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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.01685 |
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| _version_ | 1866909010651774976 |
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| author | Chen, Rong Liao, Enzi |
| author_facet | Chen, Rong Liao, Enzi |
| contents | Fix $k \in \mathbb{N}$ and let $G$ be a connected graph with treewidth at most $k$. We say that $xy \notin E(G)$ is a {\em $k$-ghost-edge} of $G$ if for every tree decomposition $(T, \cB)$ of $G$ with width at most $k$, both $x$ and $y$ are contained in a bag of $(T, \cB)$. Moreover, if $G$ does not contain any $k$-ghost-edges, then $G$ is {\em $k$-ghost-free}. Hickingbotham proposed a conjecture that every connected $k$-ghost-free graph $G$ has a tree decomposition $(T, \cB)$ with width at most $k$ such that $T$ is a subgraph of $G$. In this paper, we prove that Hickingbotham's conjecture is false for all $k\geq3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_01685 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On tree decompositions whose trees are subgraphs Chen, Rong Liao, Enzi Combinatorics Fix $k \in \mathbb{N}$ and let $G$ be a connected graph with treewidth at most $k$. We say that $xy \notin E(G)$ is a {\em $k$-ghost-edge} of $G$ if for every tree decomposition $(T, \cB)$ of $G$ with width at most $k$, both $x$ and $y$ are contained in a bag of $(T, \cB)$. Moreover, if $G$ does not contain any $k$-ghost-edges, then $G$ is {\em $k$-ghost-free}. Hickingbotham proposed a conjecture that every connected $k$-ghost-free graph $G$ has a tree decomposition $(T, \cB)$ with width at most $k$ such that $T$ is a subgraph of $G$. In this paper, we prove that Hickingbotham's conjecture is false for all $k\geq3$. |
| title | On tree decompositions whose trees are subgraphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2605.01685 |