofadiscretegraph... 108

4.2.2 Discretizationoftheproblem ... 112

4.3 Parallelohedronnormsintheplane... 113

4.3.1 Theregularhexagon... 113

4.3.2 GeneralVorono¨ıhexagons... 116

4.4 ThenormsinducedbytheVorono¨ıcellsofAnandDn ... 121

4.4.1 ThelatticeAn ... 121

4.4.2 ThelatticeDn ... 124

4.5 ThechromaticnumberofG(Rn_{, ·P)... 126}

4.6 Conclusion ... 127

5 Optimal weightedindependenceratio 129 5.1 Introduction... 129

5.2 Ourapproach... 130

5.2.1 Optimalweightedindependenceratio ... 130

5.2.2 Weighteddiscretizationlemma ... 132

5.2.3 Fractionalchromaticnumber ... 133

5.3 Generalnorms ... 136

5.3.2 Thealgorithm ... 138

5.3.3 TheEuclideanplane... 140

5.4 Parallelohedronnorms... 143

5.4.1 Λ-classesandk-regularity... 143

5.4.2 Thealgorithm ... 149

5.4.3 Buildingfinitegraphs ... 152

5.4.4 Thetruncatedoctahedron... 154

5.5 Conclusion ... 155

6 Complexityoflocally-injectivehomomorphismstotournaments 157 6.1 Introduction... 157

6.1.1 Ourproblem ... 157

6.1.2 Knownresults ... 160

6.2 Ios-injectivehomomorphisms ... 161

6.2.1 Ios-injectiveT4-colouring ... 161

6.2.2 Ios-injectiveT5-colouring ... 166

6.2.3 Dichotomytheorem ... 168

6.3 Iot-injectivehomomorphisms ... 171

6.3.1 Iot-injectiveT4-colouring ... 171

6.3.2 Iot-injectiveT5-colouring ... 175

6.3.3 Dichotomytheorem ... 178

6.4 Conclusion ... 180

7 Conclusionandfurther work 181 7.1 Separationonlanguagesandtraffic monitoring ... 181

7.1.1 Reduciblelanguages ... 181

7.1.2 Planarinstances ... 184

7.2 Minimumconnectingtransitionsets ... 184

7.2.1 Sparsegraphs... 185

7.2.2 Stretchofthesolution... 185

7.3 Setsavoidingdistance1 ... 185

7.3.1 TheEuclideanplaneandErd˝os’conjecture ... 186

7.3.2 ParallelohedraandBachoc-Robins’conjecture... 186

7.3.3 Powerandlimitationofweightedsubgraphs... 187

7.4 Locally-injectivedirectedhomomorphisms... 188 A ComputationalboundintheEuclideanplane 191

Index 193

Introduct

ion(enfran¸

ca

is)

Cetteth`ese´etudiedesprobl`emeetdesnotionsli´ees`alastructure math´ematique degraphe. Lesgraphesetgraphesorient´essontun mod`elepuissantquipermet ded´ecriren’importequellerelationbinairesurunensemble. Les´el´ementsdecet ensemblesontappel´esdessommetsetlespairesd’´el´ementsli´es(ouadjacent)sont appel´eesarˆetes. Grˆace`aleurexpressivit´e,lesgraphestrouventdesapplications dansd’innombrablesdomaines:syst`emesd’information,t´el´ecommunications,b io-informatique,traitementd’images,r´eseauxdetransports,r´eseauxsociaux,plan ifi-cation...etbiend’autres.Lesgraphesont´et´eintroduitsilyapresquetroissi`ecles etl’int´erˆetquileurestport´en’acess´edegrandirdepuis,surtoutavecl’´emergence del’informatique.

Pluspr´ecis´ement,cetteth`eses’int´eresse`alanotionde marcheetdedistance danslesgraphes,ainsiqu’auxaspectscombinatoires,algorithmiqueset` alacom-plexit´edeprobl`emesli´es. Une marchedansungrapheestunesuitedesommets adjacentsquipermetderelierdeuxsommets.Lenombred’´el´ementsdanslamarche permetded´efinirsalongueuretlalongueurdesmarchesentredeuxsommetsd´efinit leurdistance. Nous´etudions´egalementdepr`eslesgraphesg´eom´etriques,quisont desgraphesdontlessommetssontdespointsdel’espacer´eel. Larelat ionen-treladistanced´efinieparl’adjacencedanslegrapheetladistanceg´eom´etrique nousint´eresseratoutparticuli`erement. Nostravauxfontaussibeaucoupintervenir d’autresnotionsconnuesdeth´eoriedesgraphes,dontcellesd’ensemblestable,de colorationetd’homomorphisme.

Lesprobl`emesetr´esultatspr´esent´esdanscetteth`esepeuventsediviserentrois parties.

Transitionsdanslesgraphes

Lapremi`erepartiesepenchesurle mod`eledegraphes`atransitionsinterdites. La puissancedes mod`elesdegrapheetde marcheenfaitles mod`elesdechoixpour ´

etudierdesprobl`emesderoutagedansdenombreuxcontextes. Parexemple,le r´eseauroutierd’unevillepeutˆetremod´elis´eparungraphedanslequelchaqueendroit d’int´erˆetetchaquecroisementsontrepr´esent´espardessommetseto`ul’existence d’uneroutedirecteentredeuxsommetsesttraduiteparunearˆete(ouunarcdansle casd’uneroute`asensunique). Cefaisant,nousd´efinissonsimplic itementunensem-bledesmarchesentrechaquepairedesommetsquipeut-ˆetreutilis´epourr´esoudrede nombreuxprobl`emes. Onpeutparexempler´esoudredesprobl`emesd’optimisation pourtrouverlepluscourtcheminentredeuxpoints,´eviterdesembouteillages,ou

chercherdesattributsquidistinguentunensemblede marches. Cependant,dansla pratique,beaucoupdes marchesd´efiniesparlegrapherepr´esententdesitin´eraires qu’unconducteurn’apasledroitdeprendre. Pour mod´eliserunesituationo`uun conducteurn’estpasautoris´e`atourner`agaucheou`adroite`auneintersection,nous d´efinissonslestransitionsdanslegraphecommedespairesd’arˆetescons´ecutives. L’´etudedesgraphes`atransitionsinterdites,o`ulad´efinitiondugrapheinclutun ensembledetransitionsautoris´eesouinterdites,estundomainedelath´eoriedes graphesqui´emergerapidement.Ilexisteaussiplusieursautres mod`elesproches, parmilesquelsles marchesproprementcolor´eessurdesgraphesarˆetes-color´esoules graphes`asous-cheminsinterdits.

Lepremierdesdeuxprincipauxprobl`emesli´es`alanotiondetransitionsinterdites quenous´etudionsdanscetteth`eseestleprobl`emedetraffic monitoring. Dansle cadredeceprobl`eme,onnousdonneungraphedanslequeldesobjetssed´eplacentet nousconnaissons`al’avanceunensembledemarchespossiblesquecesobjetspeuvent prendre. Nousavonslapossibilit´edeplacerdescapteurssurlesarcsdugraphequi nousindiquentquandunobjetpasseparunarc´equip´e. Ainsi,pourchaqueobjet, nousconnaissonslasuiteordonn´eedesarcs´equip´esqu’ilautilis´e. Notreobjectif estdetrouvercommentplaceraussipeudecapteursquepossiblesurlegraphede fa¸con`acequelesinformationsqu’ilsrenvoientsuffisentquandmˆeme`areconstituer exactementl’itin´erairedes marcheurs.Prenonsparexemplelegrapherepr´esent´een figure1,o`ulesarcstu,vwetwxsont´equip´esdecapteursquenousappelonsa,betc respectivement. Unobjetquisuitla marche(t,u,v,w,x,z)activerasuccessivement lescapteursa,betc. Unobjetquisuitla marche(t,u,w,x,y,v,w,z)activerales mˆemecapteursmaisactiveracavantb. Ainsi,l’ensembledecapteurs{a,b,c}permet dedistinguercesdeux marches.

t z u w x v y a b c

Figure1: Ungraphedonttroisarcs(repr´esent´esenbleu)sont´equip´esdecapteur. Lacomplexit´edeceprobl`emead´ej`a´et´e´etudi´edans[85],maisavantnostravaux, lesseulsalgorithmescon¸cuspourceprobl`eme[88]sepla¸caientdanslecasdesgraphes acycliques,danslequellesmarchespossiblessonttr`eslimit´ eesetnepeuventnotam-mentpaspasserplusieursfoisparle mˆemesommetoule mˆemearc.

Leprobl`emedetraffic monitoringrequiertdetrouverdes moyensefficacesde d´ecriredesensemblesdes marchesdansungraphe,cequenousfaisonsgrˆaceades outilsdeth´eoriedeslangages. Nousfaisonsapparaˆıtredesliensentreleprobl`emede traffic monitoringetleprobl`emedecodes´eparateurgrˆace`aunnouveau mod`elede s´eparationbas´esurleslangages,qu inouspermetdeprendreencomptelenom-bredefoisetl’ordredanslequellescapteurssontactiv´es.

Nousessayonsen-suited’identifierlestypesd’instanceslesplusutilesenpratiqueetd’exploiterleur sp´ecificit´espourd´evelopperdesalgorithmesaussiefficacequepossible. C’estici quele mod`eledetransitionsinterditesprendtoutesonimportance. Nous´etudions plusieurstypesd’instances,dontcertainsfontapparaˆıtredestransitionsinterdites, etnousd´evelopponsdesalgorithmespourreformulerleprobl`emedetraffic mon i-toringsouslaformed’unprogrammelin´eaireennombresentiers[9].

