Reduction of minimal connected domination enumeration to transversal enumeration

Determine whether enumerating minimal connected dominating sets in graphs reduces to enumerating minimal transversals of hypergraphs, thereby resolving whether the connected domination variant is equivalent to Trans-Enum in this sense.

Background

Minimal connected dominating sets can be represented as minimal transversals of a hypergraph whose hyperedges are the graph’s minimal separators. However, this separator hypergraph may have exponentially many edges and may be difficult to compute. The paper notes that a reduction to the standard minimal-transversal enumeration problem is unknown and that prior work conjectured such a reduction does not exist.

References

Concerning connected dominating sets, it is not known whether the problem reduces to Trans-Enum. It has in fact been conjectured that this is not the case in.

— Enumerating minimal dominating sets and variants in chordal bipartite graphs  (2502.14611 - Castelo et al., 20 Feb 2025) in Section 1, paragraph beginning “Concerning connected dominating sets”