Destroying densest subgraphs is hard ☆

We analyze the computational complexity of the following computational problems called BOUNDED-DENSITY EDGE DELETION and BOUNDED-DENSITY VERTEX DELETION: Given a graph G, a budget k and a target density t rho, are there kedges (k vertices) whose removal from G results in a graph where the densest subgraph has density at most t rho? Here, the density of a graph is the number of its edges divided by the number of its vertices. We prove that both problems are polynomial-time solvable on trees and cliques but are NP-complete on planar bipartite graphs and split graphs. From a parameterized point of view, we show that both problems are fixed-parameter tractable with respect to the vertex cover number but W[1]-hard with respect to the solution size. Furthermore, we prove that BOUNDED-DENSITY EDGE DELETION is W[1]-hard with respect to the feedback edge number, demonstrating that the problem remains hard on very sparse graphs. (c) 2025 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).