Sparse Graphs of Twin-Width 2 Have Bounded Tree-Width.

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.