A (2-dimensional) realisation of a graph G is a pair \((G,p)\), where \(p\) maps the vertices of \(G\) to \(\mathbb{R}^2\). A realisation is flexible if it can be continuously deformed while keeping the edge lengths fixed, and rigid otherwise. Similarly, a graph is flexible if its generic realisations are flexible, and rigid otherwise. We show that a minimally rigid graph has a flexible realisation with positive edge lengths if and only if it is not a 2-tree. This confirms a conjecture of Grasegger, Legersky and Schicho. Our proof is based on a characterisation of graphs with \(n\) vertices and \(2n-3\) edges and without stable cuts due to Le and Pfender. We also strengthen a result of Chen and Yu, who proved that every graph with at most \(2n-4\) edges has a stable cut, by showing that every flexible graph has a stable cut. Additionally, we investigate the number of NAC-colourings in various graphs. A NAC-colouring is a type of edge colouring introduced by Grasegger, Legersky and Schicho, who showed that the existence of such a colouring characterises the existence of a flexible realisation with positive edge lengths.

This is joint work with Clinch, Garamvolgyi, Haslegrave, Huynh and Legersky.

PDF abstract here.