Twin-width is a structural width parameter introduced by Bonnet, Kim, Thomass\'e and Watrigant [FOCS 2020]. Very briefly, its essence is a gradual reduction (a contraction sequence) of the given graph down to a single vertex while maintaining limited difference of neighbourhoods of the vertices, and it can be seen as widely generalizing several other traditional structural parameters. Having such a sequence at hand allows to solve many otherwise hard problems efficiently. Our paper focuses on a comparison of twin-width to the more traditional tree-width on sparse graphs. Namely, we prove that if a graph $G$ of twin-width at most $2$ contains no $K_{t,t}$ subgraph for some integer $t$, then the tree-width of $G$ is bounded by a polynomial function of $t$. As a consequence, for any sparse graph class $\mathcal{C}$ we obtain a polynomial time algorithm which for any input graph $G \in \mathcal{C}$ either outputs a contraction sequence of width at most $c$ (where $c$ depends only on $\mathcal{C}$), or correctly outputs that $G$ has twin-width more than $2$. On the other hand, we present an easy example of a graph class of twin-width $3$ with unbounded tree-width, showing that our result cannot be extended to higher values of twin-width.

翻译：孪宽（twin-width）是由Bonnet、Kim、Thomassé和Watrigant [FOCS 2020]引入的结构宽度参数。简而言之，其本质是通过逐步缩减（收缩序列）将给定图简化为单个顶点，同时保持顶点邻域差异的有限性，可视为多种传统结构参数的广泛推广。拥有此类序列能高效解决许多原本困难的问题。本文聚焦于稀疏图上孪宽与传统树宽的对比。具体而言，我们证明：若孪宽至多为2的图$G$不包含某整数$t$对应的$K_{t,t}$子图，则$G$的树宽受限于$t$的多项式函数。由此，对于任意稀疏图类$\mathcal{C}$，我们提出多项式时间算法：对任意输入图$G \in \mathcal{C}$，该算法要么输出宽度至多为$c$（$c$仅取决于$\mathcal{C}$）的收缩序列，要么正确判定$G$的孪宽大于2。另一方面，我们给出一个孪宽为3但树宽无界的图类简单实例，表明该结论无法推广至更高孪宽值。