L’autreprobl`emequenous´etudionsdanscettepartiedelath`eseestceluid ’ensem-bledetransitionsconnectantminimum,quipeutˆetrevucommel’analogueenterme detransitionsduprobl`emed’arbrecouvrantminimum.´Etantdonn´ eungraphecon-nexe,unarbrecouvrant minimumestunsous-ensembled’arˆetesdetaille minimum quiassurelaconnectivit´edugraphe,i.e.telqu’ilexisteune marcheentretoute pairedesommetsquin’utilisequedesarˆetesdel’arbrecouvrant minimum. Dela mˆemefa¸con,unensembledetransitionsconnectant minimumestunensemblede transitionsaussipetitquepossibletelqu’ilexisteune marcheentrechaquepaire desommetsquin’utilisequedestransitionsdel’ensembleconnectant minimum. D’apr`eslad´efinitionstandardd’unetransitiondansungraphenonorient´e,utiliser deuxfoisdesuitelamˆemearˆeten’estpasunetransitionetestdonctoujoursautoris´e. Parexemple,lafigure2repr´esenteungrapheo`ulesdeuxtransitionsautoris´eessont uvyetwvy.Ilexistedes marchesautoris´eesentrewetn’importeque lautresom-metdugraphe:les marches(w,v)et(w,x)n’utilisentaucunetransition,la marche (w,v,y)utiliseunetransitionautoris´eeet mˆemesila marche(w,v,u)estinterdite, ilesttoujourspossiblederelierw`auparlamarche(w,v,y,v,u). N´eanmoins,ilest impossibled’allerdex`auouyetl’ensembledetransitions{wvy,uvy}n’estdonc pasconnectant.Illedevientsionluiajouteparexemplelatransitionuvx.

u

x w

v y

Figure2: Ungrapheavecdeuxtransitionsautoris´ees,repr´esent´eesenvert. Dansbeaucoupdechampsd’applicationo`ule mod`eledetransitionsinterdites estpertinent(c’estlecasparexempledesr´eseauxroutiers,desr´ eseauxdetrans-portsoudesr´eseauxoptiquedet´el´ecommunications),ilpeutarriverqu’`acausede travauxde maintenanceoud’un malfonctionnement,certainestransitionsdev ien-nentinutilisables.Leprobl`emederobustesse`alasuppressiondetransitionsad´ej`a ´

et´e´etudi´epourplusieurspropri´et´esdesgraphes[105]etlaconnexit´ eestunepro-pri´et´efondamentalequ’onattenddesr´eseauxdanstousleschampsd’application destransitionsinterdites. Toutefois,cequenous´etudionsicin’estpaslep luspe-titnombredetransitions`aenleverpourd´econnecterlegraphe, mais`al’inverse,le pluspetitnombredetransitions`aassurerpourgarantirlaconnexit´edugraphe. Sansdonnerd’informationpertinentesurlarobustesseauxpannesdur´eseau,cette grandeurpermettoutefoisdemettreen´evidencequellepartiedur´eseausontlesplus importantes`asonbonfonctionnementetpeutdoncs’av´ererutiledanslaconception

der´eseaurobuste,commelesarbrescouvrantssontutilesdanslesdomaineso`ules pannesaffectentlesarˆetesetnonpaslestransitions.

Cetteth`esepr´esentelesr´esultatsd’untravailcommunavecBenjam inBergoug-nouxet´etudiediff´erentsaspectsdeceprobl`eme. Nousana lysonslesaspectsstruc-turelsdesensemblesconnectants minimaux,etenressortonsunereformulationdu probl`emesousformed’unprobl`emeded´ecompositiondegraphequenousappelons hypergraphesconnectantsoptimaux. Cenouveauprobl`emes’av`ereplusfacile`a ma-nipuleretsemontred’unegrandeaidedanslespreuvesdesr´esultatssuivants. Nous nousint´eressonsensuite`al’aspectalgorithmiqueduprobl`emeet`asacomplexit´eet nosr´esultatsprincipauxsontlapreuvedeNP-compl´etudeduprobl`emeetla mise aupointd’une3

2-approximationpolynomiale[10].

Graphesg´eom´etriquesetensemble´evitantladistance1

Ladeuxi`emepartiedecetteth`esepr´esentelesr´esultatsdeprojetsmen´esavecChr is-tineBachoc,Philippe Moustrou, ArnaudPˆecheretAntoineSedillotsurladensit´e maximalequipeutˆetreatteintepardesensemblesdepointsAdel’espacer´eelquine contiennentpasdeuxpoints`adistanceexactement1.Intuitivement,ladensit´eest laportiond’espacequ’occupeA. Laplusgrandedensit´eatteignabled´ependaussi deladistancedonton munitRn<sub>. Danscetteth`ese,nousneconsid´eronsquedes</sub> distancesinduitespardesnormesetnousnotonscettequantit´em1(Rn,·)o`u · estnotrenormesurRn_{.Ladensit´e maximaledesensembles´evitantladistance1a} ´

et´e´etudi´eedepuisau moinsled´ebutdesann´ees1960[91],surtoutdanslecasdu planeuclidien, maiss’av`ereˆetreunprobl`emetr`esdifficilequiestencoretr`esouvert. Prenonsunempilementdedisquesderayon1(i.e.unensemblededisquesde rayon1deux`adeuxdisjoints). Unexempled’ensemblequinecontientpasdeux points`adistance1estl’uniondedisquesouvertsderayon1

2etde mˆemecentre quelesdisquesdenotreempilement. Ladensit´e maximaled’unempilementde disquedansleplaneuclidienestd’environ0.9069etonsaitainsiquem1(R2)

0.9069

4 0.2267. Unensembler´ealisantcetteborneestillustr´eenfigure3. La meilleureborneinf´erieureest`apeine meilleure;ellea´et´eobtenuepar Croften 1967enaffinantlaconstructionpr´ec´edenteetestd’environ0.2293. `Al’inverse, lesbornessup´erieuresont´et´eam´elior´eesplusieursfoisaufildesann´ees maissont toujoursloindelaborneinf´erieure. Avantnostravaux,lameilleurebornesup´erieure connue´etaitdem1(R2) 0.258795eta´et´e´etabliepar Keletietal.en2015[70] `

al’aided’uneapprochebas´eesurl’analyseharmoniquetr`esdiff´erentedelanotre. Danscetteth`ese,nousd´evelopponsunenouvelleapprocheetam´elioronslaborne enm1(R2) 0.256828.

Unobjectifimportantdestravauxportantsurcesu jetestdeprouverunecon-jectured’Erd˝osselonlaquellem1(R2)<1₄.Remarquonsquesilanormeesttelleque l’ensembledespoints`adistance1del’origineestunpolytopequipaveparfaitement l’espace(c’est`adirequ’ilexisteunempilementdepolytopesunit´ededensit´e1), laconstructionpr´esent´eedansleparagraphepr´ec´edentproduitunensemble´evitant ladistance1dedensit´eexactement1

4.Ainsi,laconjectured’Erd˝osditquem1est pluspetitdansleplaneuclidien`acausedesespacesvidesentrelesdisquesd’un empilementoptimal.

Figure3: Unexempled’ensemble´evitantladistanceeuclidienne1(enbleu). Pourprogresserverslaconjectured’Erd˝os,BachocetRobinsont´etudi´ eladen-sit´edesensembles´evitantladistance1pourdesdistantesinduitespardesnormes pourlesquelleslepolytopeunit´epaveparfaitementl’espace(onappelledetelles normesdesnormesparall´elo`edres).Ilsontconjectur´equedanscecas ,laconstruc-tionpr´ec´edente´etaitoptimaleetdonc,quesi ·P estunenormeparall´elo`edre, m1(Rn,·P)=₂1n.Danscetteth`ese,nousprouvonscetteconjectureendimension

2ainsiquepourunefamilledenormesparall´elo`edresendimensionquelconqueet nous´etablissonsdesbornessurm1(Rn,·P)pouruneautrefamilledenormes[5].

Ceprobl`emeest´etroitementli´eauc´el`ebreprobl`emede Hadwiger-Nelson,qui consiste`ad´eterminercombiendecouleurssontn´ecessairespourcolorertousles pointsduplandesortequedeuxpoints`adistanceexactement1n’aientjamais la mˆemecouleur. Lafigure4illustreunecolorationd’uneportionduplan. Par exemple,untriangle´equilat´eraldecˆot´e1dessin´edansunplanproprementcolor´e auraforc´ementsestroissommetsdecouleursdiff´erentes.

Figure4: Unecolorationd’uneportionduplaneuclidien.

Denombreusesvariantesdeceprobl`emeont´egalement´et´e´etudi´ees, parmi lesquelleslacolorationd’espaces munisdedistancesnon-euclidiennes,lacoloration mesurable(o`uonimposequelesclassesdecouleursoient mesurables)ouencorela colorationfractionnaire,o`ul’oncherchelepluspetitnombrea

btelqueacouleurs diff´erentessoientsuffisantespourdonner`achaquepointduplanbcouleursdistinctes defa¸con`acequedeuxpoints`adistance1n’aientaucunecouleurencommun.

Leprobl`emedeHadwiger-Nelsonafaitl’objetdenombreuxtravauxdepuisau moins1960[52].Lesbornesinf´erieuresetsup´erieuresde4et7ont´et´erapidement ´

etablies, maispersonnen’asulesam´eliorerjusqu’`acequeDeGreyprouveenavril 2018dans[34]qu’au moins5couleurssontn´ecessairespourcolorerproprementle planeuclidien.

