For each $d\leq3$, we construct a finite set $F_d$ of multigraphs such that for each graph $H$ of girth at least $5$ obtained from a multigraph $G$ by subdividing each edge at least two times, $H$ has twin-width at most $d$ if and only if $G$ has no minor in $F_d$. This answers a question of Berg\'{e}, Bonnet, and D\'{e}pr\'{e}s asking for the structure of graphs $G$ such that each long subdivision of $G$ has twin-width $4$. As a corollary, we show that the $7\times7$ grid has twin-width $4$, which answers a question of Schidler and Szeider.

翻译：对于每个$d\leq3$，我们构造一个有限多重图集合$F_d$，使得对于任意围长至少为$5$的图$H$（该图通过对多重图$G$的每条边进行至少两次细分得到），$H$的双宽度至多为$d$当且仅当$G$不包含$F_d$中的任何子式。这回答了Bergé、Bonnet和Déprés提出的关于“具有何种结构的图$G$，其任意充分细分后的图都具有双宽度$4$”的问题。作为推论，我们证明$7\times7$网格图的双宽度为$4$，从而回答了Schidler和Szeider提出的一个问题。