Generalize the staircase reduction to larger diameter bounds

Determine whether the staircase-based reduction rule for 2-Club Cluster Edge Deletion on interval graphs can be generalized from diameter bound s=2 to arbitrary values of s greater than or equal to 2, or establish that fundamentally new structural ideas are required.

Background

The interval-graph kernelization uses reduction rules that apply to s-Club Cluster Edge Deletion for all s greater than or equal to 2, except for the staircase-clique rule. That rule relies on the fact that every pair of vertices in a 2-club must have a common neighbor.

For larger diameter bounds, the paper states that these local common-neighbor arguments are insufficient because more complex distance interactions arise. Extending this part of the kernelization is therefore left as an unresolved structural problem.