Complexity of recognizing globally noncrossing graphs

Ascertain the complexity class of the decision problem that, given a graph with edge‑length constraints, asks whether all of its realizations are noncrossing (i.e., whether the graph is globally noncrossing). In particular, determine whether this problem is ∀R‑complete.

Background

The paper introduces and studies globally noncrossing graphs, proving strong hardness results for realization, rigidity, and global rigidity within this class. However, the meta‑problem of recognizing whether a given constrained graph is globally noncrossing is not classified.

Their ∃R‑completeness result shows that distinguishing unrealizable graphs from realizable globally noncrossing graphs is already ∃R‑complete, but the full recognition problem could be harder; clarifying its status would deepen understanding of noncrossing constraints in computational geometry.

References

Table 1 settles most problems in this area, but a few interesting open problems remain. We also introduced the class of globally noncrossing graphs. Is it ∀R-complete to determine whether a graph with edge-length constraints is globally noncrossing, that is, whether all its realizations are noncrossing?

— Who Needs Crossings?: Noncrossing Linkages are Universal, and Deciding (Global) Rigidity is Hard  (2510.17737 - Abel et al., 20 Oct 2025) in Section 7, Open Problems