Birigidity in the Plane

The two-dimensional generic rigidity matroid $R(G)$ of a graph G is considered. The notions of vertex and edge birigidity are introduced. It is proved that vertex birigidity of G implies the connectivity of $R(G)$ and that the connectivity of $R(G)$ implies the edge birigidity of G. These implications are not equivalences.A class of minimal vertex birigid graphs is exhibited and used to show that $R(G)$ is not representable over any finite field.