Touscesprobl`emespeuventˆetrereformul´escommedesprobl`emesdansdes graphesg´eom´etriques.Legrapheg´eom´etriquedontlessommetssonttouslespoints del’espaceetdontlesarˆetessontlespairesdepoints`adistanceg´eom´etrique1est appel´elegraphedistance-unit´e. Ladensit´edesensembles´evitantladistance1,le probl`emedeHadwiger-Nelsonetsesvariantesreviennenttous`ad´eterminerlavaleur d’invariantsdegraphesbienconnus,telsqueletauxdestabilit´e ,lenombrechroma-tiqueoulenombrechromatiquefractionnaire,surlegraphedistance-unit´eduplan euclidien.

Lesgraphesdistance-unit´eontuneinfinit´enond´enombrab ledesommetsetsor-tentducadredes math´ematiquesdiscr`etesdanslequels’inscrittraditionnellement lath´eoriedesgraphes. Cependant,beaucoupd’informationssurlegraphed istance-unit´epeuventˆetred´eduitesdesessous-graphes.Parexemple,letriangle´equilat´eral repr´esent´eenfigure4d´efinitunsous-graphecompletdetaille3pourlequeltrois couleurssontn´ecessaires. Onprouveainsiqu’au moinstroiscouleurssontrequises pourcolorerleplaneuclidienentier. Nousverronsquedespropri´et´essimilairessont v´erifi´eesparladensit´edesensemblesstables(quisontdesensemblesdesommets deux-`a-deuxnon-adjacentsetsontdoncdirectementreli´es`am1).

Nous´etudionsalgorithmiquementlanotiondetauxdestabilit´epond´er´eoptimal etl’utilisonspour´etudierlesprobl`emesd´ecritspr´ec´edemment. Cettem´ethodeestau cœurdeplusieursprojetsencoursetad´ej`aamen´edesr´esultatsint´eressants.Parmi cesr´esultats,onpeutciterl’am´eliorationdelabornesup´erieuresurm1(R2), mais aussil’am´eliorationdelaborneinf´erieuresurlenombrefractionnairechromatique duplande3.61904[31]`a3.89366etunebornesup´erieuresurm1(R3,·,P)quand Pestunparall´elo`edrer´egulier[4][11].

Complexit´edeshomomorphismesdegraphes

Latroisi`emepartiedecetteth`eseportesurlacomplexit´edeprobl`emesd ’homomor-phismesdegraphes,etplusparticuli`erementsurleshomomorphismeslocalement injectifsdegraphesorient´es. Unhomomorphismedegrapheestunefonctiondes sommetsd’ungrapheverslessommetsd’unautretellequel’imaged’unearˆeteest unearˆete,oudanslecasorient´e,tellequel’imaged’unarcestunarc.Parexemple, danslafigure5,lafonctionquiassocieb,fetg`au,dete`av,a`awetc`axestun homomorphismedugrapherepr´esent´eenfigure5averslegraphedelafigure5b.

b a d c f e g (a) v u x w (b)

Figure5: Onattribueunecouleur`achaquesommetdugraphecible(`adroite).La couleurdessommetsdugraphedegaucheindiqueleurimageparl’homomorphisme.

Leshomomorphismessontuneg´en´eralisationdelacolorationdegraphes. Dans lecasnon-orient´e,lacolorationestlecasparticulierd’homomorphismeo`uchaque

sommetdugraphecibleestreli´e`achaqueautre, maispas`alui-mˆeme. Danslecas orient´e,lescolorationssontd´efiniescommedeshomomorphismesversdesgraphes quicontiennentexactementunarcentrechaquepairedesommetsdistincts. Detels graphess’appellentdestournoisetsonttr`esimportantsdansl’´ etudedeshomomor-phismesorient´es.

Leshomomorphismeslocalementinjectifssontuncasparticulierd ’homomor-phismeso`uonajoutelacontraintequel’homomorphismedoitˆetreinjectifsurlevois i-nagedessommetsdugrapheenentr´ee. Leshomomorphismeslocalementinjectifs sontint´eressantsdufaitqu’ilspr´eserventlastructurelocaledugrapheenentr´eebien mieuxquenelefontleshomomorphismesstandards. Ainsi,leshomomorphismes localement-injectifstrouventdesapplicationsdansunegrandevari´et´ededomaines, parmilesquelslad´etectionde motifsenanalysed’image,labio-informatiqueou encorelath´eoriedescodes. Cependant,enfonctiondecequel’onveut mod´eliser, noscontraintesnesontpasexactementles mˆemesetl’ontrouveainsiplusieurs d´efinitionsdel’injectivit´elocaledanslalitt´erature. Parexemple,onpeutvouloir quel’homomorphismesoitinjectifsurlevoisinageouvertdessommetsousurleur voisinageferm´e(quiinclutlesommetlui-mˆeme). Enfonctiondelad´efinition,le faitquesurlafigure5,lesommetgaitla mˆemeimagequesonvoisinfpeutˆetre compatibleounonaveclecrit`ered’injectivit´elocale. Dela mˆemefa¸cononpeut imposerquel’homomorphismesoitinjectifsurlesvoisinagesentrantsetsortants dessommetss´epar´ementousurleurunion.Iciencore,enfonctiondelad´efinition, lefaitqu’unvoisinentrant(lesommetb)etqu’unvoisinsortant(f)dedaientla mˆemeimagepeutˆetreautoris´eounon. Onchoisitenfonctiondesapplicationsles propri´et´esdugrapheinitialquidoiventˆetrepr´eserv´eesetonchoisitlad´efinition d’injectivit´elocalequilepermet. Parexemple,sil’homomorphismeestinjectifsur l’uniondesvoisinagesentrantsetsortantsdessommets,onassurequel’imaged’un chemin(une marcheo`ulessommetssontdeux`adeuxdistincts)delongueur2sera ´

egalementunchemindelongueur2. N´eanmoins,renforcerle mod`elepeutrendre algorithmiquementplusdifficileleprobl`emeded´eterminerl’existenced ’unhomo-morphismelocalementinjectifentredeuxgraphes. C’estcequenous´etudionsdans cettepartiedelath`ese.

L’objectifhabitueldes´etudessurlacomplexit´edesprobl`emescommela k-colorationoul’existenced’unhomomorphismeversungrapheGdonn´eestlamiseau pointd’unth´eor`emededichotomie:onpartitionnelesprobl`emesendeuxclasses,on prouvequeceuxdelapremi`ereclassesontpolynomiauxetqueceuxdeladeuxi`eme sont NP-complets. Parexemple,danslecasnon-orient´e,i lestconnuquelak-colorationestpolynomialepourk 2etNP-compl`etepourk>2. HelletNeˇsetˇril ontprouv´edans[61]queleprobl`emed’homomorphismeversungraphenon-orient´e GestpolynomialsiGposs`edeuneboucleouestbipartietNP-completdanslecas contraire. Cesr´esultatssontdesth´eor`emesdedichotomie.

Notrepointded´epartdansl’´etudedelacomplexit´edeshomomorph ismeslo-calementinjectifsestlecaso`ulacibleestuntournoi. Eneffet,lacomplexit´edes variantesd’homomorphismes´etudi´eesdanslalitt´eratured´ependsouventdunombre chromatiquedelacible.Parexemple,dansleth´eor`emedeHell-Neˇsetˇril,lesgraphes bipartissontexactementlesgraphes2-colorables. Parmilesquatred´efinitionsnon ´

complexit´edetroisestouvertedanslecasdestournois. Cestroisd´efinitionssont ´

equivalentesdanslecaso`ulacibleneposs`edepasdebouclesetl’uned’ellesne tientpascomptedesbouclessurlacible.Sacomplexit´eestdoncimpliqu´eeparcelle desdeuxautresetilnousrestedonc`a´etudierdeuxd´efinitionsd’injectivit´elocale. Souscesd´efinitions,Campbell,Clarkeet MacGillivrayontd´ej`a´etablidans[20]que leprobl`emeestpolynomialversletournoir´eflexif`adeuxsommetsetNP-complet verslesdeuxtournoisr´eflexifs`atroissommets. Cependant,aucunth´eor`emede dichotomien’avait´emerg´esuruneclassedetournoisinfinieavantnostravaux.

AvecStefanBard,ChristopherDuffy, Gary MacGillivrayetFeiran Yang,nous avons´etabliunth´eor`emededichotomiesurlacomplexit´edeshomomorphismes localementinjectifsverslestournoisr´eflexifspournosdeuxd´efinitionsd’injectivit´e locale:danslesdeuxcas,leprobl`emeestpolynomialsilacibleadeuxsommets ou moinsetNP-completsilacibleenatroisouplus[7]. Nosr´esultatsimpliquent ´egalementunth´eor`emededichotomiesurlacomplexit´edescolorationsr´eflexives localementinjectives.

Contenudelath`ese

Lechapitre 1pr´esentelesnotionsdebasequenousutilisonstoutaulong delath`ese. Cesnotionsviennentdedomainesvari´esdes math´ematiquesetde l’informatiquedontlath´eoriedesgraphes,delacomplexit´e,lag´eom´etrie,lath´eorie deslangagesetlaprogrammation math´ematique.

Lechapitre2introduitleprobl`emedetraffic monitoring,pr´esentelemod`eleet lesoutilsquenousd´evelopponspourl’´etudieret montrecommentlesutiliserpour r´esoudreleprobl`emesurplusieurstypesd’instancesint´eressantesenpratique.

Lechapitre3pr´esenteleprobl`emed’ensembledetransitionsconnectant m in-imum,l’´etudiesurplusieursclassesdegraphesetpr´esenteunereformulation,une approximationpolynomialeetunepreuvedeNP-compl´etudedanslecasg´en´eral.

