Shallow Minors, Graph Products and Beyond Planar Graphs

TL;DR

This paper extends the graph product structure theorem to various beyond planar graph classes using shallow minors, leading to new bounds on their queue-number, chromatic numbers, and coloring properties.

Contribution

It introduces shallow minor-based product structure theorems for several beyond planar graphs, generalizing previous planar graph results.

Findings

Beyond planar graphs can be represented as shallow minors of strong products of planar graphs and small complete graphs.

These classes have bounded queue-number, nonrepetitive chromatic number, and polynomial p-centered chromatic numbers.

Some classes, like k-gap planar graphs, have exponential local treewidth and cannot be represented as subgraphs of certain product graphs.

Abstract

The planar graph product structure theorem of Dujmovi\'{c}, Joret, Micek, Morin, Ueckerdt, and Wood [J. ACM 2020] 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 colourings, centered colourings, and adjacency labelling 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 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 $p$-centred chromatic numbers, linear strong colouring numbers, and cubic weak colouring 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.