Birigidity in the Plane

We consider the 2-dimensional generic rigidity matroid R(G) of a graph G. The notions of vertex and edge birigidity are introduced. We prove 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.