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 "small conjecture" 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.

中文翻译：

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双宽II：小班

图$G$的孪生宽度是最小整数$d$，使得$G$具有$d$-收缩序列，即$|V(G)|-1$迭代顶点标识的序列对于其中入射到单个顶点的红色边的总体最大数量最多为 $d$，其中如果在 $G$ 中两组已识别的顶点之间不是同质的，则红色边会出现在它们之间。我们证明，如果一个图承认一个 $d$-收缩序列，那么它也有一个 $f(d)$-收缩的线性树，对于某些函数 $f$。首先，这允许表明每个有界双宽度类都很小，即对于某些常量 $c$，最多有 $n!c^n$ 个标记为 $[n]$ 的图。这统一并扩展了有界树宽图的相同结果 [Beineke 和 Pippert，JCT '69]，置换图的正确子类 [Marcus 和 Tardos，JCTA '04]，和适当的无未成年人课程 [Norine et al., JCTB '06]。第二个结果是有界双宽度图的 $O(\log n)$-邻接标记方案，证实了隐式图猜想的几种情况。然后我们探索“小猜想”，相反，每个小的遗传类都有有界孪生宽度。受对数深度排序网络的启发，我们证明 $\log_{\Theta(\log \log d)}n$-$K_n$ 的细分（$d$ 恒定时的一个小类）最多具有孪生宽度$d$。我们通过令人惊讶的直接证明获得了相当尖锐的对比：$K_n$ 的 $\log_{d+1}n$-细分具有至少 $d$ 的双宽度。其次，具有有界堆栈或队列数（也是小类）的图具有有界孪生宽度。第三，我们展示了通过从 $K_4$~[Bilu 和 Linial，Combinatorica '06] 也有界孪生宽度。我们建议在小猜想和群论之间建立有希望的联系。最后，我们定义了一个稳健的稀疏孪生宽度概念，并讨论了它与其他稀疏类的比较。