Weighted Treedepth is NP-complete on Graphs of Bounded Degree

TL;DR

This paper studies the weighted treedepth problem, showing it remains NP-hard on graphs of maximum degree $10$ via a reduction from Vertex Cover, extending hardness beyond trees. It also establishes positive results: weighted treedepth is computable in polynomial time on paths and on $1$-subdivided stars, and the problem is fixed-parameter tractable when parameterized by the instance's weighted treedepth. The work clarifies the complexity landscape by linking weighted treedepth to classical parameters, and raises open questions about the exact thresholds for hardness (e.g., $2$-subdivided stars) and the status on planar graphs or with respect to unweighted treedepth parameterization. Overall, the results provide a strong baseline for understanding where weighted treedepth transitions from tractable to intractable and informs future algorithmic and structural investigations.

Abstract

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.