Fixed-parameter tractability of Independent Set by the H-size parameter

Establish whether Independent Set is in FPT for non-crossing H-graphs when parameterized solely by the size \(\Vert H\Vert\) of the defining multigraph.

Background

The paper proves that Independent Set is fixed-parameter tractable on non-crossing H-graphs when parameterized by $\Vert H\Vert+k$, where k is the requested independent-set size. It remains unresolved whether the solution-size parameter k can be removed and whether parameterization by $\Vert H\Vert$ alone suffices. The paper notes that first-order model checking cannot generally be fixed-parameter tractable under this parameter alone unless $\mathsf{P}=\mathsf{PSPACE}$, but Independent Set may still behave differently.