Tree-partitions of graphs with bounded tree-depth

Abstract

Wood~ recently showed that every graph $G$ of pathwidth $h$ and $Δ(G)\ge1$ admits a $T$-partition of width at most $4(h+1)^2Δ(G)$ for some tree $T$ with $pw(T)\leq2h+1$. In this paper, we establish an analogous result for tree-depth, which is a stronger parameter than pathwidth. We prove that every connected graph with tree-depth $h$ admits a $T$-partition of width at most $\mathrm{max}\{1, (4h-10)Δ(G)+1\}$ for some tree $T$ with $\operatorname{rad}(T)\leq h-1$.