Tree-Partitions with Small Bounded Degree Trees

Abstract

A "tree-partition" of a graph $G$ is a partition of $V(G)$ such that identifying the vertices in each part gives a tree. It is known that every graph with treewidth $k$ and maximum degree $Δ$ has a tree-partition with parts of size $O(kΔ)$. We prove the same result with the extra property that the underlying tree has maximum degree $O(Δ)$ and $O(|V(G)|/k)$ vertices.