Removing the number of colors from the Swap-CD parameter

Determine whether Swap-CD is fixed-parameter tractable when parameterized solely by the vertex-cover number, thereby removing the number of colors from the proven parameterization $vc(G)+q$.

Background

The paper establishes fixed-parameter tractability for Swap-CD parameterized by the combined parameter $vc(G)+q$, where $vc(G)$ is the vertex-cover number and $q$ is the number of colors. The conclusion identifies as unresolved whether the dependence on $q$ can be eliminated from this parameterization.

This question concerns a strengthening of the paper’s Swap-CD algorithmic result; it is not answered by the paper’s hardness results, which address other parameter combinations.