On tree decompositions whose trees are minors

Pub Date : 2024-02-11 DOI:10.1002/jgt.23083

Pablo Blanco, Linda Cook, Meike Hatzel, Claire Hilaire, Freddie Illingworth, Rose McCarty

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Abstract

In 2019, Dvořák asked whether every connected graph $G$ has a tree decomposition $(T, B)$ so that $T$ is a subgraph of $G$ and the width of $(T, B)$ is bounded by a function of the treewidth of $G$ . We prove that this is false, even when $G$ has treewidth 2 and $T$ is allowed to be a minor of $G$ .

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关于树为未成年人的树分解

2019 年，德沃夏克提出了一个问题：是否每个连通图 G$G$ 都有一个树分解 (T,B)$(T,{\rm{ {\mathcal B} }})$，从而 T$T$ 是 G$G$ 的子图，并且 (T,B)$(T,{\rm{ {\mathcal B} }})$ 的宽度受 G$G$ 树宽的函数约束？我们证明，即使 G$G$ 的树宽为 2 且允许 T$T$ 是 G$G$ 的次要图，这也是错误的。

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