Minimal counterexamples to Hendrickson's conjecture on globally rigid graphs

TL;DR

This paper investigates minimal counterexamples to Hendrickson's conjecture on global rigidity, identifying the complete bipartite graph K_{5,5} as the smallest such counterexample through computational methods.

Contribution

The paper characterizes the smallest counterexamples to Hendrickson's conjecture, specifically proving K_{5,5} is minimal among graphs that are redundantly rigid, highly connected, but not globally rigid.

Findings

K_{5,5} is the smallest counterexample to the conjecture.

Few graphs in the class exist for a given number of vertices.

Computational evidence supports the minimality of K_{5,5}.

Abstract

In this paper we consider the class of graphs which are redundantly $d$-rigid and $(d+1)$-connected but not globally $d$-rigid, where $d$ is the dimension. This class arises from counterexamples to a conjecture by Bruce Hendrickson. It seems that there are relatively few graphs in this class for a given number of vertices. Using computations we show that $K_{5,5}$ is indeed the smallest counterexample to the conjecture.