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{ "pk": 64848, "title": "Twin-width II: small classes", "subtitle": null, "abstract": "The recently introduced twin-width of a graph $G$ is the minimum integer $d$ such that $G$ has a $d$-contraction sequence, that is, a sequence of $\\left| V(G) \\right|-1$ iterated vertex identifications for which the overall maximum number of red edges incident to a single vertex is at most $d$, where a red edge appears between two sets of identified vertices if they are not homogeneous in $G$ (not fully adjacent nor fully non-adjacent). We show that if a graph admits a $d$-contraction sequence, then it also has a linear-arity tree of $f(d)$-contractions, for some function $f$. Informally if we accept to worsen the twin-width bound, we can choose the next contraction from a set of $\\Theta(\\left| V(G) \\right|)$ pairwise disjoint pairs of vertices. This has two main consequences. First it permits to show that every bounded twin-width class is small, i.e., has at most $n!c^n$ graphs labeled by $[n]$, for some constant $c$. This unifies and extends the same result for bounded treewidth graphs [Beineke and Pippert, JCT '69], proper subclasses of permutations graphs [Marcus and Tardos, JCTA '04], and proper minor-free classes [Norine et al., JCTB '06]. It implies in turn that bounded-degree graphs, interval graphs, and unit disk graphs have unbounded twin-width. The second consequence is an $O(\\log n)$-adjacency labeling scheme for bounded twin-width graphs, confirming several cases of the implicit graph conjecture. We then explore the small conjecture that, conversely, every small hereditary class has bounded twin-width. The conjecture passes many tests. Inspired by sorting networks of logarithmic depth, we show that $\\log_{\\Theta(\\log \\log d)}n$-subdivisions of $K_n$ (a small class when $d$ is constant) have twin-width at most $d$. We obtain a rather sharp converse with a surprisingly direct proof: the $\\log_{d+1}n$-subdivision of $K_n$ has twin-width at least $d$. Secondly graphs with bounded stack or queue number (also small classes) have bounded twin-width. These sparse classes are surprisingly rich since they contain certain (small) classes of expanders. Thirdly we show that cubic expanders obtained by iterated random 2-lifts from $K_4$ [Bilu and Linial, Combinatorica '06] also have bounded twin-width. These graphs are related to so-called separable permutations and also form a small class. We suggest a promising connection between the small conjecture and group theory. Finally we define a robust notion of sparse twin-width. We show that for a hereditary class $\\mathcal C$ of bounded twin-width the five following conditions are equivalent: every graph in $\\mathcal C$ (1) has no $K_{t,t}$ subgraph for some fixed $t$, (2) has an adjacency matrix without a $d$-by-$d$ division with a 1 entry in each of the $d^2$ cells for some fixed $d$, (3) has at most linearly many edges, (4) the subgraph closure of $\\mathcal C$ has bounded twin-width, and (5) $\\mathcal C$ has bounded expansion. We discuss how sparse classes with similar behavior with respect to clique subdivisions compare to bounded sparse twin-width.\n \nMathematics Subject Classifications: 68R10, 05C30, 05C48\n \nKeywords: Twin-width, small classes, expanders, clique subdivisions, sparsity", "language": "en", "license": { "name": "Creative Commons Attribution 4.0", "short_name": "CC BY 4.0", "text": "Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.\n\nNo additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.", "url": "https://creativecommons.org/licenses/by/4.0" }, "keywords": [ { "word": "Twin-width" }, { "word": "small classes" }, { "word": "expanders" }, { "word": "clique subdivisions" }, { "word": "sparsity" } ], "section": "Research Articles", "is_remote": true, "remote_url": "https://escholarship.org/uc/item/9cs265b9", "frozenauthors": [ { "first_name": "Édouard", "middle_name": "", "last_name": "Bonnet", "name_suffix": "", "institution": "Univ Lyon, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR5668, France", "department": "" }, { "first_name": "Colin", "middle_name": "", "last_name": "Geniet", "name_suffix": "", "institution": "Univ Lyon, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR5668, France", "department": "" }, { "first_name": "Eun", "middle_name": "Jung", "last_name": "Kim", "name_suffix": "", "institution": "Université Paris-Dauphine, PSL University, CNRS UMR7243, LAMSADE, Paris, France", "department": "" }, { "first_name": "Stéphan", "middle_name": "", "last_name": "Thomassé", "name_suffix": "", "institution": "Univ Lyon, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR5668, France", "department": "" }, { "first_name": "Rémi", "middle_name": "", "last_name": "Watrigant", "name_suffix": "", "institution": "Univ Lyon, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR5668, France", "department": "" } ], "date_submitted": "2022-06-25T19:34:43Z", "date_accepted": "2022-06-25T19:34:43Z", "date_published": "2022-06-30T07:00:00Z", "render_galley": null, "galleys": [ { "label": "", "type": "pdf", "path": "https://journalpub.escholarship.org/combinatorial_theory/article/64848/galley/49658/download/" } ] }