New Classes of Counterexamples to Hendrickson’s Global Rigidity Conjecture

Authors:
Samuel Frank;Jiayang Jiang
Affiliations:
Columbia University, Department of Mathematics, 10027, New York, NY, USA;Columbia University, Department of Mathematics, 10027, New York, NY, USA
Venue:
Discrete & Computational Geometry
Year:
2011

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Generic global rigidity of body-bar frameworks

Journal of Combinatorial Theory Series B

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Abstract

We examine the generic local and global rigidity of various graphs in ℝd . Bruce Hendrickson showed that some necessary conditions for generic global rigidity are (d+1)-connectedness and generic redundant rigidity, and hypothesized that they were sufficient in all dimensions. We analyze two classes of graphs that satisfy Hendrickson’s conditions for generic global rigidity, yet fail to be generically globally rigid. We find a large family of bipartite graphs for d3, and we define a construction that generates infinitely many graphs in ℝ5. Finally, we state some conjectures for further exploration.