Sufficient Conditions for the Global Rigidity of Graphs

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