Fixed-parameter sub-exponential algorithms for Dominating Set or Connected Dominating Set parameterized by 2ccvd

Determine whether either Dominating Set or Connected Dominating Set admits a fixed-parameter sub-exponential algorithm with respect to the 2-Club Cluster Vertex Deletion parameter (2ccvd), where 2ccvd is the minimum number of vertices whose deletion results in a disjoint union of graphs of diameter at most two (2-clubs).

Background

In 2-club graphs, Dominating Set and Connected Dominating Set can be solved in sub-exponential time because small dominating structures exist; for instance, the paper notes a bound on the size of a minimum connected dominating set leading to a sub-exponential algorithm for Connected Dominating Set in 2-clubs, and analogous results are known for Dominating Set.

The authors consider parameterization by the 2-Club Cluster Vertex Deletion parameter (2ccvd), defined as the minimum number of vertices whose removal leaves a disjoint union of 2-clubs, and ask whether fixed-parameter sub-exponential algorithms exist for these two problems under this parameterization.