Completeness of recognizing non-crossing H-graphs

Determine whether recognizing non-crossing H-graphs is complete for some connected irreflexive multigraph H.

Background

The paper establishes algorithmic results for non-crossing H-graphs assuming that a non-crossing H-representation is provided. The authors note that recognition of H-graphs and proper H-graphs is complete for certain graphs H, but that no analogous completeness result is known for non-crossing H-graphs. The open problem asks whether such a graph H exists.

References

However, we do not know any graph $H$ for which recognizing non-crossing $H$-graphs is -complete and whether boundedness of proper mixed-thinness would be of help.

— 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