Tree-Partitions with Small Bounded Degree Trees

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 $\mathrm{\Delta }$ has a tree-partition with parts of size $O(k\mathrm{\Delta })$. We prove the same result with the extra property that the underlying tree has maximum degree $O(\mathrm{\Delta })$ and $O(\mathrm{\mid }V(G)\mathrm{\mid }\mathrm{/}k)$ vertices.