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L’étude développe la notion de twin-width via des séquences de contraction et de partition, puis établit des propriétés fondamentales et des caractérisations à partir de matrices d’adjacence. Le texte relie ces paramètres à d’autres largeurs de graphes, examine la croissance des classes et des schémas de labellisation, et fournit des applications algorithmiques, y compris en logique du premier ordre et pour le calcul de plus courts chemins.",{"@graph":58,"@context":115},[59,76,98],{"@type":60,"itemListElement":61},"BreadcrumbList",[62,67,70,73],{"item":63,"name":64,"@type":65,"position":66},"https://docshare.wps.com","Home","ListItem",1,{"item":68,"name":9,"@type":65,"position":69},"https://docshare.wps.com/fr/document/",2,{"item":71,"name":30,"@type":65,"position":72},"https://docshare.wps.com/fr/document/recherche-rapport/",3,{"item":74,"name":54,"@type":65,"position":75},"https://docshare.wps.com/fr/document/twin-width-and-contraction-sequences-habilitation-a-diriger-des-recherches-report/135386/",4,{"url":74,"name":54,"@type":77,"image":78,"author":83,"headline":54,"publisher":86,"fileFormat":89,"inLanguage":52,"description":56,"dateModified":90,"datePublished":91,"encodingFormat":89,"isAccessibleForFree":92,"interactionStatistic":93},"DigitalDocument",{"url":79,"@type":80,"width":81,"height":82},"https://docshare.wps.com/thumbnails/twin-width-and-contraction-sequences-habilitation-a-diriger-des-recherches-report/135386.png","ImageObject",300,407,{"name":84,"@type":85},"Evangeline","Person",{"url":63,"name":87,"@type":88},"DocShare","Organization","application/pdf","2026-10-01","2026-08-21",true,{"@type":94,"interactionType":95,"userInteractionCount":97},"InteractionCounter",{"@type":96},"ViewAction",9,{"@type":99,"mainEntity":100},"FAQPage",[101,107,111],{"name":102,"@type":103,"acceptedAnswer":104},"Qu’est-ce que la twin-width et comment est-elle définie dans le rapport ?","Question",{"text":105,"@type":106},"Le rapport présente la twin-width à travers des séquences de contraction (et de partition), puis en étudie le lien avec des structures et décompositions adaptées aux graphes.","Answer",{"name":108,"@type":103,"acceptedAnswer":109},"Quelles sont les méthodes de caractérisation principales utilisées ?",{"text":110,"@type":106},"Les caractérisations passent notamment par des matrices d’adjacence, en combinant des résultats sur des mineurs (dont des grilles) et des théorèmes comme celui de Marcus–Tardos, ainsi que des variantes impliquant des mineurs mixtes.",{"name":112,"@type":103,"acceptedAnswer":113},"Le rapport fournit-il des applications concrètes, au-delà des propriétés théoriques ?",{"text":114,"@type":106},"Oui : il couvre des applications algorithmiques (algorithmes paramétrés, approximation, plus courts chemins) et des résultats en logique du premier ordre, notamment la préservation de la twin-width bornée sous des transductions.","https://schema.org",{"og:url":74,"og:type":117,"og:title":54,"og:site_name":87,"og:description":56},"article",{"robots":119,"canonical":74},"index,follow",{"doc_id":121,"site_id":51},135386,1787310513,{"code":4,"msg":5,"data":124},{"doc_id":121,"user_id":125,"nickname":84,"user_avatar":126,"doc_module":4,"category_id":29,"category_name":30,"doc_title":54,"doc_description":56,"doc_content":127,"file_id":128,"file_url":129,"file_type":130,"file_size":131,"view_count":97,"is_deleted":4,"is_public":66,"is_downloadable":66,"audit_status":66,"page_count":132,"language":133,"language_code":52,"site_id":51,"html_lang":52,"table_of_contents":134,"faqs":135,"seo_title":136,"seo_description":56,"update_tm":122,"read_time":137},13056703019662,"https://ap-avatar.wpscdn.com/avatar/be000253a8e92610077?_k=1778726343310543188","HABILITATION À DIRIGER DES RECHERCHES  \nde l'École Normale Supérieure de Lyon  \nPrésentée le 19 avril 2024  \npar Édouard Bonnet  \nTwin-Width and Contraction Sequences  \nRapporteurs :  \nThomas Colcombet   DR CNRS, Université Paris Cité David Eppstein   Professeur, Université de Californie, Irvine Frédéric Havet   DR CNRS, Université Nice–Sophia-Antipolis  \nExaminateur·rices :  \nAngela Bonifati .................... Professeur, Université Lyon 1  \nClaire Mathieu ................. DR CNRS, Université Paris Cité  \nBruno Salvy ............................ DR INRIA, ENS de Lyon  \nTable of Contents  \nAcknowledgments .................................. 1  \nChapter 1. Introduction ........................... 3  \n1. The origin of twin-width ........................ 3  \n2. The Guillemot–Marx permutation width on graphs ..... 4  \n3. Motivations ................................. 8  \n4. Overview of the thesis .......................... 9  \nChapter 2. Background and Twin-Width ............... 14  \n1. Sets, partitions, functions ........................ 14  \n2. Graphs ..................................... 14  \n3. First-order and monadic second-order logic ........... 19  \n4. Model checking ............................... 20  \n5. Interpretations, transductions, and dependence ........ 21  \n6. Contraction or partition sequences, and twin-width ..... 23  \nChapter 3. First Properties ........................ 26  \n1. Operations preserving bounded twin-width ........... 26  \n2. Split sequences and proper colorings ................ 27  \n3. Twin-decompositions ........................... 