Lechapitre4pr´esenteleprobl`emedeladensit´edesensembles´evitantlad is-tance1ainsiquesoncontexte,etnotammentleprobl`emedeHadwiger-Nelson. Nous ypr´esentonsune m´ethodequiutilisedesbornessurletauxdestabilit´edegraphes g´eom´etriquesinfinisetl’utilisonspourprouverlaconjecturedeBachoc-Robinssur plusieursfamillesdeparall´elo`edresdonttousceuxdedimension2,etnous´etablissons desbornessup´erieuressurm1(Rn, ·)pourd’autres.

Lechapitre5pr´esentelanotiondetauxdestabilit´epond´er´eoptimal,l’´etudie surdesgraphesg´eom´etriquesetl’utilisepour´etendrelesr´esultatsduchapitre pr´ec´edent. Cechapitrecontientdesam´eliorat ionssurlabornedunombrefrac-tionnairechromatiqueduplaneuclidienetsurladensit´edesensembles´evitantla distance1dansleplaneuclidienetdansl’espaceentroisdimensions´equip´ede normesparall´elo`edresr´eguli`eres.

Lechapitre6pr´esenteleprobl`emed’homomorphismeslocalementinjectifset lesdeuxd´efinitionsquenousconsid´eronsetprouvepourlesdeuxunth´eor`emede dichotomiesurlestournoisr´eflexifs.

Lechapitre7conclutlath`eseenr´esumantlescontributionsprincipaleseten pr´esentantdesprobl`emesouvertsquenostravauxsoul`eventetquipourraientfaire l’objetdefutursprojets.

Introduct

ion

Thisthesisstudiesproblemsandnotionsthatrevolvearoundthe mathematica lob-jectscalledgraphs. Graphsanddirectedgraphsareverypowerfulmodelsthatallow todescribeanybinaryrelationonaset. Theelementsofthissetarecal ledver-ticesandthepairofrelated(oradjacent)verticesarecallededges. Becauseoftheir expressiveness,graphsfindapplicationinacountlessvarietyoffields:information systems,telecommunication,bio-informatics,imageprocessing,transportat ionsys-tems,socialnetworks,scheduling....among manyothers. Graphswereintroduced almostthreecenturiesagoandhavereceivedever-increasingattentionsince ,espe-ciallywiththeemergenceofcomputerscience.

Moreprecisely,thisthesispayscloseattentionstothenotionsof walksand distancesingraphsandtothecombinatorial,algorithmicandcomplexityaspectsof someoftherelatedproblems. Awalkinagraphisasequenceofadjacentvertices thatallowstoconnecttwovertices. Thenumberofelementsinthe walkallows todefineitslengthandthelengthofthewalksbetweentwoverticesleadstothe definitionoftheirdistance. Wecloselystudygeometricgraphs, whicharegraphs whoseverticesarepointoftherealspace. Theconnectionbetweenthedistance definedbytheadjacencyinthegraphandthegeometricdistancewillbeofgreat interesttous. Ourworksalsoinvolveextensivelyotherwell-knownnotionsofgraph theory,suchasindependentsets,graphcolouringsandhomomorphisms.

Theproblemsandresultspresentedinthisthesiscanbedividedintothreeparts.

Transitionsingraphs

Thefirstpartfocusesonthe modelofforbidden-transitiongraphs. Thestrength ofthe modelsofgraphsand walks makesthemthe modelofchoicetoaddress routingproblemsinvariouscontexts. Forexample,theroadnetworkinacitycan bemodelledbyagraphwhereeverypotentialdestinationorcrossroadisdenotedby avertexandroadsbetweentwovertices,byedges(orarcsiftheroadishalfway). Wetherebyimplicitlydefineasetofpossible walksbetweeneachpairofvertices thatcanthenbeusedfor manypurposes. Examplesincludeseveraloptimization problem,suchasfindingtheshortestwalkbetweentwopoints,avoidingtrafficjams, or monitoringandseparatingasetofpossiblewalks. However,inpractice, many ofthewalksthatthegraphdefinesdenoteroutesthatdriversarenotallowedto take. To modelasituation whereadriver maynotturnleftorrightatagiven crossroad, wedefineatransitioninagraphasapairofconsecutiveedges. The studyofforbidden-transitiongraphs,wheregraphsaredefinedtogetherwithasetof

edge-colouredgraphsandgraphswithforbiddensubpaths.

Thefirstofthetwomainproblemsrelatedtoforbiddentransitionsthatwestudy inthisthesisiscalledtrafficmonitoring. Here,wearegivenagraphinwhichobjects canwalkandweknowwhichwalkstheseobjectscanuse. Wehavethepossibilityto placecensorsonthearcsofthegraphthatindicatewhenanobjectgoesthroughan equippedarc. Hence,foreachobject,weknowtheorderedsequenceofequippedarcs theyhaveused. Ourobjectiveistofindhowtoequipasfewarcsaspossiblewith censorsinsuchawaythattheinformationthatthecensorsreturnisstillsufficient todetermineexactlywhichroutetheobjecthastaken. Forexample,considerthe graphdepictedinFigure1andletthearcstu,vwandwxbeequippedwithcensors that wecalla,bandcrespectively. Anobjectthatusesthewalk(t,u,v,w,x,z) wouldactivatesuccessivelythecensors a,bandc. Anobjectthatusesthe walk (t,u,w,x,y,v,w,z)activatesthesamecensorsbutactivatesthecensorcbeforeb. Thus,thesetofcensors{a,b,c}allowstodistinguishthosetwowalks.

t z u w x v y a b c

Figure1: Agraphwiththree monitoredarcsdepictedinblue.

Thecomplexityofthisproblemhadalreadybeenstudiedin[85],butpriorto ourwork,theonlyalgorithmsthathadbeendesigned[88]focusedonthespecial caseofacyclicgraph,inwhichwalksareverylimitedandcannotusetwicethesame vertexorarc.

Thetraffic monitoringproblemrequirestofindefficientwaystodescribesetsof walksinagraph,whichwedobyusingtoolsstemmingfromlanguagetheory. We drawparallelsbetweentheproblemoftrafficmonitoringandthewell-knownproblem ofseparatingcodebydevelopinganew modelofseparationbasedonlanguages, whichallowsustotakeintoaccountthenumberoftimesandtheorderinwhich thecensorsareactivated. Wethentrytoidentifywhichkindofinstancescanbe relevantforthepracticalapplicationsoftraffic monitoringandusetheirspecificity todevelopsolutionsasefficientaspossible. Thisiswherethe modelofforbidden transitionscomesuseful. Westudyseveralkindsofinstances,someofwhichinvolve forbiddentransitionsandwedevelopalgorithmstoreformulatetrafficmonitoringas anintegerlinearprogram[9].

Theotherproblemthatwestudyinthispartofthethesisiscalled minimum connectingtransitionsetandcanbeseenasanadaptationwithtransitionsofthe well-knownminimumspanningtrees. Givenaconnectedgraph,aminimumspanning treeisasubsetofedgesof minimumsizethatkeepsthegraphconnectedi.e.such

thatthereexistsawalkbetweeneverypairofverticesthatonlyusesedgesofthe minimumspanningtree.Similarly,a minimumconnectingtransitionsetisasetof transitionassmallaspossiblesuchthatthereexistsawalkbetweeneverypairof verticesthatonlyusestransitionsofthe minimumconnectingtransitionset. Note thataccordingtothestandarddefinitionoftransitionsinundirectedgraphs,using twicethesameedgeinarowisnotatransitionandisalwayspermitted.Forexample, considerthegraphdepictedinFigure2wherethetwopermittedtransitionsareuvy andwvy. Therearepermittedwalksleadingfromthevertexwtoanyothervertex ofthegraph:thewalks(w,v)and(w,x)donotevenrequireatransition,thewalk (w,v,y)usesapermittedtransitionandwhilethewalk(w,v,u)isforbidden,one canstillgofromwtoubyusingthewalk(w,v,y,v,u). However,thereisnoway togofromxtouoryandthetransitionset{wvy,uvy}isthereforenotconnecting. Thiscanbefixedforexamplebyaddingthetransitionuvx.

u

x w

v y

Figure2: Agraphwithtwopermittedtransitionsdepictedingreen.

Inmanyoftheapplicationfieldswherethemodeloftransitionsisrelevant(such asroadnetworks,transportationsystemsoropticaltelecommunicationnetworks), itispossiblethatbecauseofmaintenanceworkormalfunction,sometransit ionsbe-comeunusable. Theproblemofrobustnesstotheremovaloftransitionshasalready beenstudiedwithseveralgraphproperties[105]andconnectivityisanimportant requirementofthenetworksineveryapplicationfieldsofgraphs withforbidden transitions. Here,whatwestudyisnotthesmallestnumberoftransitionsthathas tobreakdowntodisconnectthegraphbutthesmallestnumberoftransitionswe havetosecureforthegraphtostayconnected. Whilethisdoesnotprovidean interesting measureoftherobustnessofthenetwork,ithighlightswhichpartsofa networkarethe mostusefulforitsproperfunctioningandcanthushelpdesigning robustnetwork,like minimumspanningtreeshelpinthecaseswherebreakdowns impactedgesandnottransitions.

