Generically globally rigid zeolites in the plane

A d-dimensional zeolite is a d-dimensional body-and-pin framework with a (d+1)-regular underlying graph G. That is, each body of the zeolite is incident with d+1 pins and each pin belongs to exactly two bodies. The corresponding d-dimensional combinatorial zeolite is a bar-and-joint framework whose graph is the line graph of G. We show that a two-dimensional combinatorial zeolite is generically globally rigid if and only if its underlying 3-regular graph G is 3-edge-connected. The proof is based on a new rank formula for the two-dimensional rigidity matroid of line graphs.