Every connected graph of tree-depth h admits a tree-partition of width at most max(1, (4h-10)Δ+1) whose indexing tree has radius at most h-1.
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Tree-partitions of graphs with bounded tree-depth
Every connected graph of tree-depth h admits a tree-partition of width at most max(1, (4h-10)Δ+1) whose indexing tree has radius at most h-1.