Recognition complexity of non-crossing H-graphs

Determine whether recognizing non-crossing H-graphs is complete for some graph H.

Background

The paper establishes fixed-parameter tractability results for several problems on non-crossing H-graphs but does not resolve the computational complexity of recognizing these graphs. Recognition of general H-graphs and proper H-graphs is known to be complete for certain graphs H, whereas no graph H is known for which recognition of non-crossing H-graphs has the corresponding completeness classification. The authors also question whether the bounded proper mixed-thinness proved in the paper could assist with recognition.

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 4, Conclusions

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.

Is recognizing non-crossing $H$-graphs -complete for some $H$?

— 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 (Section 5; Open Problem immediately following the introductory paragraph on recognition)