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.

翻译：树分解和树宽在结构图论与算法图论中具有基础重要性。树分解的“扩散度”定义为最小整数 $s$，使得每个顶点至多属于 $s$ 个袋子。扩散度为 2 的树分解称为“多米诺”树分解，这是扩散度最小的有意义取值。为使扩散度 1 变得有意义，可将树分解的定义放松为“树划分”，即允许每条边的两个端点位于同一袋子或相邻袋子中，同时要求每个顶点恰好出现在一个袋子里。Ding 与 Oporowski [1995] 证明：任意树宽为 $k$、最大度为 $Δ$ 的图 $G$ 均存在宽度为 $O(kΔ)$ 的树划分。我们证明了相同结论，但常数更优，且额外满足：底层树的最大度为 $O(Δ)$，顶点数为 $O(|V(G)|/kΔ)$。该结果改进了 Bodlaender [1999] 关于多米诺树宽为 $O(kΔ^2)$ 的已知最优上界（同时常数更优）。此外，我们通过构造满足 $k \geqslant 2$ 且多米诺树宽为 $Ω(kΔ^2)$ 的图，证明了该上界是最优的，从而解决了 Bodlaender 提出的一个开放问题。另一方面，允许扩散度随 $k$ 变化时，我们证明了宽度 $O(kΔ)$ 可以实现。该结果利用了与弦完备化的关联，我们证明该关联是最优的，这一结论本身具有独立意义。