page 18 : genus $k$ upper bounds twin-width by $\mathcal O(k)$ – The twin-width of every graph $G$ of Euler genus $g \ge 1$ is at most … $18 \sqrt{47g}+O(1)$.
page 2 : twin-width – … we consider a sequence of graphs $G_n,G_{n-1},\dots,G_2,G_1$, where $G_n$ is the original graph $G$, $G_1$ is the one-vertex graph, $G_i$ has $i$ vertices, and $G_{i-1}$ is obtained from $G_i$ by performing a single contraction of two (non-necessarily adjacent) vertices. For every vertex $u \in V(G_i)$, let us denote by $u(G)$ the vertices of $G$ which have been contracted to $u$ along the sequence $G_n,\dots,G_i$. A pair of disjoint sets of vertices is \emph{homogeneous} if, between these sets, there are either all possible edges or no edge at all. The red edges … consist of all pairs $uv$ of vertices of $G_i$ such that $u(G)$ and $v(G)$ are not homogeneous in $G$. If the red degree of every $G_i$ is at most $d$, then $G_n,G_{n-1},\dots,G_2,G_1$ is called a \emph{sequence of $d$-contractions}, or \emph{$d$-sequence}. The twin-width of $G$ is the minimum $d$ for which there exists a sequence of $d$-contractions.
page 15 : grid upper bounds twin-width by a constant – Theorem 4.3. For every positive integers $d$ and $n$, the $d$-dimensional $n$-grid has twin-width at most $3d$.
