Existence of globally rigid unit‑distance graphs beyond triangles

Determine whether there exist globally rigid unit‑distance graphs (graphs embedded in the plane with all edges of unit length and crossings allowed) having more than three edges, in order to enable hardness classifications for the global‑rigidity decision problem on unit‑edge‑length graphs.

Background

The paper proves strong ∀R‑completeness for global rigidity when edge lengths are in {1,2}, but the analogous result for unit‑distance graphs remains out of reach. The authors remark that they could not even find any example of a globally rigid equilateral linkage larger than a triangle, making the existence question central to progress on the complexity classification.

Resolving this existence question would likely allow the methods in the paper to produce hardness results for global rigidity under unit‑length constraints, thereby settling the only unresolved cell in their summary table.