Treewidth 2 in the Planar Graph Product Structure Theorem

We prove that every planar graph is contained in $H_1\boxtimes H_2\boxtimes K_2$ for some graphs $H_1$ and $H_2$ both with treewidth 2. This resolves a question of Liu, Norin and Wood [arXiv:2410.20333]. We also show this result is best possible: for any $c \in \mathbb{N}$, there is a planar graph $...

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Published inDiscrete Mathematics and Theoretical Computer Science Vol. 27:2; no. Graph Theory; p. 1
Main Authors Distel, Marc, Hendrey, Kevin, Karol, Nikolai, Wood, David R., Yip, Jung Hon
Format Journal Article
LanguageEnglish
Published DMTCS 01.08.2025
Discrete Mathematics & Theoretical Computer Science
Subjects
computer science - discrete mathematics
Decomposition (Mathematics)
Graph theory
mathematics - combinatorics
Australia
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ISSN1365-8050
1462-7264
1365-8050
DOI10.46298/dmtcs.14785

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Abstract We prove that every planar graph is contained in $H_1\boxtimes H_2\boxtimes K_2$ for some graphs $H_1$ and $H_2$ both with treewidth 2. This resolves a question of Liu, Norin and Wood [arXiv:2410.20333]. We also show this result is best possible: for any $c \in \mathbb{N}$, there is a planar graph $G$ such that for any tree $T$ and graph $H$ with $\text{tw}(H) \leqslant 2$, $G$ is not contained in $H \boxtimes T \boxtimes K_c$. Comment: arXiv admin note: text overlap with arXiv:2410.20333
AbstractList We prove that every planar graph is contained in [H.sub.1] [??] [H.sub.2] [??] [K.sub.2] for some graphs [H.sub.1] and [H.sub.2] both with treewidth 2. This resolves a question of Liu, Norin and Wood [arXiv:2410.20333]. We also show this result is best possible in the following sense: for any c [member of] N, there is a planar graph G such that for any tree T and graph H with tw(H) [??] 2, G is not contained in H [??] T [??] [K.sub.c].
We prove that every planar graph is contained in $H_1\boxtimes H_2\boxtimes K_2$ for some graphs $H_1$ and $H_2$ both with treewidth 2. This resolves a question of Liu, Norin and Wood [arXiv:2410.20333]. We also show this result is best possible: for any $c \in \mathbb{N}$, there is a planar graph $G$ such that for any tree $T$ and graph $H$ with $\text{tw}(H) \leqslant 2$, $G$ is not contained in $H \boxtimes T \boxtimes K_c$.
We prove that every planar graph is contained in $H_1\boxtimes H_2\boxtimes K_2$ for some graphs $H_1$ and $H_2$ both with treewidth 2. This resolves a question of Liu, Norin and Wood [arXiv:2410.20333]. We also show this result is best possible: for any $c \in \mathbb{N}$, there is a planar graph $G$ such that for any tree $T$ and graph $H$ with $\text{tw}(H) \leqslant 2$, $G$ is not contained in $H \boxtimes T \boxtimes K_c$. Comment: arXiv admin note: text overlap with arXiv:2410.20333
We prove that every planar graph is contained in [H.sub.1] [??] [H.sub.2] [??] [K.sub.2] for some graphs [H.sub.1] and [H.sub.2] both with treewidth 2. This resolves a question of Liu, Norin and Wood [arXiv:2410.20333]. We also show this result is best possible in the following sense: for any c [member of] N, there is a planar graph G such that for any tree T and graph H with tw(H) [??] 2, G is not contained in H [??] T [??] [K.sub.c]. Keywords: planar graph, product structure
Audience Academic
Author Yip, Jung Hon
Karol, Nikolai
Wood, David R.
Distel, Marc
Hendrey, Kevin
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Snippet We prove that every planar graph is contained in $H_1\boxtimes H_2\boxtimes K_2$ for some graphs $H_1$ and $H_2$ both with treewidth 2. This resolves a...
We prove that every planar graph is contained in [H.sub.1] [??] [H.sub.2] [??] [K.sub.2] for some graphs [H.sub.1] and [H.sub.2] both with treewidth 2. This...
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SubjectTerms computer science - discrete mathematics
Decomposition (Mathematics)
Graph theory
mathematics - combinatorics
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Title Treewidth 2 in the Planar Graph Product Structure Theorem
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