Treewidth 2 in the Planar Graph Product Structure Theorem
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866910881393147904 |
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| author | Distel, Marc Hendrey, Kevin Karol, Nikolai Wood, David R. Yip, Jung Hon |
| author_facet | Distel, Marc Hendrey, Kevin Karol, Nikolai Wood, David R. Yip, Jung Hon |
| contents | 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 $H \boxtimes T \boxtimes K_c$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_00343 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Treewidth 2 in the Planar Graph Product Structure Theorem Distel, Marc Hendrey, Kevin Karol, Nikolai Wood, David R. Yip, Jung Hon Combinatorics Discrete Mathematics 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 $H \boxtimes T \boxtimes K_c$. |
| title | Treewidth 2 in the Planar Graph Product Structure Theorem |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2411.00343 |