Parameterized Complexity of s-Club Cluster Edge Deletion: When Is the Diameter Bound Necessary?

We study the parameterized and kernelization complexity of the s-Club Cluster Edge Deletion problem, a distance-bounded generalization of Cluster Edge Deletion. 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 in the resulting graph has diameter at most s. This captures a broad class of distance-constrained graph modification problems that lie between clustering and connectivity control. We prove that for s = 2 the problem is NP-hard already on split graphs, closing the gap between the polynomially solvable cases s = 1 and s = 3. For this setting we give a cubic vertex kernel parameterized by k, the first polynomial kernel for 2-Club Cluster Edge Deletion on split graphs. On the structural side, we show that the problem is W[1]-hard when parameterized by pathwidth (and hence treewidth), implying that the diameter bound s is crucial for fixed-parameter tractability. In contrast, the problem is FPT when parameterized by treedepth, neighborhood diversity, or the cluster vertex deletion number. Finally, we design an FPT bicriteria approximation scheme that, for graphs excluding long induced cycles, runs in time f(k, 1/epsilon) * n^{O(1)} and outputs a solution of size at most k whose components have diameter at most (1 + epsilon) * s. We further present an exact FPT algorithm for interval graphs parameterized by k and a polynomial-time algorithm for unit interval graphs. We also introduce the directed variant s-Club Cluster Arc Deletion and show it is W[1]-hard when parameterized by k, even on DAGs.