New Classes of Counterexamples to Hendrickson's Global Rigidity Conjecture

TL;DR

This paper identifies new classes of graphs that satisfy Hendrickson's necessary conditions for global rigidity but are not globally rigid, challenging previous conjectures in rigidity theory.

Contribution

It introduces specific counterexamples in bipartite graphs and a construction method for R^5, providing new insights into the limitations of Hendrickson's conditions.

Findings

Found bipartite graphs satisfying Hendrickson's conditions but not globally rigid

Developed a construction generating infinitely many non-rigid graphs in R^5

Proposed conjectures for future research in rigidity theory

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 R^5. Finally, we state some conjectures for further exploration.