Graph Product Structure for $h$-Framed Graphs

Graph product structure theory expresses certain graphs as subgraphs of the strong product of much simpler graphs. In particular, an elegant formulation for the corresponding structural theorems involves the strong product of a path and of a bounded treewidth graph, and allows to lift combinatorial results for bounded treewidth graphs to graph classes for which the product structure holds, such as to planar graphs [Dujmović et al., J. ACM, 67(4), 22:1-38, 2020]. In this paper, we join the search for extensions of this powerful tool beyond planarity by considering the $h$-framed graphs, a graph class that includes $1$-planar, optimal $2$-planar, and $k$-map graphs (for appropriate values of $h$). We establish a graph product structure theorem for $h$-framed graphs stating that the graphs in this class are subgraphs of the strong product of a path, of a planar graph of treewidth at most $3$, and of a clique of size $3\lfloor \frac{h}{2}\rfloor + \lfloor \frac{h}{3}\rfloor -1$. This allows us to improve over the previous structural theorems for $1$-planar and $k$-map graphs. Our results lead to significant progress over the previous bounds on the queue number, non-repetitive chromatic number, and $p$-centered chromatic number of these graph classes, e.g., we lower the currently best upper bound on the queue number of $1$-planar graphs and $k$-map graphs from $115$ to $82$ and from $\lfloor \frac{33}{2}(k+3\lfloor \frac{k}{2}\rfloor -3)\rfloor$ to $\lfloor \frac{33}{2}(3\lfloor \frac{k}{2}\rfloor + \lfloor \frac{k}{3}\rfloor -1)\rfloor$, respectively. We also employ the product structure machinery to improve the current upper bounds on the twin-width of $1$-planar graphs from $O(1)$ to $72$. All our structural results are constructive and yield efficient algorithms to obtain the corresponding decompositions.