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

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