An alternative characterisation of graphs quasi-isometric to graphs of bounded treewidth
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908732796960768 |
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| author | Distel, Marc |
| author_facet | Distel, Marc |
| contents | Quasi-isometry is a measure of how similar two graphs are at `large-scale'. Nguyen, Scott, and Seymour [arXiv:2501.09839] and Hickingbotham [arXiv:2501.10840] independently gave a characterisation of graphs quasi-isometric to graphs of treewidth $k$. In this paper, we give a new characterisation of such graphs. Specifically, we show that such graphs $G$ are characterised by the existence of a partition whose quotient has treewidth at most $k$ and such that each part has bounded weak diameter in $G$. The primary contribution of our characterisation is a structural description of graphs that admit such a quasi-isometry. This differs from the characterisation mentioned above, which primarily shows the existence of such a quasi-isometry. The characterisations are complementary, and neither immediately implies the other. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_21946 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An alternative characterisation of graphs quasi-isometric to graphs of bounded treewidth Distel, Marc Combinatorics Quasi-isometry is a measure of how similar two graphs are at `large-scale'. Nguyen, Scott, and Seymour [arXiv:2501.09839] and Hickingbotham [arXiv:2501.10840] independently gave a characterisation of graphs quasi-isometric to graphs of treewidth $k$. In this paper, we give a new characterisation of such graphs. Specifically, we show that such graphs $G$ are characterised by the existence of a partition whose quotient has treewidth at most $k$ and such that each part has bounded weak diameter in $G$. The primary contribution of our characterisation is a structural description of graphs that admit such a quasi-isometry. This differs from the characterisation mentioned above, which primarily shows the existence of such a quasi-isometry. The characterisations are complementary, and neither immediately implies the other. |
| title | An alternative characterisation of graphs quasi-isometric to graphs of bounded treewidth |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2512.21946 |