• Corpus ID: 244527650

Shallow Minors, Graph Products and Beyond Planar Graphs

@inproceedings{Hickingbotham2021ShallowMG,
  title={Shallow Minors, Graph Products and Beyond Planar Graphs},
  author={Robert Hickingbotham and David R. Wood},
  year={2021}
}
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… 
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References

SHOWING 1-10 OF 88 REFERENCES
Graph product structure for non-minor-closed classes
TLDR
It is proved that every planar graph is a subgraph of the strong product of a graph of bounded treewidth and a path and implies, amongst other results, that $k$-planar graphs have non-repetitive chromatic number upper-bounded by a function of $k$.
Planar Graphs have Bounded Queue-Number
TLDR
It is proved that every proper minor-closed class of graphs has bounded queue-number, and it is shown that every planar graph is a subgraph of the strong product of a path with some graph of bounded treewidth.
Diameter and Treewidth in Minor-Closed Graph Families
TLDR
It is shown that treewidth is bounded by a function of the diameter in a minor-closed family, if and only if some apex graph does not belong to the family, and the O(D) bound above can be extended to bounded-genus graphs.
Structural Properties of Graph Products
Dujmović, Joret, Micek, Morin, Ueckerdt, and Wood [J. ACM 2020] established that every planar graph is a subgraph of the strong product of a graph with bounded treewidth and a path. Motivated by this
Notes on Graph Product Structure Theory
TLDR
This paper characterises when a graph class defined by a cartesian or strong product has bounded or polynomial expansion, and explores graph product structure theorems for various geometrically defined graph classes.
Book Embeddings of Graph Products
TLDR
It is shown that the stack number is bounded for the strong product of a path and (i) agraph of bounded pathwidth or (ii) a bipartite graph of bounded treewidth and bounded degree.
Layered separators in minor-closed graph classes with applications
Defective Colouring of Graphs Excluding A Subgraph or Minor
TLDR
A common generalisation of these theorems with a weaker assumption about excluded subgraphs is proved, which leads to new defective colouring results for several graph classes, including graphs with linear crossing number, graphs with given thickness, and graphs excluding a complete bipartite graph as a topological minor.
Graph Minors. II. Algorithmic Aspects of Tree-Width
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