Realizability of Graphs and Linkages

We show that deciding whether a graph with given edge lengths can be realized by a straight-line drawing has the same complexity as deciding the truth of sentences in the existential theory of the real numbers, ETR; we introduce the class ∃R that captures the computational complexity of ETR and many other problems. The graph realizability problem remains ∃R-complete if all edges have unit length, which implies that recognizing unit distance graphs is ∃R-complete. We also consider the problem for linkages: in a realization of a linkage vertices are allowed to overlap and lie on the interior of edges. Linkage realizability is ∃R-complete and remains so if all edges have unit length. A linkage is called rigid if any slight perturbation of its vertices which does not break the linkage (i.e. keeps edge-lengths the same) is the result of a rigid motion of the plane. Testing whether a configuration is not rigid is ∃R-complete.