ThisthesispresentstheresultsofjointworkwithBenjaminBergougnouxand studiesseveralaspectsoftheproblem. Weinvestigatestructuralaspectsofminimum connectingsets,whichleadstoareformulationoftheproblemasaproblemofgraph decompositionthatwecalloptimalconnectinghypergraph. Thisnewproblemturns outtobeeasierto manipulateandofgreathelpinthesubsequentproofs. We thenstudytheproblemunderitsalgorithmicandcomplexityaspectsandour main resultsaretheestablishmentofitsNP-completenessandthedesignofapolynomial

3

ThesecondpartofthisthesispresentstheresultsofjointprojectswithChristine Bachoc,Philippe Moustrou,ArnaudPˆecherandAntoineSedillotonthe maximum densitythatcanbeachievedbyasetofpointsA oftherealspacethatdoesnot containtwopointsatgeometricdistanceexactly1.Intuitively,thedensitydenotes theportionofspacethatAfills. Thismaximumdensityalsodependsonthedistance weuseonRn_{. Throughoutthisthesis,weonlyconsiderdistancesinducedbynorms} andwedenotethisnumberbym1(Rn, ·)where · isthenormRnisequipped with. The maximumdensityofsetsavoidingdistance1hasbeenstudiedsinceat leasttheearly1960’s[91]especiallyintheEuclideanplane,butturnsouttobea verydifficultproblemandisstillwideopen.

Considerapackingofdiscsofradius1(i.e.asetofdiscsofradius1thatdo notoverlap). Anexampleofsetthatdoesnotcontaintwopointsatdistance1is theunionofopendiscsofradius1

2andofsamecenterasthediscsofourpacking. TheoptimaldensityofapackingofdiscsintheEuclideanplaneisabout0.9069 andwethusknowthatm1(R2) 0.9069₄ 0.2267. Asetachievingthisdensityis illustratedinFigure3. Thebestknownlowerboundisbarelybetter;itwasobtained byCroftin1967byrefiningthepreviousconstructionandisofabout0.2293. On theotherhand,upperboundshavebeenimproved moreoftenthroughtheyears butarestillfarfromthelowerbounds. Priortoourworks,thebestupperbound wasm1(R2) 0.258795andwasestablishedby Keletietal.in2015[70]through anapproachbasedonharmonicanalysisverydifferentfromours.Inthisthesis,we developanewapproachandimprovetheboundtom1(R2) 0.256828.

Figure3: AnexampleofsetavoidingEuclideandistance1(inblue). Animportantobjectiveoftheworksthathavebeencarriedoutonthistopicisto proveaconjecturebyErd˝osthatm1(R2)<1₄.Notethatifthenormissuchthatthe setofpointsatdistance1fromtheoriginisapolytopethattilesthespaceperfectly (i.e.thereexistsapackingofunitpolytopesofdensity1),theconstructiondescribed inthepreviousparagraphleadstoasetavoidingdistance1ofdensityexactly1

4. Thus,theconjectureofErd˝ossaysthatm1islowerwiththeEuclideannormbecause oftheemptyspacebetweentheunitdiscsofanoptimalpacking,

Asasteptoward Erd˝os’conjecture, Bachocand Robinsstartedstudyingthe densityofsetsavoidingdistance1fordistanceinducedbynormswhoseunitpo ly-topesperfectlytilethespace(suchnormsarecalledparallelohedronnorms). They conjecturedthatinthiscase,theconstructiondescribedpreviouslyofsetsavo id-ingdistance1 wasoptimal,andthus,thatif ·P isaparallelohedronnorm,

m1(Rn, ·P)=₂1n.Inthisthesis,weprovethisconjectureindimension2aswell

asforafamilyofn-dimensionalparallelohedronnorms,andestablishboundson m1(Rn, ·P)foranother[5].

Thisproblemiscloselyrelatedtothefamous Hadwiger-Nelsonproblemthat aimsatdetermininghow manycoloursareneededtocolourallthepointsofthe Euclideanplaneinsuchawaythatnotwopointsatdistanceexactlyonereceivethe samecolour. Figure4depictsacolouringofaportionoftheplane. Forexample, everyequilateraltriangleofdiameter1drawninaproperly-colouredplanehasits threeverticesofdifferentcolours.

Figure4: AcolouringofaportionoftheEuclideanplane.

Manyvariantsofthisproblemhavealsobeenstudied,includingthecolouring ofotherspacesorofspaceequippedwithnon-Euclideandistances,the measurable colouring(wherewerequirethecolourclassestobe measurable),andthefractional colouringwherewelookforthesmallestnumbera

bsuchthatwecangivebdifferent colourstoallthepointsofaspacechosenamongasetofpossiblecoloursofsizea, insuchawaythatnotwopointsatdistanceexactlyoneshareacommoncolour.

Theproblemof Hadwiger-Nelsonhasreceivedalotofattentionsinceatleast 1960[52].Lowerandupperboundsof4and7werequicklyestablished,butnoone hasbeenabletoimprovethemuntil De GreyprovedinApril2018in[34]thatat least5coloursareneededtoproperlycolourtheEuclideanplane.

Alltheseproblemscanbereformulatedasproblemsingeometricgraphs. The geometricgraphwhoseverticesareallthepointsofthespaceandwhoseedgesare thepairsofpointsatgeometricdistance1iscalledtheunit-distancegraph. The densityofsetsavoidingdistance1,the Hadwiger-Nelsonproblemanditsvariants allcomedowntodeterminingthevaluesofwell-knowngraphinvariantssuchasthe independenceratio,thechromaticnumberorthefractionalchromaticnumberof theunit-distancegraphoftheEuclideanplane.

Unit-distancegraphshaveuncountably manyverticesandbringusoutofthe rangeofdiscrete mathematicsinwhichgraphtheoryistraditionallyconsideredto belong. However, manyinformationontheunit-distancegraphcanbeinferredfrom itssubgraphs.Forexample,theequilateraltriangledepictedinFigure4providesa completesubgraphofsize3,forwhichthreecoloursarerequired. Thisalsoproves thatatleastthreecoloursareneededtocolourtheentireEuclideanplane. Wewill seethatsimilarpropertiesholdsforthedensityofindependentsets(setsofpairwise non-adjacentvertices,whicharethusdirectlyrelatedtom1).

coreofseveralstillongoingprojectsandhasalreadyprovidedinterestingresults. Thoseresultsincludetheimprovementoftheupperboundonm1(R2)butalsothe improvementofthebestlowerboundonthefractionalchromaticnumberofthe planefrom3.61904[31]to3.89366andanupperboundonm1(R3,·,P)whenPis aregularparallelohedron[4][11].

Complexityofgraphhomomorphisms

Thethirdpartofthisthesisstudiesthecomplexityofgraphhomomorph ismprob-lems,andespeciallyoflocally-injectivedirectedgraphhomomorphisms. Agraph homomorphismisafunctionfromthevertexsetofagraphtothevertexofasecond graphsuchthattheimageofanedgeisanedge,orinthedirectedcase,suchthat theimageofanarcisanarc. Forexample,inFigure5,thefunctionthat mapsb, fandgtou,dandetov,atowandctoxisahomomorphismfromthegraph depictedinFigure5atothegraphdepictedinFigure5b.

b a d c f e g (a) v u x w (b)

Figure5: Weattributeacolourtoeachvertexofthetargetgraph(ontheright). Thecolouroftheverticesofthegraphontheleftindicatestheirimagebythe homomorphism.

Homomorphismsgeneralizegraphcolouring.Intheundirectedcase,colouring isaspecificcaseofgraphhomomorphismwhereeveryvertexofthetargetgraph isconnectedtoeveryotherbutnottoitself.Inthedirectedcase,colouringsare definedashomomorphismstographswherethereisexactlyonearcbetweeneach pairofdistinctvertices.Suchgraphsarecalledtournamentsandareveryimportant inthestudyofdirectedhomomorphisms.

Locally-injectivehomomorphismsareaspecificcaseofhomomorphismswherewe addtheconstraintthatthehomomorphismmustbeinjectiveontheneighbourhood oftheverticesoftheinputgraph.Locally-injectivehomomorphismsareinteresting becausetheypreservethelocalstructureoftheinputgraph muchbetterthanstan-dardhomomorphisms. Hence,locally-injectivehomomorphismsfindapplicationina widerangeofareasincludingpatterndetectioninimageprocessing,bio-informatics orcodingtheory. However,dependingonwhatwewantto model,ourconstraints arenotexactlythesameandwecanthusfindseveraldefinitionsoflocalinjectivity intheliterature.Forexample,wecanaskthehomomorphismtobeinjectiveeither ontheopenneighbourhoodoftheverticesorontheircloseneighbourhood(which includesthevertexitself). Hence,dependingonthedefinition,thefactthatinF ig-ure5,thevertexghasthesameimageasitsneighbourfmayormaynotviolatethe

injectivityrequirement.Similarly,wemayaskthehomomorphismtobeinjectiveon thein-andout-neighbourhoodsoftheverticesseparatelyorontheirunion. Here again,dependingonthedefinition,thefactthatanin-neighbour(thevertexb)and anout-neighbour(f)ofdhavethesameimage mayor maynotbeallowed. We choosedependingontheapplicationswhichpropertiesoftheinitialgraphhaveto bepreservedbythelocally-injectivehomomorphismandwechooseourdefinitionof localinjectivityaccordingly. Forexample,ifthehomomorphismisinjectiveonthe unionofthein-andout-neighbourhoodsofvertices,wecanensurethattheimage ofapath(a walk whoseverticesarepairwisedistinct)oflength2isalsoapath oflength2. However,strengtheningthe model mightalso makethealgorithmic problemofdeterminingtheexistenceofalocally-injectivehomomorphismbetween twographs moredifficult. Thisiswhatwestudyinthispartofthethesis.

Whenstudyingthecomplexityofproblemssuchas k-colouringorhomomorphism toagivengraphG,theobjectiveistoestablishwhatwecalladichotomytheorem: wepartitiontheproblemsintotwoclasses,provethattheproblemsofthefirst onearepolynomialandthattheproblemoftheotheronesareNP-complete. For example,intheundirectedcase,itis well-knownthatk-colouringispolynomial fork 2and NP-completefork >2. Helland Neˇsetˇrilhaveprovedin[61]that undirectedhomomorphismtoagraphGispolynomialifGhasalooporisbipartite andNP-completeotherwise. Theseresultsaredichotomytheorems.

