Rigidity of Graph Joins and Hendrickson's Conjecture

TL;DR

This paper extends the understanding of graph rigidity by characterizing infinitesimal flexes in joined graphs, leading to new counterexamples to Hendrickson's conjecture in higher dimensions.

Contribution

It generalizes Whiteley's characterization to joined graphs, revealing new counterexamples to Hendrickson's conjecture in dimensions five and above.

Findings

Identifies new infinite families of counterexamples in
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Provides a generalized framework for analyzing graph rigidity

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Abstract

Whiteley \cite{wh} gives a complete characterization of the infinitesimal flexes of complete bipartite frameworks. Our work generalizes a specific infinitesimal flex to include joined graphs, a family of graphs that contain the complete bipartite graphs. We use this characterization to identify new families of counterexamples, including infinite families, in $\R^5$ and above to Hendrickson's conjecture on generic global rigidity.