The recently introduced twin-width of a graph is the minimum integer such that has a -contraction sequence, that is, a sequence of iterated vertex identifications for which the overall maximum number of red edges incident to a single vertex is at most , where a red edge appears between two sets of identified vertices if they are not homogeneous in (not fully adjacent nor fully non-adjacent). We show that if a graph admits a -contraction sequence, then it also has a linear-arity tree of -contractions, for some function . Informally if we accept to worsen the twin-width bound, we can choose the next contraction from a set of 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 graphs labeled by , for some constant . 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 -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 -subdivisions of (a small class when is constant) have twin-width at most . We obtain a rather sharp converse with a surprisingly direct proof: the -subdivision of has twin-width at least . 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 [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 of bounded twin-width the five following conditions are equivalent: every graph in (1) has no subgraph for some fixed , (2) has an adjacency matrix without a -by- division with a 1 entry in each of the cells for some fixed , (3) has at most linearly many edges, (4) the subgraph closure of has bounded twin-width, and (5) has bounded expansion. We discuss how sparse classes with similar behavior with respect to clique subdivisions compare to bounded sparse twin-width.
Mathematics Subject Classifications: 68R10, 05C30, 05C48
Keywords: Twin-width, small classes, expanders, clique subdivisions, sparsity