Ourstartingpointinthestudyofthecomplexityoflocally-inject ivehomomor-phismsisthecase wheretargetsaretournaments.Indeed,thecomplexityofthe variantsofhomomorphismsstudiedintheliteratureoftendependsonthechromatic numberofthetarget. Forexample,intheHell-Neˇsetˇriltheorem,bipartitegraphs areexactlythe2-colourablegraphs. Amongthefournon-equivalentdefinitionsof localinjectivitythatweknowofinthedirectedcase,thecomplexityofthreeisopen inthecaseoftournaments. Thosethreedefinitionsareequivalentinthecaseofa looplesstargetandoneofthemdoesnottakeintoaccounttheloopsonthetarget. Itscomplexityisthereforeimpliedbythecomplexityoftheothertwoandweare leftwithtwodefinitionsoflocalinjectivitytostudy. Underthosedefinitions ,Camp-bell,Clarkeand MacGillivrayhavealreadyestablishedin[20]thattheproblemis polynomialonthereflexivetournamentontwovertices(reflexivetournamentsare theonesthathaveloopsoneveryvertex)and NP-completeonthetworeflexive tournamentsonthreevertices. However,nodichotomytheoremonaninfiniteclass oftournamentshademergedpriortoourwork.

Jointly withStefan Bard, Christopher Duffy, Gary MacGillivrayand Feiran Yang,wesuccessfullyestablishedadichotomytheoremonthecomplexityoflocal ly-injectivehomomorphismstoreflexivetournamentsforourtwodefinitionsoflocal injectivity:inbothcase,theproblemispolynomialifthetargethastwoverticesor fewerandNP-completeifthetargethasthreeverticesor more[7]. Ourresultsalso implyadichotomytheoremonthecomplexityoflocally-injectivereflexivecolouring.

Outlineofthethesis

Chapter 1 introducesbasicnotionsthat weusethroughoutthisthesis.Those notionscomefromvariousareasof mathematicsandcomputerscience,including

Chapter 2 introducestheproblemoftraffic monitoring,presentsthe modeland thetoolsthatwehavedevelopedtoaddressitandshowshowtousethemtosolve theproblemonseveralkindofinstancesofpracticalinterest.

Chapter 3 introducestheproblemof minimumconnectingtransitionset,studies itonseveralclassesofgraphs,presentsareformulation,apolynomialapproximation andaproofofNP-completenessinthegeneralcase.

Chapter 4 presentstheproblemofthedensityofsetsavoidingdistance1andits context,includingtherelatedproblemof Hadwiger-Nelson. Wepresenta method thatrequirestoboundtheindependenceratioofinfinitegeometricgraphsandwe useittoprovetheBachoc-Robinsconjectureonseveralfamiliesofparallelohedra includingallthoseindimension2andtoestablishupperboundsonm1(Rn, ·) forsomeothers.

Chapter 5 presentsthenotionofoptimalweightedindependenceratio,studiesit ongeometricgraphsandusesittoextendstheresultsofthepreviouschapter. This chaptercontainsimprovementoftheboundsonthefractionalchromaticnumberof theEuclideanplaneandonthedensityofsetsavoidingdistance1intheEuclidean planeandinthe3-dimensionalspaceequippedwithregularparallelohedronnorm. Chapter 6 presentstheproblemoflocally-injectivehomomorphismsandthetwo definitionsthatweconsiderandprovesforbothofthemadichotomytheoremon reflexivetournaments.

Chapter 7 concludesthisthesisbysummingupthe maincontribut ionsandpre-sentingsomeopenproblemsthatourworkraisesandthat maybeatthecoreof futureworks.

Pre

l

im

inar

ies

Thischapterintroducesnotionsandresultsfromdifferentareasofcomputerscience and mathematicsthat weneedthroughoutthisthesis. Itspurposeisalsotofix notationsanddefinitionsforwhichdifferentvariantscanbefoundintheliterature. Section1.1presentsgenericnotionsofgraphtheorythatareatthecoreofallthe workwepresentinthisthesis.Section1.2presentsbasicnotionsofcomplexitythat weusethroughoutthisthesisandintroducesthenotionofNP-completenessthatis fundamentalinChapters3and6.Section1.3extendsSection1.1bypresentingbasic notionsofgraphtheorybutalsopresentstheemergingfieldofforbidden-transition graphs. ThosenotionsarecrucialinChapters2and3.Section1.4introduceskey notionsofgeometryandnumbertheoryandespeciallystudiesparallelohedra. The notionsdevelopedinthissectionplayanimportantroleinChapters4and5.Section 1.5presentsfundamentalsoflanguagetheoryandnotablyintroducestoolsthatare extremelyusefulinChapter2todescribesetsofwalksinagraph.Finally,Section 1.6introducesaverypowerfuloptimizationtechniquecalledlinearprogramming. Duetotheirefficiencyandexpressivepower,linearprogramsareusedinpracticein awiderangeofareasandareveryimportantinChapters2and5ofthisthesis.

Contents

1.1 Fundamentalsofgraphtheory... 18 1.2 Elementsofcomplexity... 30 1.3 Walks,connectivityandtransitions ... 33 1.4 Polytopesandlattices... 38 1.5 Rationallanguagesandautomata... 45 1.6 Linearprogramming... 50

Common mathematicalnotations

ThecardinalityofasetSisnoted|S|anditspowersetisnotedP(S). Weuse thenotation[a,b],]a,b[and[a,b[or]a,b]forclosed,openandsemi-openintervals respectively.Integerintervalsarenoted[[a,b]].

Wesaythataset Sthathasagivenpropertyismaximal(respectively minimal) ifnoneofitssupersets(resp.subsets)possessthisproperty. Wesayitis maximum

(resp. minimum)ifnosetofstrictlybigger(resp. smaller)cardinalityhasthis property.

Whenworkingin Rn_{,we maydenoteby0thevector(0,0,...,0).}

Wedenoteby P(A)theprobabilityofaneventAandbyE[X]theexpectedvalue ofarandomvariableX.

ThelimitsuperiorofasequenceU isnotedlimsup

n→∞ Unandisthesupremumof thelimitsachievedbysubsequenceofU.

1

.1 Fundamenta

lsofgraphtheory

Forastandardintroductiontographtheory,wereferthereaderto[36].

1.1.1 Coredefinitions

Undirectedgraphs. AgraphG(sometimesreferredtoasasimplegraphtoavoid anyambiguity)isanorderedpair(V,E)whereV isanon-emptyfinitesetwhose elementsarecalledverticesandE isasetofunorderedpairsofvertices whose elementsarecallededges. Wecansimplywriteuvtodenotetheedge{u,v}. The verticesuandvaretheendpointsoftheedgeuv. Theedgeuvisincidenttothe verticesuandv. Toavoidambiguities, we maydenoterespectivelybyV(G)and E(G)thesetsofverticesandedgesofagraphG.

Avertex uisaneighbourofavertexvifandonlyif{u,v}∈E. Theopen neighbourhoodofavertexv,notedN(v),istheset{u:uv∈E}andtheclosed neighbourhoodofavertexv,notedN[v],isthesetN(v)∪{v}.If{u,v}∈E,uand vareadjacentandtwoedgesareadjacentiftheyhaveacommonendpoint.

The degreeofavertexvinasimplegraphisnotedd(v)andisitsnumber ofneighbours. Themaximumdegree andminimumdegree ofagraphG,denoted respectivelyby ∆(G)andδ(G),arethe maximumand minimumdegreeofavertex ofthegraph.

A multigraph isagraphthat mayhaveloopsandparalleledges. Aloopisan edgewhosetwoendpointsarethesamevertexandparalleledges,alsocalledmultiple edges,areedgesthathavethesameendpoints.Formally,amultigraphisalsodefined asanorderedpair(V,E)buthere,edgesare multisetsofverticesofcardinality2 andEisa multisetofedges.

Example1.1.

LetusconsiderthegraphGdepictedinFigure1.1. Here,V(G)={u,v,w,x,y} andE(G) ={uv,uw,vx,wx,wy}. TheopenneighbourhoodofvisN(v) ={u,x} andvthereforehasdegree2. Themaximumandminimumdegreesofthegraphare respectively3and1andareachievedbywandyrespectively.

Theworkpresentedinthisthesisalsoinvolvesdirectedgraphs. Wenowtranslate thebasicnotionsofundirectedgraphsintothecaseofdirectedgraphs.

Directedgraphs. A directedgraphG (orsimpledirectedgraph)isanordered pair(V,A)whereVisanon-emptyfinitesetwhoseelementsarecalledverticesand

u v

w x

y

Figure1.1: AsimplegraphG. Theneighboursofvaredepictedinred. Aisasetoforderedpairsofverticeswhoseelementsarecalledarcs. Aspreviously, wemaysimplywriteuvtodenotethearc(u,v)andwedenoterespectivelybyV(G) andA(G)thesetsofverticesandarcsofagraphG. Thevertexuistheoriginof thearcuvandvisitstarget. Twoarcsaandbareconsecutiveifandonlyifthe targetofaistheoriginofb. Thearcs(u,v)and(v,u)areopposite.

Avertexuisanin-neighbourofavertexvifandonlyif(u,v)∈Aanduisan out-neighbourofavertexvifandonlyif(v,u)∈A. Thevertexuisaneighbour ofvifandonlyifuisanin-oranout-neighbourofv. Theopenneighbourhood, openin-neighbourhoodandopenout-neighbourhoodo favertexvaredenotedre-spectivelybyN(v),N−_(v)andN+_{(v). TheclosedneighbourhoodN[v],theclosed} in-neighbourhoodN−_{[v]andtheclosedout-neighbourhoodN}+_[ v]aredefinedaspre-viously.

