Improved product structure for graphs on surfaces

Marc, Distel,; Hickingbotham, Robert; Huynh, Tony; Wood, David R.

doi:10.48550/arxiv.2112.10025

Cited by 3 publications

(3 citation statements)

References 9 publications

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“…They have been the key tool to resolve several open problems regarding queue layouts [15], nonrepetitive colourings [13], centered colourings [11], clustered colourings [14], adjacency labellings [5,12,18], vertex rankings [7], twin-width [6], odd colourings [16], and infinite graphs [23]. Similar product structure theorems are known for other classes including graphs with bounded Euler genus [10,15], apex-minor-free graphs [15], (g, d)-map graphs [17], (g, δ)-string graphs [17], (g, k)-planar graphs [17], powers of planar graphs [17,21], k-semi-fan-planar graphs [21] and k-fan-bundle planar graphs [21].…”

Section: Introductionmentioning

confidence: 88%

The product structure of squaregraphs

Hickingbotham¹,

Jungeblut²,

Merker³

et al. 2022

Preprint

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A squaregraph is a plane graph in which each internal face is a 4-cycle and each internal vertex has degree at least 4. This paper proves that every squaregraph is isomorphic to a subgraph of the semi-strong product of an outerplanar graph and a path. We generalise this result for infinite squaregraphs, and show that this is best possible in the sense that "outerplanar graph" cannot be replaced by "forest".

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Section: Introductionmentioning

confidence: 88%

The product structure of squaregraphs

Hickingbotham¹,

Jungeblut²,

Merker³

et al. 2022

Preprint

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“…The interested reader is referred to [13][14][15] for extensions of this result to graphs of bounded genus, and other natural generalizations of planar graphs.…”

Section: Subgraphs Of Strong Productsmentioning

confidence: 99%

Colouring Strong Products

Esperet,

Wood

2022

Preprint

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Recent results show that several important graph classes can be embedded as subgraphs of strong products of simpler graphs classes (paths, small cliques, or graphs of bounded treewidth). This paper develops general techniques to bound the chromatic number (and its popular variants, such as fractional, clustered, or defective chromatic number) of the strong product of general graphs with simpler graphs classes, such as paths, and more generally graphs of bounded treewidth. We also highlight important links between the study of (fractional) clustered colouring of strong products and other topics, such as asymptotic dimension in metric theory and topology, site percolation in probability theory, and the Shannon capacity in information theory.

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“…The building blocks typically have bounded treewidth, which is the standard measure of how similar a graph is to a tree. Examples of graphs classes that can be described this way include planar graphs [27,72], graphs of bounded Euler genus [23,27], graphs excluding a fixed minor [27], and various non-minor-closed classes [29,41]. These results have been the key to solving several open problems regarding queue layouts [27], nonrepetitive colouring [26], p-centered colouring [24], adjacency labelling [25,35], and twin-width [3,11].…”

Section: Introductionmentioning

confidence: 99%

Product structure of graph classes with bounded treewidth

Campbell¹,

Clinch²,

Marc³

et al. 2022

Preprint

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We show that many graphs with bounded treewidth can be described as subgraphs of the strong product of a graph with smaller treewidth and a bounded-size complete graph. To this end, define the underlying treewidth of a graph class G to be the minimum non-negative integer c such that, for some function f , for every graph G ∈ G there is a graphWe introduce disjointed coverings of graphs and show they determine the underlying treewidth of any graph class. Using this result, we prove that the class of planar graphs has underlying treewidth 3; the class of K s,t -minor-free graphs has underlying treewidth s (for t max{s, 3}); and the class of K t -minor-free graphs has underlying treewidth t − 2. In general, we prove that a monotone class has bounded underlying treewidth if and only if it excludes some fixed topological minor. We also study the underlying treewidth of graph classes defined by an excluded subgraph or excluded induced subgraph. We show that the class of graphs with no H subgraph has bounded underlying treewidth if and only if every component of H is a subdivided star, and that the class of graphs with no induced H subgraph has bounded underlying treewidth if and only if every component of H is a star.

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Improved product structure for graphs on surfaces

Abstract: Dujmović, Joret, Micek, Morin, Ueckerdt and Wood [J. ACM 2020] proved that for every graph G with Euler genus g there is a graph H with treewidth at most 4 and a path P such that G ⊆ H P K max{2g,3} . We improve this result by replacing "4" by "3".

Cited by 3 publications

References 9 publications

The product structure of squaregraphs

The product structure of squaregraphs

Colouring Strong Products

Product structure of graph classes with bounded treewidth

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