On the Parameterized Complexity of Bounded-Density Vertex Deletion

2026-06-24

AI Summary

This study addresses the Bounded-Density Vertex Deletion problem: determining whether at most $k$ vertices can be removed from a graph so that the density of its densest subgraph falls below a given threshold. By leveraging parameterized complexity theory, structural graph properties, and refined reduction techniques, the authors establish for the first time that the problem is W[1]-hard when parameterized by treedepth or feedback vertex set size. Conversely, they show that when the target density is a constant, the problem admits a fixed-parameter tractable algorithm parameterized by clique-width. These results comprehensively delineate the parameterized complexity landscape of the problem across prominent graph parameters, precisely demarcating the boundary between tractable and intractable parameter regimes.

Abstract

We explore the parameterized complexity of Bounded Density Vertex Deletion (BDVD): given a graph $G$, an integer budget $k$, and a target density $τ_ρ$, the task is to determine whether the density (i.e. number of edges divided by number of vertices) of the densest subgraph of $G$ can be reduced to at most $τ_ρ$ by deleting at most $k$ vertices. Our primary focus is on structural graph parameters related to treewidth, as the parameterized complexity of BDVD with respect to treewidth was left as open question by Bazgan et al. [JCSS, 2025]. We resolve this question by showing W[1]-hardness with respect to various parameters, including treedepth and feedback vertex number. These results imply W[1]-hardness with respect to treewidth. We obtain positive results for parameters larger than treedepth and feedback vertex number, namely we show BDVD is in FPT parameterized by the max leaf number or vertex integrity. Under the assumption that the target density $τ_ρ$ is a fixed constant the parameterized complexity landscape of BDVD changes drastically, allowing a fixed-parameter tractable algorithm even for parameters smaller than treewidth, namely cliquewidth. Altogether, our results provide a refined complexity landscape for Bounded Density Vertex Deletion, sharply distinguishing between tractable and intractable parameter regimes under structural parameterizations.

Problem

• Bounded Density Vertex Deletion
• parameterized complexity
• treewidth
• density
• vertex deletion

Innovation

• parameterized complexity
• bounded-density vertex deletion
• W[1]-hardness
• fixed-parameter tractability
• structural graph parameters