On the Parameterized Complexity of the $s$-Club Cluster Edge Deletion Problem

TL;DR

This paper investigates the parameterized complexity of the s-Club Cluster Edge Deletion problem, proving it is fixed-parameter tractable when parameterized by s and the input graph's treewidth.

Contribution

The paper establishes that the s-Club Cluster Edge Deletion problem is fixed-parameter tractable with respect to s and treewidth, despite being NP-hard for s=2.

Findings

FPT algorithm exists for the problem when parameterized by s and treewidth.

NP-hardness of the problem for s=2 is acknowledged.

The approach leverages treewidth to achieve fixed-parameter tractability.

Abstract

We study the parameterized complexity of the $s$-Club Cluster Edge Deletion problem: Given a graph $G$ and two integers $s \ge 2$ and $k \ge 1$, is it possible to remove at most $k$ edges from $G$ such that each connected component of the resulting graph has diameter at most $s$? This problem is known to be NP-hard already when $s = 2$. We prove that it admits a fixed-parameter tractable algorithm when parameterized by $s$ and the treewidth of the input graph.