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 Jordán and that of body-bar graphs in ${\mathbb{R}}^d$ 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 ${\mathbb{R}}^d$. 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 $v\in V$.

References in corpus (2)

• The number of equivalent realisations of a rigid graph
• Combining Globally Rigid Frameworks