Extending the flip algorithm from distance to complete to distance to cluster

Extend the fixed-parameter algorithm for Flip-CD parameterized by distance to complete to an algorithm parameterized by distance to cluster, where the graph becomes a disjoint union of cliques after deleting the modulator.

Background

For the color-flipping variant, the paper gives an exact fixed-parameter algorithm parameterized by distance to complete, namely the minimum number of vertices whose deletion leaves a complete graph. The conclusion asks whether this algorithmic technique can be generalized to the distance-to-cluster parameter.

Distance to cluster allows the remaining graph to consist of several disjoint cliques. The unresolved issue is that multiple cliques may each use the same color once, creating interactions absent from the single-clique setting.

References

Finally, it is worthwhile to establish whether the flip algorithm for distance to complete can be extended to the distance to cluster (that is, the disjoint union of cliques), where several cliques may each use the same color once.

— On the Parameterized Complexity of Coloring Discovery  (2609.09837 - Decker et al., 9 Sep 2026) in Section 6 (Conclusion)