Fixed-parameter tractability of Feedback Vertex Set

Determine whether Feedback Vertex Set is in FPT for proper H-graphs and non-crossing H-graphs when parameterized by \(\|H\|\).

Background

Feedback Vertex Set is presented as a candidate problem whose complexity may benefit from the structural properties of non-crossing H-graphs. For general H-graphs, the problem is known to be in XP but W[1]-hard when parameterized by ∥H∥\|H\|. The open problem asks whether restricting to proper or non-crossing H-graphs yields fixed-parameter tractability.

References

Is Feedback Vertex Set in $\mathsf{FPT}$ for proper $H$-graphs and non-crossing $H$-graphs when parameterized by $\Vert H\Vert$?

— 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