Improved product structure for graphs on surfaces

Dujmovi\'c, Joret, Micek, Morin, Ueckerdt and Wood [J. ACM 2020] proved that for every graph $G$ with Euler genus $g$ there is a $H$ treewidth at most 4 path $P$ such $G\subseteq H \boxtimes P K_{\max\{2g,3\}}$. We improve this result by replacing "4" "3" planar. in fact prove more general terms of so-called framed graphs. This implies $(g,d)$-map contained $ P\boxtimes K_\ell$, some planar $3$, where $\ell=\max\{2g\lfloor \frac{d}{2} \rfloor,d+3\lfloor\frac{d}{2}\rfloor-3\}$. It also $(g,1)$-planar (that is, graphs can be drawn surface one crossing per edge) $H\boxtimes K_{\max\{4g,7\}}$, $3$.