Graph product structure for non-minor-closed classes

TL;DR

This paper extends graph product structure theorems to non-minor-closed classes like k-planar graphs, showing they can be represented as subgraphs of products involving graphs with bounded treewidth, with implications for coloring and other properties.

Contribution

It introduces the first product structure theorem for non-minor-closed classes, specifically k-planar graphs, and generalizes results to various graph classes using shortcut systems.

Findings

k-planar graphs are subgraphs of products of bounded treewidth graphs and paths

k-planar graphs have non-repetitive chromatic number bounded by a function of k

results extend to graphs on arbitrary surfaces and other graph classes

Abstract

Dujmovi\'c et al. [\emph{J.~ACM}~'20] recently proved that every planar graph is isomorphic to a subgraph of the strong product of a bounded treewidth graph and a path. Analogous results were obtained for graphs of bounded Euler genus or apex-minor-free graphs. These tools have been used to solve longstanding problems on queue layouts, non-repetitive colouring, $p$-centered colouring, and adjacency labelling. This paper proves analogous product structure theorems for various non-minor-closed classes. One noteable example is $k$-planar graphs (those with a drawing in the plane in which each edge is involved in at most $k$ crossings). We prove that every $k$-planar graph is isomorphic to 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 more general setting based on so-called shortcut systems, which are of independent interest. This leads to analogous results for certain types of map graphs, string graphs, graph powers, and nearest neighbour graphs.