Thenumberofin-andout-neighboursofavertexvarenotedd−_(v)andd+_(v) respectivelyandarecalledin-degreeandout-degreeofv.

Anarcfromavertextoitselfiscalledaloop. Observethatevensimpledirected graphshaveloopssince(u,u)isanorderedpairofvertices. Theexistenceofan arcfromavertexutoavertexvdefinesarelationonthevertexsetofG. Thus,a graphisirreflexiveifnovertexhasaloopandreflexiveifeveryvertexhasaloop. Adirectedgraphis symmetricif∀(u,v)∈A,(v,u)∈A too. Wedefinedirected multigraphs asanorderedpair(V,A)whereAisa multi-setofarcs(andarcsare stillorderedpairofvertices). Thisdefinitionthusallowsparallelsarcs.

Thefollowingsubclassofdirectedgraphsplaysanimportantroleinthestudy ofgraphhomomorphisms.

Definition1.2. Orientedgraphs:

Adirectedgraphisan orientedgraphifandonlyifithasnopairofopposite arcsi.e.iftherearenoverticesu=vsuchthatuvandvu∈A. Anorientedgraph −_{G isanor}→ _{ientationofanundirectedgraphG ifandonlyifV(}−_{G) =V(G)and}→ E(G)={{u,v}:(u,v)∈A(−G)or(v→ ,u)∈A(−G)}.→

Example1.3.

InthegraphG1depictedinFigure1.2a,wisanin-neighbouro fv,uisanout-neighbourofvandxisboth. Thereisapairofoppositearcsbetweenvandxand anotherbetweenwandy. Hence,thegraphG1isnotoriented.Ifweremoveanarc

u v w x y (a)AdirectedgraphG1. u v w x y (b)AnorientedgraphG2. Figure1.2

ineachpairofoppositearcs,G1becomesorientedbutisstillnotanorientationof thegraphGdepictedinFigure1.1becauseGhasnoedgebetweenwandv.

ThegraphG2depictedinFigure1.2bisanorientationofthegraphGdepicted inFigure1.1.

Wenowpresentthenotionsofsubgraphsandcomplementarygraphsthatapply tobothdirectedandundirectedgraphs.

Subgraphsandinducedsubgraphs. LetG beagraph. ThegraphH isa subgraphofGifandonlyifV(H)⊂V(G)andE(H)⊂E(G).

LetS⊂V beasubsetofvertexofG. ThesubgraphofG inducedbyS,noted G[S],isthegraphwhosevertexsetisSandwhoseedgesetis{uv∈E(G):(u,v)∈ S2_{}. ThegraphH isaninducedsubgraphofGifandonlyifH =G[V(H)].}

Letv∈V. WedenoterespectivelybyG−SandG−vthegraphsG[V\S]and G[V\{v}]. Example1.4. u v w x y (a)AgraphG. u v w (b)AgraphH1. u v x (c)AgraphH2. u v x (d)AgraphH3. Figure1.3

ThegraphH1depictedinFigure1.3bisasubgraphofthegraphG inFigure 1.3abutnotaninducedsubgraph.Indeed,thevertexset{u,v,w}inducestheedge uw. ThegraphH2depictedinFigure1.3cisthesubgraphofGinducedby{u,v,x}. ThegraphH3inFigure1.3dcontainstheedgeuxwhichisnotinG;H3istherefore notasubgraphofG.

Complementofagraph. LetG beagraph. ThecomplementarygraphofG, notedG,isthegraphdefinedbyV(G)=V(G)andE(G)={uv:uv/∈E(G)}. Example1.5. Figure1.4depictsagraphanditscomplement.

u v w x y (a)AgraphG. u v w x y (b)ItscomplementarygraphG. Figure1.4

Wenowintroducesomekeyfamiliesofgraphsthatareusefulthroughoutthis thesis.

Somebasicgraphfamilies.

•Apathofnvertices,notedPn,isthegraphdefinedbyV(Pn)={v0,...,vn−1} andE(Pn)={vivi+1:0 i n−2}.

•Acycleofnvertices,notedCn,isthegraphdefinedbyV(Cn)={v0,...,vn−1} andE(Cn)={viv(i+1) mod n:0 i n−1}.

•Acompletegraphofnvertices,notedKn,isthegraphdefinedbyV(Kn) = {v0,...,vn−1}andE(Kn)={vivj:i=j}.

•AgraphG=(V,E)isbipartiteifthereexistsapartitionofV intotwosets V1andV2suchthateveryedgeofG hasoneendpointinV1andoneinV2. Ifinaddition,∀u∈V1,∀v∈V2,uv∈E,thegraphiscompletebipartite. We denotebyKi,jthecompletebipartitegraphwith|V1|=iand|V2|=j. •Astarofn+1verticesorstarwithnbranches,notedSn,isthegraphdefined

byV(Sn)={c,v1,v2,...,vn}andE(Sn)={cvi:i∈[[1,n]]}. Thevertexcis thecenterofthestar. NotethatSn=K1,n.

•Atournamentonnverticesisanorientationofthecompletegrapho fnver-ticesKn.Ifn 3,thereareseveraltournamentsonnvertices. Notethat tournamentsarebydefinitionirreflexivebutwecanalsodefinereflex ivetour-namentsastournamentswhereweaddalooponeveryvertex.

Example1.6. Figure1.5illustratestheaforementionedgraphclasses. Weconcludethissubsectionbyintroducingthenotionofplanarity.

v3 v4 v1 v2 (a)K4. v3 v4 v1 v2 c (b)S4=K1,4. v4 v3 v5 v2 v1 (c)P5. v3 v4 v1 v2 (d)C4=K2,2. v3 v4 v1 v2 (e)Atournamenton4vertices. Figure1.5

Definition1.7. Planargraphs:

A planarembeddingofagraphG isadrawingofthegraphintheplanesuch thatnotwoedgesofGintersecteachother. Aplanargraphisagraphthatcanbe embeddedintheplane. Aco-planargraphisagraphwhosecomplementisplanar. Example1.8.

ThedrawingofK4depictedinFigure1.5aisnotaplanarembeddingsincethe edgev1v4crossestheedgev2v3. However,thegraphK4isstillplanar,asillustrated byFigure1.6a.

Figure1.6bdepictsaplanarembeddingofK2,4andwecanprovemoregenerally thatK2,nisplanarforalln.

ThegraphsK5andK3,3arefamousexamplesofnon-planargraphs.

v3 v4

v1

v2

(a)AplanarembeddingofK4.

v1 v5 v4 v3 v2 v6

(b)AplanarembeddingofK2,4.

1.1.2 Homomorphismsandcolouring

Thissubsectionpresentsthenotionofgraphhomomorphismanditsrelationto graphcolourings,whichiscentraltoChapter6.

Definition1.9.Undirectedgraphhomomorphisms:

A homomorphismfromanundirectedgraphG1=(V1,E1)toG2=(V2,E2), alsocalledG2-colouringofG1,isafunctionf:V1→ V2suchthat∀{u,v}∈ E1, {f(u),f(v)}∈E2. ThegraphG2iscalledthetargetofthehomomorphism.

Thefamousproblemofgraphcolouringisaspecificcaseofhomomorphism. Definition1.10. Undirectedn-colouring:

A n-colouringofanundirectedgraphG =(V,E),sometimescal ledpropern-colouringtoavoidanyambiguity,isafunctionc:V →{c1,...,cn}suchthat ∀{u,v}∈E,c(u)=c(v). Theelementsc1,...,cnarecalledcolours.

Findingan-colouringofagraphGcomesdowntofindingahomomorphismfrom GtotheirreflexivecompletegraphonnverticesKn.

Ifthereexistsan-colouringofagraphG,Gisn-colourable. Thesmallestnumber nsuchthatGisn-colourableiscalledthechromaticnumberofGandisnotedχ(G). IfGisn-colourablebutnot(n−1)-colourable,Gisn-chromatic.

ThesetsCi={v∈V:c(v)=ci}ofverticesofthesamecolourarecalledcolour classes. Example1.11. v0 v1 v4 v3 v2 v5

Figure1.7: A2-colouringofanundirectedgraphG.

Figure1.7presentsa2-colouringofagraphG.SinceGisnot1-colourable,itis 2-chromatic.

Whilethedefinitionofhomomorphismisstraightforwardtogeneralizetothe orientedcase(theimageofanarcmustbeanarc),thegeneralizationofcolouringis alittle moresubtle.Indeed,an-colouringofanund irectedgraphGisahomomor-phismfromGtoKnandan-colouringofanorientedgraphGisahomomorphism fromG toanorientationofKn(i.e.atournamentonnvertices). However,while thereisonlyonecompletegraphofnvertices,thereareexponentially manytour-naments. Colouringofanorientedgraphconsistsoffindingnotonlythecolouring functionbutalsothetargettournament. Forexample,oncewehavecolouredthe twoendpointsofanarc(u,v),weknowthattheremustbeanarcfromf(u)tof(v)

inthetargettournament. Hence,therecannotbeanarcfromf(v)tof(u). Weend upwiththefollowingequivalentdefinition:

Definition1.12. Orientedgraphhomomorphismsandcolouring:

An homomorphismfromanorientedgraphG1=(V1,A1)toG2=(V2,A2), alsocalledG2-colouringofG1,isafunctionf:V1→ V2suchthat∀(u,v)∈ A1, (f(u),f(v))∈A2.

