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 graph H with treewidth at most 4 and a path P such that G⊆ H P K_max\2g,3\. We improve this result by replacing "4" by "3" and with H planar. We in fact prove a more general result in terms of so-called framed graphs. This implies that every $(g,d)$-map graph is contained in H P K_, for some planar graph H with treewidth $3$, where =max\2g d/2 ,d+3/2-3\. It also implies that every $(g,1)$-planar graph (that is, graphs that can be drawn in a surface of Euler genus g with at most one crossing per edge) is contained in H P K_max\4g,7\, for some planar graph H with treewidth $3$.