Abstract
The 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 $G$. 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 $f$. First this 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]. 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 that, conversely, every small hereditary class has bounded twin-width. 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. Thirdly we show that cubic expanders obtained by iterated random 2-lifts from $K_4$~[Bilu and Linial, Combinatorica '06] have bounded twin-width, too. We suggest a promising connection between the small conjecture and group theory. Finally we define a robust notion of sparse twin-width and discuss how it compares with other sparse classes.
Highlights
A trigraph is a graph with two disjoint edge sets: black edges and red edges
The graph induced by the red edges is called the red graph
We proved that many classes such as, bounded rankwidth graphs, proper minor-free classes, proper subclasses of permutation graphs, and posets with antichains of bounded size have bounded twin-width
Summary
We continue to develop the theory of twin-width, a novel graph and matrix invariant introduced in the first paper of the series [7]. If the conjecture is true, it gives a universal explanation for the single-exponential growth (up to isomorphism) of combinatorial classes: Translate the objects into graphs or matrices, a bound or lack thereof in the twin-width of the class decides the existence of such a bound in the growth. Another by-product of the contraction tree is that we can always contract in parallel a linear number of disjoint pairs of vertices. Do classes with polynomial expansion have bounded twin-width? We discuss (possible) containments and strict containments of established sparse classes with respect to bounded sparse twin-width
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