🤖 AI Summary
This study addresses the parameterized complexity of the $T_{h+1}$-Free Edge Deletion problem, which asks whether at most $k$ edges can be removed from a graph so that every connected component has size at most $h$. By integrating tools from parameterized complexity theory, integer linear programming, bicriteria approximation algorithms, and specialized algorithms for restricted graph classes such as split graphs and interval graphs, the work systematically delineates the tractability frontier of the problem under various structural parameters. Key contributions include establishing new W[1]-hardness results for parameters like treewidth and pathwidth, proving fixed-parameter tractability for the first time when parameterized by cluster vertex deletion number plus $h$ and neighborhood diversity plus $h$, and presenting the first parameterized bicriteria approximation algorithm. Additionally, the paper shows that the natural directed extension of the problem remains intractable even on DAGs.
📝 Abstract
We study the parameterized complexity of the T(h+1)-Free Edge Deletion problem. Given a graph G and integers k and h, the task is to delete at most k edges so that every connected component of the resulting graph has size at most h. The problem is NP-complete for every fixed h at least 3, while it is solvable in polynomial time for h at most 2. Recent work showed strong hardness barriers: the problem is W[1]-hard when parameterized by the solution size together with the size of a feedback edge set, ruling out fixed-parameter tractability for many classical structural parameters. We significantly strengthen these negative results by proving W[1]-hardness when parameterized by the vertex deletion distance to a disjoint union of paths, the vertex deletion distance to a disjoint union of stars, or the twin cover number. These results unify and extend known hardness results for treewidth, pathwidth, and feedback vertex set, and show that several restrictive parameters, including treedepth, cluster vertex deletion number, and modular width, do not yield fixed-parameter tractability when h is unbounded. On the positive side, we identify parameterizations that restore tractability. We show that the problem is fixed-parameter tractable when parameterized by cluster vertex deletion together with h, vertex deletion set into a clique and also when parameterized by neighborhood diversity together with h via an integer linear programming formulation. We further present a fixed-parameter tractable bicriteria approximation algorithm parameterized by k. Finally, we show that the problem admits fixed-parameter tractable algorithms on split graphs and interval graphs, and we establish hardness for a directed generalization even on directed acyclic graphs.