Twin-width II: small classes

The recently introduced twin-width of a graph \(G\) is the minimum integer \(d\) such that has \(d\)-contraction sequence, is, sequence \(\left| V(G) \right|-1\) iterated vertex identifications for which overall maximum number red edges incident to single at most \(d\), where edge appears between two sets identified vertices if they are not homogeneous in (not fully adjacent nor non-adjacent). We show admits then it also linear-arity tree \(f(d)\)-contractions, some function \(f\). Informally we accept worsen bound, can choose next contraction from set \(\Theta(\left| \right|)\) pairwise disjoint pairs vertices. This main consequences. First permits every bounded class small, i.e., \(n!c^n\) graphs labeled by \([n]\), constant \(c\). unifies and extends same result treewidth [Beineke Pippert, JCT '69], proper subclasses permutations [Marcus Tardos, JCTA '04], minor-free classes [Norine et al., JCTB '06]. It implies turn bounded-degree graphs, interval unit disk have unbounded twin-width. second consequence an \(O(\log n)\)-adjacency labeling scheme confirming several cases implicit conjecture. explore small conjecture that, conversely, hereditary passes many tests. Inspired sorting networks logarithmic depth, \(\log_{\Theta(\log \log d)}n\)-subdivisions \(K_n\) (a when constant) \(d\). obtain rather sharp converse with surprisingly direct proof: \(\log_{d+1}n\)-subdivision least Secondly stack or queue (also classes) These sparse rich since contain certain (small) expanders. Thirdly cubic expanders obtained random 2-lifts \(K_4\) [Bilu Linial, Combinatorica '06] related so-called separable form class. suggest promising connection group theory. Finally define robust notion \(\mathcal C\) five following conditions equivalent: (1) no \(K_{t,t}\) subgraph fixed \(t\), (2) adjacency matrix without \(d\)-by-\(d\) division 1 entry each \(d^2\) cells (3) linearly edges, (4) closure twin-width, (5) expansion. discuss how similar behavior respect clique subdivisions compare twin-width.Mathematics Subject Classifications: 68R10, 05C30, 05C48Keywords: Twin-width, classes, expanders, subdivisions, sparsity