  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 non-rigid 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 globally 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 global rigid. Our proofs are based on discharging arguments.
