Twin-width of subdivisions of multigraphs

Abstract: 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.