An-colouringorpropern-colouringofanorientedgraphG=(V,E)isafunction c:V→{c1,...,cn}suchthat

∀(u,v)∈A,c(u)=c(v)

∀(u,v)∈A,∀(x,y)∈A,c(u)=c(y)orc(v)=c(x)

Thenotionofn-colourability,n-chromaticity,chromaticnumberandcolourclasses aredefinedsimilarlyasintheundirectedcase.

OrientedcolouringshavebeenintroducedbyCourcelletostudymonadicsecond orderlogicingraphs[30].Itspurposewastocreatealabellingoftheverticesofthe graphthatwoulddeterminetheorientationofthearcs.

Example1.13. v0 v1 v4 v3 v2 v5

(a)A4-colouringofanorientedgraphG. (b)ThetargettournamentT. Figure1.8

LetustrytocolourthegraphGdepictedinFigure1.8a. Wecanpickarbitrarily thecolourofv1,sayblue. Thevertexv0canthenbeanycolourbutblue,letussay itisred. Atthispoint,weknowthatthearcbetweentheblueandredvertexin thetargettournamentgoesfromtheredvertextotheblue. Thus,theverticesv2, v3andv4cannotbered. Wepickthecolouryellowforv2. Thevertexv5cannotbe yellow(ithasayellowneighbour)norblue(ithasayellowin-neighbourwhichisnot compatible withtheorientationoftheblue-yellowarcinthetargettournament), butitcanbered. Thevertexv3hasthesamein-anout-neighbourasv2andcan thereforehavethesamecolour. However,thevertexv4canneitherberednorblue becauseitalreadyhasblueandredneighboursanditcannotbeyellowbecause ithasaredin-neighbour. Hence, weneedafourthcolourforv4. Wecanpick arbitrarilytheorientationofthearcbetweentheyellowandthegreenvertexinthe targettournament(depictedinFigure1.8b). Wehaveexhibiteda4-colouringofG andprovedthatG cannotbecoloured withthreecoloursorlessandistherefore 4-chromatic.

Notethatadirectedgraphthathasoppositearcsbetweentheverticesuandv cannotbecolouredbecauseitwouldrequirethetargettournamenttohaveanarc fromf(u)tof(v)andonefromf(v)tof(u), whichisimpossible. Finally,note thatG2-colouringistrivialifG2hasalooponavertexv.Indeed,thefunctionf that mapseveryvertextovisahomomorphism. Thisiswhydirectedcolouringis definedasahomomorphismfromanorientedgraphtoanirreflexivetournament.

Wenowpresentavariantofcolouringcallededge-colouring. Thisvariantis especiallyusefultostudythecomplexityofhomomorphismsproblems.

Definition1.14. Edge-colouring:

Ak-edge-colouring(sometimescalledproperk-edge-colouring)ofagraphG= (V,E)isafunctionc:E →{c1,...,cn}suchthatiftwoedgese1ande2are adjacent,thenc(e1)=c(e2). Thek-edge-colourability,thek-edge-chromaticityand theedge-chromaticnumberaredefinedsimilarly.

LetGbeagraph. ThelinegraphofGisthegraphH suchthatV(H)=E(G) andtwoverticesofH areadjacentinH ifandonlyiftheydenoteadjacentedgesin G. Onecannoticethatedge-colouringGcomesdowntocolouringitslinegraph. Example1.15.

Figure1.9illustratestheconnectionbetweencolouringandedge-colouring.

e1 e2 _e 3 e4 e5 e6 e9 e8 e7 (a)AgraphG. e1 e3 e2 e6 e5 e4 e9 e7 e8 (b)ThelinegraphofG.

Figure1.9: Anillustrationofaproper3-edge-colouringo fagraphGandtheasso-ciated3-colouringofitslinegraph.

Finally, wepresentthenotionofautomorphismwhichgreatlyhelpsstudythe symmetryofagraph.

Definition1.16. Isomorphisms,automorphismsandorbits:

An isomorphismfromagraphG1=(V1,E1)toG2=(V2,E2)isabijection f:V1→ V2suchthat∀(u,v)∈V12,{u,v}∈E1ifandonlyif{f(u),f(v)}∈E2. NotethattheinversefunctionofanisomorphismfromG1toG2isanisomorphism fromG2toG1. Twographsaresaidtobeisomorphicifandonlyifthereexistsan isomorphismfromonetotheother.

An automorphismisanisomorphismfromagraphG =(V,E)toitself. The automorphismsofagraphformagroupundercomposition. Twoverticesuandv areinthesameorbitifandonlyifthereexistsanautomorphismfofG suchthat f(u)=v. Onecaneasilyprovethatthisisanequivalencerelationandsinceorbits areequivalenceclasses,theydefineapartitionoftheverticesofagraph.Ifallthe verticesofthegraphareinthesameorbit,thegraphisvertex-transitive.

Wecandefineanalogouslyorbitofedgesbystatingthattwoedges{u,v}and {w,x}areinthesameorbitifthereexistsanautomorphismsuchthat{f(u),f(v)}= {w,x}.Ifalltheedgesofthegraphareinthesameorbit,thegraphisedge-transitive. Example1.17.

v6

v2 v3 v1

v5 v4

(a) AgraphG andarepresentationofits orbits.

w6

w2 w3 w1

w5 w4

(b)AgraphH isomorphictoG. Figure1.10

Thefunction fsuchthatf(v1) =w1,f(v2) =w5,f(v3) =w3,f(v4) =w6, f(v5)=w2andf(v6)=w4isanisomorphismfromthegraphGdepictedinFigure 1.10atothegraphH depictedinFigure1.10b. Thefunctionf1:V(G)→ V(G)that transposesv1withv3andv4withv6isanautomorphismofG.Soarethefunction f2thattransposesv1withv6,v2withv5andv3withv4andthefunctionf3=f1◦f2. Thoseautomorphismsprovethatv1,v3,v4andv6areinthesameorbitandthatv2 andv5areinthesameorbit.Sincedeg(v1)=deg(v2),noautomorphismcan map v1tov2andGhasexactlytwoorbits. Theorbitofedgesare{v1v2,v2v3,v4v5,v5v6}, {v1v6,v3v4}and{v2v5}.

ThegraphdepictedinFigure 1.9isvertex-transitivebutnotedge-transitive. Indeed,theedgese4,e5ande6belongtonotrianglewhilethesixothersdo. The graphS4(seeFigure1.5b)isedge-transitivebutnotvertex-transitive. Thegraphs K4andC4(seeFigures1.5aand1.5d)arebothvertex-andedge-transitive.

1.1.3 Specialvertexsets

Thissubsectionpresentsthenotionsofindependence,dominationandseparationin graphs.

Definition1.18. Cliques,independentsetsanddominatingsets:

LetG =(V,E)beagraph. AcliqueisasetofverticesS ⊂ V suchthat ∀(u,v)∈S2_{,{u,v}∈E. AnindependentsetisasubsetofverticesS⊂ V such} that∀(u,v)∈S2_{,{u,v}/∈E. AdominatingsetisasetofverticesS⊂V suchthat} ∀u∈V\S,∃v∈S,{u,v}∈E.

ThesizeofthebiggestcliqueinG,notedω(G),iscalledthecliquenumberofG. ThesizeofthebiggestindependentsetinG,notedα(G),iscalledtheindependence numberofG. ThesizeofthesmallestdominatingsetinG,notedγ(G),iscalled thedominationnumberofG.

Proposition1.19.

•Theemptyvertexsetisacliqueandanindependentsetineverygraph. More generally,anysubsetofaclique(resp. independentset)isaclique(resp. independentset)aswell. ThesetV(G)ofverticesofagraphG isalwaysa dominatingset. Anysupersetofadominatingsetisdominatingtoo.

•Givenagraph G andapropercolouringconG,thecolourc lassesareinde-pendentsetsbydefinition. Hence,χ(G)×α(G) V(G)sinceeveryvertex belongstoacolourclass.

•Inacolouringofagraph,alltheverticesofacliquemusthavedifferentcolour. Hence,ω(G) χ(G). Thisboundisnotalwaystight.

•Acliqueinagraphisanindependentsetinitscomplement. Hence,ω(G)= α(G). Example1.20. v0 v1 v2 v3 v4 v5 v6 v7 v8 v9 Figure1.11: AgraphG.

Theset{v2,v3,v4}isacliqueinthegraphGdepictedinFigure1.11.SinceGis 3-colourable(seethefigureforanexampleof3-colouring),weknowthatω(G)=3. Thecolourclassesareindependentsetandnotably{v0,v4,v6,v9 }isanindepen-dentsetofsize4. ThesameargumentappliedtoG indicatesthatsinceG canbe partitionedinto4cliques(circledingrey),α(G) =4(anindependentsethasat mostonevertexofeachclique).

Onecancheckthattheset {v0,v4,v8}isa minimumdominatingsetonG and γ(G)=3.

Thecycleof5verticesC5cannotbecolouredwithlessthan3coloursbutcontains nocliqueofsizemorethan2. Thisprovesthattheboundω(G) χ(G)isnottight. SinceC5=C5,wealsoobservethatC5cannotbepartitionedinlessthan3cliques butitsindependencenumberis2.

Finally, wepresentthenotionsofseparatingandidentifyingcodes, whichare studiedinChapter2.

Definition1.21.Separatingandidentifyingcodes:

LetG=(V,E)beagraphandCbeasetofverticesofG. Thesignatureofa vertexv∈V isthesetsign(v)=N[x]∩CandCissaidtobeaseparatingcodeof GifandonlyifalltheverticesofVhavepairwisedistinctsignatures. AsetCthat isbothseparatinganddominatingiscalledanidentifyingcode.Inthiscase,the signatureofalltheverticesarepairwisedistinctandnon-empty.