Tree-partitions and small-spread tree-decompositions
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866915920496033792 |
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| author | Distel, Marc Kaul, Neel Kaul, Raj Wood, David R. |
| author_facet | Distel, Marc Kaul, Neel Kaul, Raj Wood, David R. |
| contents | Tree-decompositions and treewidth are of fundamental importance in structural and algorithmic graph theory. The "spread" of a tree-decomposition is the minimum integer $s$ such that every vertex lies in at most $s$ bags. A tree-decomposition is "domino" if it has spread 2, which is the smallest interesting value of spread. So that spread 1 becomes interesting, one can relax the definition of tree-decomposition to "tree-partition", which allows the endpoints of each edge to be in the same bag or adjacent bags, while demanding that each vertex appears in exactly one bag. Ding and Oporowski [1995] showed that every graph $G$ with treewidth $k$ and maximum degree $Δ$ has a tree-partition with width $O(kΔ)$. We prove the same result with an improved constant, and with the extra property that the underlying tree has maximum degree $O(Δ)$ and $O(|V(G)|/kΔ)$ vertices. This result implies (with an improved constant) the best known upper bound on the domino treewidth of $O(kΔ^2)$, due to Bodlaender [1999]. Moreover, solving an open problem of Bodlaender, we show this upper bound is best possible, by exhibiting graphs with domino treewidth $Ω(kΔ^2)$ for $k\geqslant 2$. On the other hand, allowing the spread to be a function of $k$, we show that width $O(kΔ)$ can be achieved. This result exploits a connection to chordal completions, which we show is best possible, a result of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_05690 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Tree-partitions and small-spread tree-decompositions Distel, Marc Kaul, Neel Kaul, Raj Wood, David R. Combinatorics Discrete Mathematics Tree-decompositions and treewidth are of fundamental importance in structural and algorithmic graph theory. The "spread" of a tree-decomposition is the minimum integer $s$ such that every vertex lies in at most $s$ bags. A tree-decomposition is "domino" if it has spread 2, which is the smallest interesting value of spread. So that spread 1 becomes interesting, one can relax the definition of tree-decomposition to "tree-partition", which allows the endpoints of each edge to be in the same bag or adjacent bags, while demanding that each vertex appears in exactly one bag. Ding and Oporowski [1995] showed that every graph $G$ with treewidth $k$ and maximum degree $Δ$ has a tree-partition with width $O(kΔ)$. We prove the same result with an improved constant, and with the extra property that the underlying tree has maximum degree $O(Δ)$ and $O(|V(G)|/kΔ)$ vertices. This result implies (with an improved constant) the best known upper bound on the domino treewidth of $O(kΔ^2)$, due to Bodlaender [1999]. Moreover, solving an open problem of Bodlaender, we show this upper bound is best possible, by exhibiting graphs with domino treewidth $Ω(kΔ^2)$ for $k\geqslant 2$. On the other hand, allowing the spread to be a function of $k$, we show that width $O(kΔ)$ can be achieved. This result exploits a connection to chordal completions, which we show is best possible, a result of independent interest. |
| title | Tree-partitions and small-spread tree-decompositions |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2604.05690 |