Powers of planar graphs, product structure, and blocking partitions

Distel, Marc; Hickingbotham, Robert; Seweryn, Michał T.; Wood, David R.

doi:10.5802/igt.4

Distel, Marc ¹; Hickingbotham, Robert ²; Seweryn, Michał T.³; Wood, David R.¹

¹ School of Mathematics Monash University Melbourne, Australia
² Laboratoire de l’Informatique du Parallélisme École Normale Supérieure de Lyon Lyon, France
³ Computer Science Institute Charles University Prague, Czech Republic

Innovations in Graph Theory, Volume 1 (2024), pp. 39-86.

Abstract

We prove that the $k$ -power of any planar graph $G$ is contained in $H ⊠ P ⊠ K_{f (Δ (G), k)}$ for some graph $H$ with bounded treewidth, some path $P$ , and some function $f$ . This resolves an open problem of Ossona de Mendez. In fact, we prove a more general result in terms of shallow minors that implies similar results for many ‘beyond planar’ graph classes, without dependence on $Δ (G)$ . For example, we prove that every $k$ -planar graph is contained in $H ⊠ P ⊠ K_{f (k)}$ for some graph $H$ with bounded treewidth and some path $P$ , and some function $f$ . This resolves an open problem of Dujmović, Morin and Wood. We generalise all these results for graphs of bounded Euler genus, still with an absolute bound on the treewidth.

At the heart of our proof is the following new concept of independent interest. An $ℓ$ -blocking partition of a graph $G$ is a partition of $V (G)$ into connected sets such that every path of length greater than $ℓ$ in $G$ contains at least two vertices in one part. We prove that for some constant $ℓ \geq 1$ every graph of Euler genus $g$ has an $ℓ$ -blocking partition with parts of size bounded by a function of $Δ (G)$ and $g$ . Motivated by this result, we study blocking partitions in their own right. We show that every graph $G$ has a $2$ -blocking partition with parts of size bounded by a function of $Δ (G)$ and $tw (G)$ . On the other hand, we show that 4-regular graphs do not have $ℓ$ -blocking partitions with bounded size parts.

Received: 2023-09-21
Accepted: 2024-09-03
Published online: 2024-11-26

DOI: 10.5802/igt.4

Classification: 05C10
Keywords: graph, planar graph, product structure, power, blocking partition, surface, minor

Author's affiliations:

Distel, Marc ¹; Hickingbotham, Robert ²; Seweryn, Michał T. ³; Wood, David R. ¹

¹ School of Mathematics Monash University Melbourne, Australia
² Laboratoire de l’Informatique du Parallélisme École Normale Supérieure de Lyon Lyon, France
³ Computer Science Institute Charles University Prague, Czech Republic

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Distel, Marc and Hickingbotham, Robert and Seweryn, Micha{\l} T. and Wood, David R.},
     title = {Powers of planar graphs, product structure, and blocking partitions},
     journal = {Innovations in Graph Theory},
     pages = {39--86},
     publisher = {Stichting Innovations in Graph Theory},
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     year = {2024},
     doi = {10.5802/igt.4},
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Distel, Marc; Hickingbotham, Robert; Seweryn, Michał T.; Wood, David R. Powers of planar graphs, product structure, and blocking partitions. Innovations in Graph Theory, Volume 1 (2024), pp. 39-86. doi : 10.5802/igt.4. https://igt.centre-mersenne.org/articles/10.5802/igt.4/

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