30  \n4. Technical lemmas ............................. 31  \nChapter 4. Characterization via Adjacency Matrices ....... 33  \n1. Grid minors and the Marcus–Tardos theorem ......... 33  \n2. Mixed minors and characterization via mixed number ... 36  \n3. Applications of the characterization ................ 38  \n4. Versatile twin-width and balanced sequences .......... 40  \n5. Oriented twin-width ........................... 41  \nChapter 5. Which Classes Have Bounded Twin-Width? .... 43  \n1. Classical graph widths .......................... 43  \n2. Subdivisions, grids, and expanders ................. 45  \n3. Intersection graphs ............................ 48  \n4. Sparse classes ................................ 51  \nii TABLE OF CONTENTS  \nChapter 6. Other Parameters based on Contraction Sequences 54  \n1. Characterization of classical width parameters ......... 55  \n2. New parameters between clique-width and twin-width ... 57  \n3. Separation of the reduced parameters ............... 59  \nChapter 7. Algorithmic Applications .................. 61  \n1. Parameterized algorithms ........................ 62  \n2. Approximation algorithms ....................... 69  \n3. Shortest paths ................................ 73  \nChapter 8. First-Order Logic and Twin-Width ........... 77  \n1. First-order transductions preserve bounded twin-width ... 78  \n2. Permutations strike back ........................ 80  \n3. Delineation .................................. 82  \nChapter 9. Growth of Classes and Labeling Schemes ...... 86  \n1. Small and tiny classes .......................... 87  \n2. Labeling schemes and universal graphs .............. 88  \n3. Complex tiny classes ........................... 90  \nChapter 10. Ordered Graphs and Matrices .............. 93  \n1. Rank minors, rank number, and rich divisions ......... 93  \n2. Equivalences ................................. 96  \n3. Unconditional algorithms ........................ 99  \nBibliography ..................................... 104  \nAcknowledgments  \nI am extremely grateful to Thomas Colcombet, David Eppstein, and Frédéric Havet for their review of the current manuscript, and to Angela Bonifati, Claire Mathieu, and Bruno Salvy for completing my habilitation jury. I crucially benefited from the help and guidance of Bora and Russ (and his infinite patience) to make this habilitation happen. A warm thank to them.  \nIn the decade following my P","cbCairaGUNXbVVHq","https://ap.wps.com/l/cbCairaGUNXbVVHq","pdf",2278254,113,"French","# Acknowledgments\n# Chapter 1. Introduction\n## The origin of twin-width\n## The Guillemot–Marx permutation width on graphs\n## Motivations\n## Overview of the thesis\n# Chapter 2. Background and Twin-Width\n## Sets, partitions, functions\n## Graphs\n## First-order and monadic second-order logic\n## Model checking\n## Interpretations, transductions, and dependence\n## Contraction or partition sequences, and twin-width\n# Chapter 3. First Properties\n## Operations preserving bounded twin-width\n## Split sequences and proper colorings\n## Twin-decompositions\n## Technical lemmas\n# Chapter 4. Characterization via Adjacency Matrices\n## Grid minors and the Marcus–Tardos theorem\n## Mixed minors and characterization via mixed number\n## Applications of the characterization\n## Versatile twin-width and balanced sequences\n## Oriented twin-width\n# Chapter 5. Which Classes Have Bounded Twin-Width?\n## Classical graph widths\n## Subdivisions, grids, and expanders\n## Intersection graphs\n## Sparse classes\n# Chapter 6. Other Parameters based on Contraction Sequences\n## Characterization of classical width parameters\n## New parameters between clique-width and twin-width\n## Separation of the reduced parameters\n# Chapter 7. Algorithmic Applications\n## Parameterized algorithms\n## Approximation algorithms\n## Shortest paths\n# Chapter 8. First-Order Logic and Twin-Width\n## First-order transductions preserve bounded twin-width\n## Permutations strike back\n## Delineation\n# Chapter 9. Growth of Classes and Labeling Schemes\n## Small and tiny classes\n## Labeling schemes and universal graphs\n## Complex tiny classes\n# Chapter 10. Ordered Graphs and Matrices\n## Rank minors, rank number, and rich divisions\n## Equivalences\n## Unconditional algorithms","[{\"question\":\"Qu’est-ce que la twin-width et comment est-elle définie dans le rapport ?\",\"answer\":\"Le rapport présente la twin-width à travers des séquences de contraction (et de partition), puis en étudie le lien avec des structures et décompositions adaptées aux graphes.\"},{\"question\":\"Quelles sont les méthodes de caractérisation principales utilisées ?\",\"answer\":\"Les caractérisations passent notamment par des matrices d’adjacence, en combinant des résultats sur des mineurs (dont des grilles) et des théorèmes comme celui de Marcus–Tardos, ainsi que des variantes impliquant des mineurs mixtes.\"},{\"question\":\"Le rapport fournit-il des applications concrètes, au-delà des propriétés théoriques ?\",\"answer\":\"Oui : il couvre des applications algorithmiques (algorithmes paramétrés, approximation, plus courts chemins) et des résultats en logique du premier ordre, notamment la préservation de la twin-width bornée sous des transductions.\"}]","Twin-Width and Contraction Sequences - Rapport d’habilitation à diriger des recherches | PDF",174]