Shallow Minors, Graph Products, and Beyond-Planar Graphs

The planar graph product structure theorem of Dujmovi\'c et al. [J. ACM, 67 (2020), 22] states that every planar graph is a subgraph of the strong product of a graph with bounded treewidth and a path. This result has been the key tool to resolve important open problems regarding queue layouts, nonrepetitive colorings, centered colorings, and adjacency labeling schemes. In this paper, we extend this line of research by utilizing shallow minors to prove analogous product structure theorems for several beyond-planar graph classes. The key observation that drives our work is that many beyond-planar graphs can be described as a shallow minor of the strong product of a planar graph with a small complete graph. In particular, we show that powers of bounded degree planar graphs, k-planar, (k, p)-cluster planar, fan-planar, and k-fan-bundle planar graphs have such a shallow-minor structure. Using a combination of old and new results, we deduce that these classes have bounded queue-number, bounded nonrepetitive chromatic number, polynomial p-centered chromatic numbers, linear strong coloring numbers, and cubic weak coloring numbers. In addition, we show that k-gap planar graphs have at least exponential local treewidth and, as a consequence, cannot be described as a subgraph of the strong product of a graph with bounded treewidth and a path.