Tree-partitions of graphs with bounded tree-depth

Rong Chen, Huayue Liu

Wood~ recently showed that every graph $G$ of pathwidth $h$ and $\Delta (G)\ge 1$ admits a $T$-partition of width at most $4(h+1{)}^2\Delta (G)$ for some tree $T$ with $pw(T)\le 2h+1$. In this pa…