Global rigidity of (quasi-)injective frameworks on the line

A realization of a graph G is pair (G,p) where p maps the vertices into Euclidean space Rd. The injective if and quasi-injective for each edge G, endpoints to different points in space. globally rigid any (G,q) Rd with same lengths congruent (G,p). In this paper we characterize graphs that have an (quasi-injective, respectively) non-globally R1, show problem recognizing these NP-complete both cases. Our characterizations are based on notion NAC-colorings, which been used previously investigate similar problems plane. We also give overview related results open rigidity theory.