Sufficient conditions for the global rigidity of graphs

We investigate how to find generic and globally rigid realizations of graphs in R-d based on elementary geometric observations. Our arguments lead to new proofs of a combinatorial characterization of the global rigidity of graphs in R-2 by Jackson and Jordan and that of body-bar graphs in R-d recently shown by Connelly, Jordan, and Whiteley. We also extend the 1-extension theorem and Connelly's composition theorem, which are main tools for generating globally rigid graphs in R-d. In particular we show that any vertex-redundantly rigid graph in R-d is globally rigid in R-d, where a graph G = (V, E) is called vertex-redundantly rigid if G - v is rigid for any v is an element of V. (C) 2015 Elsevier Inc. All rights reserved.