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.

翻译：我们研究了T_{h+1}自由边删除问题的参数化复杂性。给定一个图G以及整数k和h，该问题的任务是删除至多k条边，使得结果图的每个连通分量的大小至多为h。对于每个固定的h≥3，该问题是NP完全的，而当h≤2时可在多项式时间内求解。近期研究表明该问题存在很强的困难性障碍：当以解的大小与反馈边集的大小共同作为参数时，该问题是W[1]-难的，这排除了许多经典结构参数下固定参数可处理性的可能。我们通过证明当以到不相交路径并集的顶点删除距离、到不相交星图并集的顶点删除距离或双覆盖数作为参数时，该问题均为W[1]-难的，从而显著强化了这些负面结果。这些结果统一并扩展了关于树宽、路径宽和反馈顶点集的已知困难性结论，并表明当h无界时，包括树深、团顶点删除数和模宽在内的若干限制性参数均无法产生固定参数可处理性。在正面结果方面，我们识别了能够恢复可处理性的参数化方式。我们证明了当以团顶点删除数与h共同作为参数、以到团的顶点删除集作为参数，以及通过整数线性规划公式化以邻域多样性数与h共同作为参数时，该问题是固定参数可处理的。我们进一步提出了一种以k为参数的固定参数可处理双准则近似算法。最后，我们证明了该问题在分裂图和区间图上存在固定参数可处理算法，并建立了其有向推广版本即使在有向无环图上的困难性。