Graph product structure for non-minor-closed classes

Dujmović et al. [J. ACM '20] proved that every planar graph is isomorphic to a subgraph of the strong product bounded treewidth and path. Analogous results were obtained for graphs Euler genus or apex-minor-free graphs. These tools have been used solve longstanding problems on queue layouts, non-repetitive colouring, p-centered adjacency labelling. This paper proves analogous structure theorems various non-minor-closed classes. One noteable example k-planar (those with drawing in plane which each edge involved at most k crossings). We prove O(k5) first result this type class It implies, amongst other results, chromatic number upper-bounded by function k. All these generalise drawings arbitrary surfaces. In fact, we work more general setting based so-called shortcut systems, are independent interest. leads certain types map graphs, string powers, nearest neighbour