Weighted Treedepth is NP-complete on Graphs of Bounded Degree

TL;DR

This paper proves that determining weighted treedepth remains NP-complete even on graphs with bounded degree, but it is efficiently solvable on paths and 1-subdivided stars, highlighting complexity boundaries.

Contribution

It establishes NP-completeness of weighted treedepth on bounded degree graphs and identifies specific graph classes where the problem is efficiently solvable.

Findings

NP-complete on bounded degree graphs

Efficient algorithms for paths and 1-subdivided stars

Complexity boundaries for weighted treedepth

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.