具有有界树深度的图的树划分
Tree-partitions of graphs with bounded tree-depth
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中文总结 AI 辅助
本文针对比路宽更强的参数树深度，证明每个树深度为h的连通图，都存在半径不超过h-1的树T，其T-划分宽度不超过max{1, (4h-10)Δ(G)+1}，建立了与路宽情形类似的结论。
中文摘要 AI 辅助
伍德（Wood）最近证明，每个路宽为h且Δ(G)≥1的图G，都存在某个路宽pw(T)≤2h+1的树T，使得其T-划分的宽度不超过4(h+1)²Δ(G)。本文针对比路宽更强的参数树深度建立了类似结果，证明每个树深度为h的连通图，都存在某个半径rad(T)≤h-1的树T，使得其T-划分的宽度不超过max{1, (4h-10)Δ(G)+1}。
英文摘要
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 $\operatorname{rad}(T)\leq h-1$.
