Complexity of Clique on proper and non-crossing H-graphs

Determine the complexity of Clique for proper H-graphs and non-crossing H-graphs when parameterized by \(\Vert H\Vert\).

Background

Clique is known to be para-NP-hard on general H-graphs when parameterized by ∥H∥\Vert H\Vert. In contrast, its parameterized complexity remains unresolved for proper H-graphs and non-crossing H-graphs. The paper explicitly asks for the complexity classification of Clique on both classes under parameterization by the size of H.

References

Determine the complexity of {\sc Clique} 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

However, we leave as an open problem to determine whether {\sc Independent Set} and {\sc Clique} are in $\mathsf{FPT}$ for non-crossing $H$-graphs even when parameterized by $\Vert H\Vert$ only.

— Non-crossing $H$-graphs: a generalization of proper interval graphs admitting FPT algorithms  (2501.11192 - Bonomo-Braberman et al., 19 Jan 2025) in Section 4, Conclusions