Twin-width II: small classes

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 |V (G)| − 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 twinwidth bound, we can choose the next contraction from a set of Θ(|V (G)|) 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, con- * This work was supported by the grants from French National Agency under PRC program (project Digraphs, ANR-19-CE48-0013-01), under JCJC program (project ASSK, ANR-18-CE40-0025-01), and by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program "Investissements d'Avenir" (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR).

From χ- to χ-bounded classes

“…If uv E(H 0 ) then c(u) c(v). Thus we have a k 2 -coloring of H, and χ(H) ≤ χ 2 (G) 2 . Now assume p ≥ 3.…”

“…As a class with bounded twin-width and bounded bω has bounded expansion [2], classes with low twin-width covers and bounded bω have low bounded expansion covers hence have bounded expansion [24]. Very recently, Davies [8] announced that proper vertex-minor-closed classes are χ-bounded.…”

Twin-width VI: the lens of contraction sequences

“…We also define a notion of spanning twin-width, 4 intermediate between bounded tree-width and bounded twin-width, which exactly captures classes excluding a minor, among monotone classes. So far we explored what happens when restricting the notion of bounded twin-width.…”

“…Of course for this notion to be new, "simple" should not imply bounded twin-width. Bounded-degree and bounded-expansion are reasonably "tractable" classes with unbounded twinwidth [4]. We say that a class C is collapsible to a class D if graphs of C admit partial O(1)-sequences to (tri)graphs in D. We showcase the flexibility of the FO model-checking algorithm in [7]: Collapsible classes to bounded degree and collapsible classes to bounded expansion admit respectively a fixed-parameter tractable FO and ∃FO modelchecking algorithm, provided a corresponding partial O(1)-sequence is given.…”

“…See the work of Eppstein[14] building up on Baker's technique[2] and leading to low tree-width decompositions, and[5] for the approach with contraction sequences 4. The exact definition of spanning twin-width is somewhat technical and deferred to Section 6.…”