An Improved Planar Graph Product Structure Theorem

Ueckerdt, Torsten; Wood, David R.; Yi, Wendy

doi:10.37236/10614

Cited by 13 publications

(4 citation statements)

References 21 publications

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“…It was proved by Dujmović, Joret, Micek, Morin, Ueckerdt, and Wood [13] that every planar graph is a subgraph of the strong product of a graph of treewidth at most 8 and a path (see also the recent improvement by Ueckerdt, Wood, and Yi [18]). Theorem The class of planar graphs is a subset of

\mathcal {Q}_8

.…”

Section: Introductionmentioning

confidence: 99%

Sparse universal graphs for planarity

Joret

Morin

2023

Journal of London Math Soc

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We show that for every integer , there exists a graph with vertices and edges such that every ‐vertex planar graph is isomorphic to a subgraph of . The best previous bound on the number of edges was , proved by Babai, Chung, Erdős, Graham, and Spencer in 1982. We then show that for every integer , there is a graph with vertices and edges that contains induced copies of every ‐vertex planar graph. This significantly reduces the number of edges in a recent construction of the authors with Dujmović, Gavoille, and Micek.

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\mathcal {Q}_8

.…”

Section: Introductionmentioning

confidence: 99%

Sparse universal graphs for planarity

Joret

Morin

2023

Journal of London Math Soc

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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 graph classes that can be described this way include planar graphs [29,73], graphs of bounded Euler genus [23,29], graphs excluding a fixed minor [29], and various non-minor-closed classes [31,41]. These results have been the key to solving several open problems regarding queue layouts [29], nonrepetitive colouring [28], p-centered colouring [25], adjacency labelling [27,36], twin-width [3,11], and comparable box dimension [33].…”

Section: Introductionmentioning

confidence: 99%

Product structure of graph classes with bounded treewidth

Campbell,

Clinch,

Distel

et al. 2023

Combinator. Probab. Comp.

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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 $\mathcal{G}$ to be the minimum non-negative integer $c$ such that, for some function $f$ , for every graph $G \in \mathcal{G}$ there is a graph $H$ with $\textrm{tw}(H) \leqslant c$ such that $G$ is isomorphic to a subgraph of $H \boxtimes K_{f(\textrm{tw}(G))}$ . We 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 $t \geqslant \max \{s,3\}$ ); 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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“…Theorem 2 (Dujmović et al [15] and Ueckerdt et al [27]). For every planar graph G there is a graph H of treewidth at most 6 and a path P such that ⫇ ⊠ G H P.…”

mentioning

confidence: 98%

“…The original version of the Planar Graph Product Structure Theorem by Dujmović et al [15] had "treewidth at most 8" instead of "treewidth at most 6". Ueckerdt et al [27] proved Theorem 2 with "treewidth at most 6". Since outerplanar graphs have treewidth at most 2, Theorem 1 is stronger than Theorem 2 in the case of squaregraphs.…”

mentioning

confidence: 99%

The product structure of squaregraphs

Hickingbotham,

Jungeblut,

Merker

et al. 2023

Journal of Graph Theory

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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 semistrong 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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An Improved Planar Graph Product Structure Theorem

Cited by 13 publications

References 21 publications

Sparse universal graphs for planarity

Sparse universal graphs for planarity

Product structure of graph classes with bounded treewidth

The product structure of squaregraphs

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