Weighted Treedepth is NP-complete on Graphs of Bounded Degree

2025-10-21

AI Summary

This paper investigates the computational complexity of weighted treedepth on bounded-degree graphs. Specifically, it addresses the decision problem: given a vertex-weighted graph of bounded maximum degree, does there exist a treedepth decomposition whose total weight (sum of node weights in the decomposition) is at most a given threshold? The authors establish, for the first time, that this problem is NP-complete—via a polynomial-time reduction from 3-SAT, carefully engineered to preserve bounded maximum degree in the resulting instance. Furthermore, they identify tractable cases by showing the problem is solvable in polynomial time on paths and 1-subdivided stars, thereby delineating precise complexity boundaries. These results fill a fundamental gap in the complexity landscape of weighted structural graph parameters on sparse graphs and provide essential theoretical foundations for designing algorithms based on treedepth.

Technology Category

Knowledge Representation and Reasoning: Computational Complexity of ReasoningConstraint Satisfaction and Optimization: Satisfiability Modulo TheoriesReasoning under Uncertainty: Graphical Models

Application Category

Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphsSemantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologiesWeb Mining and Content Analysis: Models for Web evolution

Abstract

A treedepth decomposition of an undirected graph $G$ is a rooted forest $F$ on the vertex set of $G$ such that every edge $uvin E(G)$ is in ancestor-descendant relationship in $F$. Given a weight function $wcolon V(G) ightarrow mathbb{N}$, 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.

Problem

• Weighted treedepth is NP-complete for bounded degree graphs
• The problem extends NP-completeness from trees to bounded degree graphs
• It provides efficient solutions for paths and 1-subdivided stars

Innovation

• Proves NP-completeness on bounded degree graphs
• Solves weighted treedepth efficiently on paths
• Provides efficient solution for 1-subdivided stars