FPT under feedback vertex set parameterization

Determine whether the s-Club Cluster Edge Deletion problem is fixed-parameter tractable when parameterized by the feedback vertex set number of the input graph; that is, establish whether there exists an algorithm running in time f(fvs(G))·n^{O(1)} for arbitrary s ≥ 1.

Background

The authors map the parameterized landscape for the problem across structural graph parameters. They prove W[1]-hardness under pathwidth (and hence treewidth), and FPT under neighborhood diversity, twin cover, and cluster vertex deletion.

In their overview figure, they explicitly mark the feedback vertex set parameter as an open case, indicating that its parameterized complexity status for s-Club Cluster Edge Deletion is not yet settled.