W[1]-hardness of Dominating Set on proper H-graphs

Prove or refute that Dominating Set is W[1]-hard for proper H-graphs when parameterized by \(\|H\|+k\), where \(k\) is the domination-set size.

Background

The paper proves W[1]-hardness for Independent Set on proper H-graphs under the parameter ∥H∥+k\|H\|+k, and asks whether other first-order definable problems show analogous behavior. The authors specifically identify Dominating Set as unresolved because their hardness technique for Independent Set does not apply to it.

References

Is Dominating Set $\mathsf{W[1]}$-hard for proper $H$-graphs when parameterized by $\Vert H\Vert+k$?

— Non-crossing $H$-graphs: a generalization of proper interval graphs admitting FPT algorithms  (2501.11192 - Bonomo-Braberman et al., 19 Jan 2025) in Section 5, Conclusions