Fixed-parameter tractability of Independent Set parameterized only by H

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

Background

The paper proves that Independent Set is fixed-parameter tractable on non-crossing H-graphs when parameterized by ∥H∥+k\|H\|+k, where kk is the independent-set size. The stronger parameterization by ∥H∥\|H\| alone remains unresolved. The authors contrast this with the fact that Independent Set is W[1]-hard on general H-graphs under the combined parameter ∥H∥+k\|H\|+k.

References

It is still open whether {\sc Independent Set} is in $\mathsf{FPT}$ for 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 Table 1, Section 1; reiterated in Section 5, Conclusions