The structure of k-planar graphs
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Dujmović et al. (FOCS 2019) recently proved that every planar graph is a subgraph of the strong product of a graph of bounded treewidth and a path. This tool has been used to solve longstanding problems on queue layouts, non-repetitive colouring, p-centered colouring, and implicit graph encoding. We generalise this result for k-planar graphs, where a graph is k-planar if it has a drawing in the plane in which each edge is involved in at most k crossings. In particular, we prove that every k-planar graph is a subgraph of the strong product of a graph of treewidth O(k^5) and a path. This is the first result of this type for a non-minor-closed class of graphs. It implies, amongst other results, that k-planar graphs have non-repetitive chromatic number upper-bounded by a function of k. All these results generalise for drawings of graphs on arbitrary surfaces. In fact, we work in a much more general setting based on so-called shortcut systems that are of independent interest.
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