Operations preserving the global rigidity of graphs and frameworks in the plane

A straight-line realization of (or a bar-and-joint framework on) graph G in R^d is said to be globally rigid if it is congruent to every other realization of G with the same edge lengths. A graph G is called globally rigid in R^d if every generic realization of G is globally rigid. We give an algorithm for constructing a globally rigid realization of globally rigid graphs in R^2. If G is triangle-reducible, which is a subfamily of globally rigid graphs that includes Cauchy graphs as well as Grunbaum graphs, the constructed realization will also be infinitesimally rigid. Our algorithm relies on the inductive construction of globally rigid graphs which uses edge additions and one of the Henneberg operations. We also show that vertex splitting, which is another well-known operation in combinatorial rigidity, preserves global rigidity in R^2.