Globally Rigid Augmentation of Minimally Rigid Graphs in $$\mathbb {R}^2$$

The two main concepts of Rigidity Theory are rigidity, where the framework has no continuous deformation, and global rigidity, where the given distance set determines the locations of the points up to isometry. We consider the following augmentation problem. Given a minimally rigid graph \(G=(V,E)\) in \(\mathbb {R}^2\), find a minimum cardinality edge set F such that the graph \(G'=(V,E+F)\) is globally rigid in \(\mathbb {R}^2\). We provide a min-max theorem and an \(O(|V|^2)\) time algorithm for this problem.