Sufficient conditions for 2-dimensional global rigidity

TL;DR

This paper establishes new sufficient conditions based on essential connectivity for 2-dimensional global rigidity, showing that certain highly connected graphs are redundantly rigid and globally rigid.

Contribution

It introduces a novel essential connectivity criterion, proving that 3-connected essentially 9-connected graphs are globally rigid, which is optimal.

Findings

3-connected essentially 9-connected graphs are redundantly rigid and globally rigid

4-connected essentially 6-connected graphs are redundantly rigid and globally rigid

Essential connectivity thresholds are proven to be optimal

Abstract

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.