A treedepth decomposition of an undirected graph $G$ is a rooted forest $F$ on the vertex set of $G$ such that every edge $uv\in E(G)$ is in ancestor-descendant relationship in $F$. Given a weight function $w\colon V(G)\rightarrow \mathbb{N}$, the weighted depth of a treedepth decomposition is the maximum weight of any path from the root to a leaf, where the weight of a path is the sum of the weights of its vertices. It is known that deciding weighted treedepth is NP-complete even on trees. We prove that weighted treedepth is also NP-complete on bounded degree graphs. On the positive side, we prove that the problem is efficiently solvable on paths and on 1-subdivided stars.

翻译：无向图$G$的树深分解是定义在$G$顶点集上的一个有根森林$F$，使得每条边$uv\in E(G)$在$F$中均呈现祖先-后代关系。给定权重函数$w\colon V(G)\rightarrow \mathbb{N}$，树深分解的加权深度定义为从根到叶子的任意路径的最大权重，其中路径的权重为其顶点权重之和。已知即使在树上判定加权树深也是NP完全的。我们证明加权树深在有界度图上同样是NP完全的。在积极方面，我们证明该问题在路径图和1-细分星图上可高效求解。