Coarse Balanced Separators and Tree-Decompositions
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866912495858352128 |
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| author | Chudnovsky, Maria Hickingbotham, Robert |
| author_facet | Chudnovsky, Maria Hickingbotham, Robert |
| contents | A classical result of Robertson and Seymour (1986) states that the treewidth of a graph is linearly tied to its separation number: the smallest integer $k$ such that, for every weighting of the vertices, the graph admits a balanced separator of size at most $k$. Motivated by recent progress on coarse treewidth, Abrishami, Czyżewska, Kluk, Pilipczuk, Pilipczuk, and Rzążewski (2025) conjectured a coarse analogue to this result: every graph that has a balanced separator consisting of a bounded number of balls of bounded radius is quasi-isometric to a graph with bounded treewidth. In this paper, we confirm their conjecture for $K_{t,t}$-induced-subgraph-free graphs when the separator consists of a bounded number of balls of radius $1$. In doing so, we bridge two important conjectures concerning the structure of graphs that exclude a planar graph as an induced minor. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_06550 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Coarse Balanced Separators and Tree-Decompositions Chudnovsky, Maria Hickingbotham, Robert Combinatorics A classical result of Robertson and Seymour (1986) states that the treewidth of a graph is linearly tied to its separation number: the smallest integer $k$ such that, for every weighting of the vertices, the graph admits a balanced separator of size at most $k$. Motivated by recent progress on coarse treewidth, Abrishami, Czyżewska, Kluk, Pilipczuk, Pilipczuk, and Rzążewski (2025) conjectured a coarse analogue to this result: every graph that has a balanced separator consisting of a bounded number of balls of bounded radius is quasi-isometric to a graph with bounded treewidth. In this paper, we confirm their conjecture for $K_{t,t}$-induced-subgraph-free graphs when the separator consists of a bounded number of balls of radius $1$. In doing so, we bridge two important conjectures concerning the structure of graphs that exclude a planar graph as an induced minor. |
| title | Coarse Balanced Separators and Tree-Decompositions |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2505.06550 |