Sufficient Conditions for the Global Rigidity of Graphs

TL;DR

This paper provides new geometric proofs and extensions of key theorems for determining when graphs are globally rigid in Euclidean spaces, enhancing understanding of graph rigidity conditions.

Contribution

It offers new proofs of known characterizations, extends fundamental theorems, and establishes that vertex-redundant rigidity implies global rigidity in any dimension.

Findings

New proofs of graph rigidity characterizations in e^2 and e^d.

Extended 1-extension and composition theorems for global rigidity.

Vertex-redundant rigidity implies global rigidity in e^d.

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 $v\in V$.