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

TL;DR

This paper investigates the parameterized and kernelization complexity of the distance-bounded s-Club Cluster Edge Deletion problem, a generalization that enforces diameter $s$ within each component after deleting at most $k$ edges. It establishes a mixed landscape: NP-hardness for $s=2$ on split graphs with a cubic kernel, along with W[1]-hardness for pathwidth/treewidth and several FPT results for treedepth, neighborhood diversity, and cluster-vertex-deletion number; it also delivers a bicriteria FPT approximation, and exact or polynomial-time results for interval and unit interval graphs. Additionally, it studies a directed variant, showing W[1]-hardness even on DAGs, thereby clarifying the essential role of the diameter bound. Collectively, the results delineate the boundary between tractable and intractable regimes and inform when relaxing the diameter constraint can restore fixed-parameter tractability.

Abstract

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.

Figures (1)

• Figure 1: Relationship between graph parameters and our results for $s$-Club Cluster Edge Deletion. An arrow $A \to B$ indicates that there exists a function $f$ such that $f(A(G)) \ge B(G)$ for all graphs $G$. Color coding: parameters in green correspond to those for which the problem is fixed-parameter tractable (FPT), including vertex cover ($\mathop{\mathrm{vc}}\nolimits$), neighborhood diversity ($\mathop{\mathrm{nd}}\nolimits$), twin cover ($\mathop{\mathrm{tc}}\nolimits$), vertex integrity ($\mathop{\mathrm{vi}}\nolimits$), cluster vertex deletion number ($\mathop{\mathrm{cvd}}\nolimits$), and treedepth ($\mathop{\mathrm{td}}\nolimits$). Parameters in red indicate W[1]-hardness, namely pathwidth ($\mathop{\mathrm{pw}}\nolimits$), treewidth ($\mathop{\mathrm{tw}}\nolimits$), and cliquewidth ($\mathop{\mathrm{cw}}\nolimits$). Finally, parameters in gray correspond to cases that remain open, specifically the feedback vertex set and modular width.

Theorems & Definitions (34)

• Definition 1: Interval graph
• Definition 2: Unit interval graph
• Definition 3: Robertson and Seymour Neil
• Definition 4: Robertson and Seymour Neil
• Definition 5
• Definition 6
• Definition 7
• Definition 8: Ganian dmtcs:2136
• Definition 9: Ganian dmtcs:2136
• Definition 10: Lampis Lampis