Dujmović, 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".

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 $G\subseteq H \boxtimes P \boxtimes K_{\max\{2g,3\}}$. 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 \boxtimes P\boxtimes K_\ell$, for some planar graph $H$ with treewidth $3$, where $\ell=\max\{2g\lfloor \frac{d}{2} \rfloor,d+3\lfloor\frac{d}{2}\rfloor-3\}$. 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 $H\boxtimes P\boxtimes K_{\max\{4g,7\}}$, for some planar graph $H$ with treewidth $3$.

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We show that there exist a constant \(c\) and a function \(f\) such that the \(k\)-power of a planar graph with maximum degree \(\Delta\) is isomorphic to a subgraph of \(H \boxtimes P \boxtimes K_{f(\Delta, k)}\) for some graph \(H\) with treewidth at most \(c\) and some path \(P\). This is the first product structure theorem for \(k\)-powers of planar graphs, where the treewidth of \(H\) does not depend on \(k\). We actually prove a stronger result, which implies an analogous product structure theorem for other graph classes, including \(k\)-planar graphs (of arbitrary degree). Our proof uses a new concept of blocking partitions which is of independent interest. An \(\ell\)-blocking partition of a graph \(G\) is a partition of the vertex set of \(G\) into connected subsets such that every path in \(G\) of length greater than \(\ell\) contains two vertices in one set of the partition. The key lemma in our proof states that there exists a positive integer \(\ell\) such that every planar graph of maximum degree \(\Delta\) has an \(\ell\)-blocking partition with parts of size bounded in terms of \(\Delta\).