We study when the diameter bound s is essential for the tractability of the s-Club Cluster Edge Deletion problem. Given a graph G = (V, E) and integers k and s, the goal is to delete at most k edges so that every connected component of the resulting graph has diameter at most s. This problem generalizes Cluster Edge Deletion (s = 1) and captures distance-bounded clustering tasks. Montecchiani et al. (Information and Computation, 2023) proved that the problem is fixed-parameter tractable when parameterized by s + tw(G) and asked whether dependence on s is necessary. We answer negatively by showing W[1]-hardness when parameterized by pathwidth (and hence by treewidth), proving that s cannot in general be dropped. On the positive side, we show FPT algorithms parameterized by treedepth, neighborhood diversity, or cluster vertex deletion number, extending results of Italiano et al. (Algorithmica, 2023) and Komusiewicz and Uhlmann (SOFSEM, 2011). We also show that no polynomial kernel exists when parameterized by vertex cover number, even for s = 2. Classically, the problem is NP-hard on split graphs for s = 2, complementing the polynomial case s = 1. We give an FPT bicriteria approximation scheme running in f(k, 1/epsilon) * n^{O(1)} that outputs a set of at most k deletions whose components have diameter at most (1 + epsilon)s. Finally, we introduce a directed generalization, s-Club Cluster Arc Deletion, extending the undirected case to reachability distances, and show it is W[1]-hard in parameter k even on directed acyclic graphs (DAGs).

翻译：我们研究直径约束s对于s-Club聚类边删除问题可解性的必要性。给定图G = (V, E)及整数k和s，该问题的目标是通过删除至多k条边，使得结果图的每个连通分量直径不超过s。该问题推广了聚类边删除问题（s = 1），并刻画了距离约束聚类任务。Montecchiani等人（Information and Computation, 2023）证明了该问题在参数s + tw(G)下是固定参数可解的，并询问对s的依赖是否必要。我们通过证明在路径宽度（进而树宽度）参数化下的W[1]难性给出了否定回答，表明s通常不可省略。在积极方面，我们给出了在树深度、邻域多样性或聚类顶点删除数参数化下的FPT算法，扩展了Italiano等人（Algorithmica, 2023）及Komusiewicz和Uhlmann（SOFSEM, 2011）的结果。我们还证明了在顶点覆盖数参数化下（即使s = 2）不存在多项式核。经典复杂性方面，该问题在s = 2时在分裂图上是NP难的，这与s = 1时的多项式情形形成互补。我们提出了一个运行时间为f(k, 1/ε) * n^{O(1)}的FPT双准则近似方案，可输出至多k条边的删除集合，使得各分量直径不超过(1 + ε)s。最后，我们提出了有向推广版本s-Club聚类弧删除问题，将无向情形扩展至可达距离，并证明即使在有向无环图上该问题在参数k下也是W[1]难的。