Sufficient conditions for the global rigidity of graphs

We investigate how to find generic and globally rigid realizations of graphs in RdRd based on elementary geometric observations. Our arguments lead to new proofs of a combinatorial characterization of the global rigidity of graphs in R2R2 by Jackson and Jordán and that of body-bar graphs in RdRd recently shown by Connelly, Jordán, and Whiteley. We also extend the 1-extension theorem and Connelly's composition theorem, which are main tools for generating globally rigid graphs in RdRd. In particular we show that any vertex-redundantly rigid graph in RdRd is globally rigid in RdRd, where a graph G=(V,E)G=(V,E) is called vertex-redundantly rigid if G−vG−v is rigid for any v∈Vv∈V.