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 $\tau_\rho$, are there $k$ edges ($k$ vertices) whose removal from $G$ results in a graph where the densest subgraph has density at most $\tau_\rho$? 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.

翻译：我们分析了以下计算问题（称为有界密度边删除和有界密度顶点删除）的计算复杂性：给定图$G$、预算$k$和目标密度$\tau_\rho$，是否存在$k$条边（$k$个顶点）从$G$中删除后，得到的图中最密子图的密度不超过$\tau_\rho$？这里，图的密度定义为其边数除以顶点数。我们证明这两个问题在树和团上可在多项式时间内求解，但在平面二分图和分裂图上为NP完全问题。从参数化角度，我们证明这两个问题关于顶点覆盖数具有固定参数可解性，但关于解规模为W[1]-困难。此外，我们证明有界密度边删除关于反馈边数为W[1]-困难，表明该问题在非常稀疏的图上仍然困难。