Structural Properties of Graph Products

Hickingbotham, Robert; Wood, David R.

doi:10.48550/arxiv.2110.00721

Cited by 1 publication

(1 citation statement)

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“…This section explores various properties of shallow minors and graph products. For further structural results concerning graph products, see the papers [31,47] as well as the handbook [42].…”

Section: Shallow Minors and Graph Productsmentioning

confidence: 99%

Shallow Minors, Graph Products and Beyond Planar Graphs

Hickingbotham¹,

Wood²

2021

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The planar graph product structure theorem of Dujmović, 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 power of planar graphs, k-planar, (k, p)-cluster planar, k-semi-fan-planar graphs and k-fan-bundle planar graphs can be described in this manner. Using a combination of old and new results, we deduce that these classes have bounded queuenumber, 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 super-linear 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.

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Section: Shallow Minors and Graph Productsmentioning

confidence: 99%