New Classes of Counterexamples to Hendrickson's Global Rigidity Conjecture

Rigidity of Graph Joins and Hendrickson's Conjecture

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.

Necessary Conditions for the Generic Global Rigidity of Frameworks on Surfaces

A result due in its various parts to Hendrickson, Connelly, and Jackson and Jordán, provides a purely combinatorial characterisation of global rigidity for generic bar-joint frameworks in $\mathbb{R}^2$. The analogous conditions are known to be insufficient to characterise generic global rigidity in higher dimensions. Recently Laman-type characterisations of rigidity have been obtained for generic frameworks in $\mathbb{R}^3$ when the vertices are constrained to lie on various surfaces, such as the cylinder and the cone. In this paper we obtain analogues of Hendrickson's necessary conditions f…

Global Rigidity of Periodic Graphs under Fixed-lattice Representations

In 1992, Hendrickson proved that (d+1)-connectivity and redundant rigidity are necessary conditions for a generic (non-complete) bar-joint framework to be globally rigid in $\mathbb{R}^d$. Jackson and Jordan confirmed in 2005 that these conditions are also sufficient in $\mathbb{R}^2$, giving a combinatorial characterization of graphs whose generic realizations in $\mathbb{R}^2$ are globally rigid. In this paper, we establish analogues of these results for infinite periodic frameworks under fixed lattice representations. Our combinatorial characterization of globally rigid generic periodic fra…

Global rigidity of 2-dimensional linearly constrained frameworks

A linearly constrained framework in $\mathbb{R}^d$ is a point configuration together with a system of constraints which fixes the distances between some pairs of points and additionally restricts some of the points to lie in given affine subspaces. It is globally rigid if the configuration is uniquely defined by the constraint system, and is rigid if it is uniquely defined within some small open neighbourhood. Streinu and Theran characterised generic rigidity of linearly constrained frameworks in $\mathbb{R}^2$ in 2010. We obtain an analagous characterisation for generic global rigidity in $\m…

多面体赋范空间中的唯一可实现图

Uniquely realisable graphs in polyhedral normed spaces

本文研究多面体赋范空间中框架的全局刚性问题，提出了判定算法、通用框架的必要条件，并在$\ell_\infty^d$空间中获得了更强的充分条件和二维精确刻画。

关于直线上一般意义下的泛刚性

On generic universal rigidity on the line

本文研究一维情形下的一般意义下泛刚性图，通过构造反例否定Connelly的猜想性刻画，并引入两种保持或破坏泛刚性的新操作，从而得到泛刚性图规模的下界，部分回答了Jordán和Nguyen的公开问题。