Generic Rigidity

Definition 14 (Generic realization and framework)

Let \(G\) be a graph. A \(d\)-dimensional realization \(p\) of \(G\) whose coordinates are algebraically independent is called generic. A framework \((G, p)\), where \(p\) is generic, is called a generic framework.

Definition 15 (Generically rigid graph)

Let \(G\) be a graph and \(d \in \NN\). The graph \(G\) is called (generically) \(d\)-rigid if any generic d-dimensional framework \((G, p)\) is rigid; this is equivalent to \((G, p)\) being infinitesimally rigid.

PyRigi: is_rigid()

Definition 16 (Minimally generically rigid graphs)

Let \(G\) be a graph, let \(d, k \in \NN\). The graph \(G\) is called minimally (generically) \(d\)-rigid if a (equivalently, any) generic framework \((G, p)\) is minimally (infinitesimally) d-rigid.

PyRigi: is_min_rigid()

Theorem 1

A graph \(G = (V, E)\) is minimally (generically) \(2\)-rigid if and only if \(G\) is (2,3)-tight.

References: [PollaczekGeiringer27] [Lam70]