arXiv 2006.098772020-06-17cs.DMcs.DScs.LOmath.CO

双宽度II：小班

Twin-width II: small classes

Édouard Bonnet, Colin Geniet, Eun Jung Kim, Stéphan Thomassé, Rémi Watrigant

备注：: 37 pages, 9 figures

摘要：

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，对于该序列，入射到单个顶点的红色边的总最大数目至多为$d$，其中如果在$G$中两组标识的顶点不是齐次的，则红色边出现在两组标识的顶点之间。证明了如果一个图存在$d$ -压缩序列，那么对于某个函数$f$，它也存在一棵由$f(d)$ -压缩序列构成的线性树.首先，这允许表明，每一个有界双宽度类是小的，即，对于某个常数$c$，最多有标记为$[n]$的$n!c^n$图。这统一并扩展了有界树宽图[Beineke和Pippert，JCT '69]，置换图的适当子类[Marcus和Tardos，JCTA '04]和适当的无小类[Norine等人，JCTB '06]。第二个结果是一个$O(\log n)$ -邻接标号计划有界双宽度图，确认几种情况下的隐图猜想。然后，我们探讨“小猜想”，相反，每一个小的遗传类有界双宽度。受对数深度排序网络的启发，我们证明了$\log_{Θ(\log \log d)}n$ -$K_n$（$d$为常数时的一个小类）的细分在最多$d$处具有孪晶宽度。我们得到一个相当尖锐的逆与令人惊讶的直接证明：$K_n$的$\log_{d+1}n$ -细分有孪生宽度至少$d$。第二，具有有界堆栈或队列数的图（也是小类）具有有界孪生宽度。第三，我们证明了从$K_4$ ~[Bilu and Linial，Combinatorica '06]通过迭代随机2-提升得到的三次展开式也具有有界孪晶宽度。我们建议一个有前途的小猜想和群论之间的联系。最后，我们定义了一个强大的概念稀疏双宽度，并讨论它如何与其他稀疏类比较。

英文摘要：

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 $\log_{Θ(\log \log d)}n$-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.

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英文摘要：

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