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摘要翻译：
D维框架是从其顶点到E^D的图和映射。如果这样的框架是E^D中唯一具有相同的图长和边长直到刚性运动的框架，那么它是全局刚性的。对于哪些底层图是全局刚性的通用框架？我们通过证明Connelly的一个猜想来回答这个问题，他的充分条件也是必要的：一个泛型框架是全局刚性的当且仅当它有一个核维数为D+1的应力矩阵，这是可能的最小值。该条件的另一个版本来自于考虑长度平方映射l的几何：图是一般局部刚性的iff,l的秩是极大的；图是一般全局刚性的iff,l的像上的Gauss映射的秩是极大的。我们还证明了这个条件可以用随机化算法有效地检验，并证明了如果一个图不是一般全局刚性的，那么它在一维以上是柔性的。
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英文标题：
《Characterizing Generic Global Rigidity》
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作者：
Steven J. Gortler, Alexander D. Healy, Dylan P. Thurston
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最新提交年份：
2010
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分类信息：
一级分类：Mathematics 数学
二级分类：Metric Geometry 度量几何学
分类描述：Euclidean, hyperbolic, discrete, convex, coarse geometry, comparisons in Riemannian geometry, symmetric spaces
欧氏，双曲，离散，凸，粗几何，黎曼几何的比较，对称空间
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一级分类：Mathematics 数学
二级分类：Algebraic Geometry 代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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一级分类：Mathematics 数学
二级分类：Combinatorics 组合学
分类描述：Discrete mathematics, graph theory, enumeration, combinatorial optimization, Ramsey theory, combinatorial game theory
离散数学，图论，计数，组合优化，拉姆齐理论，组合对策论
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英文摘要：
A d-dimensional framework is a graph and a map from its vertices to E^d. Such a framework is globally rigid if it is the only framework in E^d with the same graph and edge lengths, up to rigid motions. For which underlying graphs is a generic framework globally rigid? We answer this question by proving a conjecture by Connelly, that his sufficient condition is also necessary: a generic framework is globally rigid if and only if it has a stress matrix with kernel of dimension d+1, the minimum possible. An alternate version of the condition comes from considering the geometry of the length-squared mapping l: the graph is generically locally rigid iff the rank of l is maximal, and it is generically globally rigid iff the rank of the Gauss map on the image of l is maximal. We also show that this condition is efficiently checkable with a randomized algorithm, and prove that if a graph is not generically globally rigid then it is flexible one dimension higher.
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PDF链接：
https://arxiv.org/pdf/0710.0926
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