Fixed-parameter tractability of Slide-CD for vertex cover plus number of colors

Determine whether Slide-CD, the Coloring Discovery variant in which colors may be swapped only across graph edges, is fixed-parameter tractable when parameterized by the sum of the vertex-cover number and the number of available colors.

Background

The paper proves that Swap-CD is fixed-parameter tractable when parameterized by the vertex-cover number plus the number of colors, using a transition-multigraph and integer-linear-programming approach. That approach does not directly apply to Slide-CD because, unlike arbitrary swaps, slide operations depend on the positions of colors in the graph.

The authors explicitly leave unresolved whether the analogous fixed-parameter tractability result holds for Slide-CD under the parameter vertex-cover number plus number of colors.