Robert Connelly

Generically globally rigid graphs have generic universally rigid\n frameworks

COMBINATORICA · 2016 · 11 citations

We show that any graph that is generically globally rigid in $\\mathbb{R}^d$\nhas a realization in $\\mathbb{R}^d$ that is both generic and universally rigid.\nThis also implies that the graph also must have a realization in $\\mathbb{R}^d$\nthat is both infinitesimally rigid and universally rigid; such a realization\nserves as a certificate of generic global rigidity.\n Our approach involves an algorithm by Lov\\'asz, Saks and Schrijver that, for\na sufficiently connected graph, constructs a ge…

Reconstruction in One Dimension from Unlabeled Euclidean Lengths

COMBINATORICA · 2024-07-11 · 2 citations

Abstract Let G be a 3-connected ordered graph with n vertices and m edges. Let $$\textbf{p}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>p</mml:mi> </mml:math> be a randomly chosen mapping of these n vertices to the integer range $$\{1, 2,3, \ldots , 2^b\}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>{</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>3</mml:mn> <mml:mo>,</mml:mo> <mml:mo>…</mml:m…

Rigidity of Circle Packings with Flexible Radii

arXiv (Cornell University) · 2022-06-14 · 1 citations

Circle packings are arrangement of circles satisfying specified tangency requirements. Many problems about packing of circles and spheres occur in nature particularly in material design and protein structure. Surprisingly, little is known about the stability and rigidity of circle packings. In this paper, we study the rigidity of circle packings representing a given planar graph. The radii of circles are flexible with equality and inequality constraints. We provide a dual condition for the packi…

Spiderwebs on the Sphere and an Isoperimetric Theorem

ArXiv.org · 2025-05-28

Here we present a rigidity result in a global (semi-global, homotopy) setting for a restrictive class of polytopes, those that can be inscribed in a unit sphere, with some additional conditions. The proof of the rigidity result for cabled frameworks on the surface of the sphere uses classical isoperimetric ideas.

Geometric Rigidity Theory

Cambridge University Press eBooks · 2025-05-03