Tree-partitions and small-spread tree-decompositions

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 $\Delta$ has a tree-partition with width $O(k\Delta)$. We prove the same result with an improved constant, and with the extra property that the underlying tree has maximum degree $O(\Delta)$ and $O(|V(G)|/k\Delta)$ vertices. This result implies (with an improved constant) the best known upper bound on the domino treewidth of $O(k\Delta^2)$, 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 $\Omega(k\Delta^2)$ for $k\geqslant 2$. On the other hand, allowing the spread to be a function of $k$, we show that width $O(k\Delta)$ can be achieved. This result exploits a connection to chordal completions, which we show is best possible, a result of independent interest.