New Classes of Counterexamples to Hendrickson's Global Rigidity Conjecture

2009/09/15 by Samuel Frank, Frank, Samuel, Jiayang Jiang +1 · 1 citation
Engineering · #Advanced Materials and Mechanics #Combinatorics (math.CO) #Dielectric materials and actuators #FOS: Mathematics #Metric Geometry (math.MG) #Structural Analysis and Optimization

paper · pdf · doi:10.48550/arxiv.0909.2893

openalex publication_date 2009/09/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We examine the generic local and global rigidity of various graphs in R^d. 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 d > 3, and we define a construction that generates infinitely many graphs in R⁵. Finally, we state some conjectures for further exploration.

Citations

• Conditions for Unique Graph Realizations

Cited by

• Rigidity of Graphs and Frameworks: A Matroid Theoretic Approach

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