FPT under solution size k parameterization

Determine whether the s-Club Cluster Edge Deletion problem admits a fixed-parameter algorithm when parameterized solely by the solution size k; specifically, decide if there exists a function f such that the problem can be solved in time f(k)·n^{O(1)} for arbitrary s ≥ 1 using only k as the parameter.

Background

The s-Club Cluster Edge Deletion problem asks whether one can delete at most k edges from a graph so that every connected component of the resulting graph has diameter at most s. It generalizes Cluster Edge Deletion (s=1).

The paper shows the problem is FPT when parameterized by s+k, but despite additional progress via a bicriteria FPT approximation (for graphs excluding long induced cycles), the exact parameterized complexity using k alone remains unresolved.