One of these nine results—that realization of unit-distance graphs is ∃ℝ-complete—was shown previously by Schaefer (2013), but the other eight are new. We strengthen several prior results. Matchstick graph realization was known to be NP-hard (Eades & Wormald 1990, or Cabello et al. 2007), but its membership in NP remained open; we show it is complete for the (possibly) larger class ∃ℝ. Global rigidity of graphs with edge lengths in {1, 2} was known to be coNP-hard (Saxe 1979); we show it is ∀ℝ-complete.
The majority of the paper is devoted to proving an analog of Kempe's Universality Theorem—informally, “there is a linkage to sign your name”—for globally noncrossing linkages. In particular, we show that any polynomial curve φ(x, y) = 0 can be traced by a noncrossing linkage, settling an open problem from 2004. More generally, we show that the nontrivial regions in the plane that may be traced by a noncrossing linkage are precisely the compact semialgebraic regions. Thus, no drawing power is lost by restricting to noncrossing linkages. We prove analogous results for matchstick linkages and unit-distance linkages as well.