Universality and complexity for globally noncrossing graphs restricted to {1} or {1,2} edge lengths

Determine, for globally noncrossing graphs whose edges are restricted to all unit length or to the set {1,2}, whether the class of drawable planar regions remains exactly the compact semialgebraic sets, whether realizability is ∃R‑complete, and whether rigidity and global rigidity are ∀R‑complete.

Background

The paper establishes universality and tight complexity for globally noncrossing graphs with constant‑bounded integer edge lengths, but does not specialize to the extreme cases of all‑unit or {1,2} edge lengths in the globally noncrossing setting.

Because their constructions use only constant‑bounded integers, the authors note that these edge‑length restrictions could be within reach, but the precise universality and completeness classifications have not been determined.

References

Table 1 settles most problems in this area, but a few interesting open problems remain. We could also consider additional graph types (rows) in Table 1. For example, we considered unit edge lengths and edge lengths in {1,2}, both when allowing crossings and when forbidding crossings, but we did not consider globally noncrossing graphs with edge lengths restricted to {1} or {1,2}. Do these linkages remain universal for compact semialgebraic sets? Is realizing them ∃R-complete? Is testing their rigidity and global rigidity ∀R-complete?

— 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