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.
Mathematics Subject Classifications: 68R10, 05C30, 05C48
Keywords: Twin-width, small classes, expanders, clique subdivisions, sparsity
