SUFFICIENT CONDITIONS FOR 2-DIMENSIONAL GLOBAL RIGIDITY

The 2-dimensional global rigidity has been shown to be equivalent to 3-connectedness and redundant rigidity by a combination of two results due to Jackson and Jord\'an, and Connelly, respectively. By the characterization, a theorem of Lov\'asz and Yemini implies that every 6-connected graph is redundantly rigid and thus globally rigid. The 6-connectedness is best possible, since there exist infinitely many 5-connected nonrigid graphs. Jackson, Servatius, and Servatius used the idea of ``essential connectivity"" and proved that every 4-connected ``essentially 6-connected"" graph is redundantly rigid and thus global rigid. Since 3-connectedness is a necessary condition of global rigidity, it is interesting to study 3-connected graphs for redundant rigidity and thus global rigidity. We utilize a different ``essential connectivity"" and prove that every 3-connected essentially 9-connected graph is redundantly rigid and thus globally rigid. The essential 9-connectedness is best possible. Under this essential connectivity, we also prove that every 4-connected essentially 6-connected graph is redundantly rigid and thus globally rigid. Our proofs are based on discharging arguments.