Bounding twin-width for bounded-treewidth graphs, planar graphs, and bipartite graphs

Twin-width is a newly introduced graph width parameter that aims at generalizing a wide range of “nicely structured” graph classes. In this work, we focus on obtaining good bounds on twin-width $\text{tww}(G)$ for graphs $G$ from a number of classic graph classes. We prove the following:

• $\text{tww}(G)\le 3\cdot 2^{\text{tw}(G)-1}$, where $\text{tw}(G)$ is the treewidth of $G$,
• $\text{tww}(G)\le \max (4\text{bw}(G),\frac{9}{2}\text{bw}(G)-3)$ for a planar graph $G$ with $\text{bw}(G)\ge 2$, where $\text{bw}(G)$ is the branchwidth of $G$,
• $\text{tww}(G)\le 183$ for a planar graph $G$,
• the twin-width of a universal bipartite graph $(X,2^X,E)$ with $\mathrm{\mid }X\mathrm{\mid }=n$ is $n - \log_2(n) + $\mathcal{O}$(1)$ . An important idea behind the bounds for planar graphs is to use an embedding of the graph and sphere-cut decompositions to obtain good bounds on neighbourhood complexity.