Tree-partitions and small-spread tree-decompositions

TL;DR

This paper improves bounds on tree-partitions and domino treewidth in graphs, establishing optimal upper bounds and connecting these concepts to chordal completions.

Contribution

It provides an improved constant bound on tree-partitions with specific properties and proves the optimality of the domino treewidth upper bound, solving an open problem.

Findings

Improved constant bound on tree-partitions with maximum degree properties.

Established that the upper bound on domino treewidth is tight and best possible.

Connected the concept to chordal completions, showing the bound is optimal.

Abstract

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.