A proof of Connelly's conjecture on 3-connected generic cycles

A graph G = (V;E) is called a generic cycle if jEj = 2jVj 2 and every X V with 2 j Xj j Vj 1 satises i(X) 2jXj 3. Here i(X) denotes the number of edges induced by X. The operation extension subdivides an edge uw of a graph by a new vertex v and adds a new edge vz for some vertex z 6 u;w. R. Connelly conjectured that every 3-connected generic cycle can be obtained from K4 by a sequence of extensions. We prove this conjecture. As a corollary, we also obtain a special case of a conjecture of Hendrickson on generically globally rigid graphs.