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 $d\leq3$，我们构造了一个有限的多重图集合 $F_d$，使得对于任意由多重图 $G$ 的每条边至少细分两次得到的围长至少为5的图 $H$，$H$ 的孪生宽度至多为 $d$ 当且仅当 $G$ 不包含 $F_d$ 中的任意一个子式。这回答了 Bergé、Bonnet 和 Déprés 提出的问题：求所有满足每条长细分图均有孪生宽度4的图 $G$ 的结构。作为推论，我们证明了 $7\times7$ 网格图的孪生宽度为4，这回答了 Schidler 和 Szeider 的问题。