Rigidity theory studies the properties of graphs that can have rigid embeddings in a euclidean space ${\mathbb{R}}^d$ or on a sphere and other manifolds which in addition satisfy certain edge length constraints. One of the major open problems in this field is to determine lower and upper bounds on the number of realizations with respect to a given number of vertices. This problem is closely related to the classification of rigid graphs according to their maximal number of real embeddings.

In this paper, we are interested in finding edge lengths that can maximize the number of real embeddings of minimally rigid graphs in the plane, space, and on the sphere. We use algebraic formulations to provide upper bounds. To find values of the parameters that lead to graphs with a large number of real realizations, possibly attaining the (algebraic) upper bounds, we use some standard heuristics and we also develop a new method inspired by coupler curves. We apply this new method to obtain embeddings in ${\mathbb{R}}^3$. One of its main novelties is that it allows us to sample efficiently from a larger number of parameters by selecting only a subset of them at each iteration.

Our results include a full classification of the 7-vertex graphs according to their maximal numbers of real embeddings in the cases of the embeddings in ${\mathbb{R}}^2$ and ${\mathbb{R}}^3$, while in the case of $S^2$ we achieve this classification for all 6-vertex graphs. Additionally, by increasing the number of embeddings of selected graphs, we improve the previously known asymptotic lower bound on the maximum number of realizations. The methods and the results concerning the spatial embeddings are part of the proceedings of ISSAC 2018 (Bartzos et al., 2018).

刚度理论研究在欧式空间中可以具有刚性嵌入的图的性质 ${\mathbb{[R}}^d$或者在球体和其他满足某些边长限制的歧管上。该领域的主要开放问题之一是相对于给定数量的顶点确定实现数量的上限和下限。根据实际图的最大实际嵌入数量，此问题与刚性图的分类密切相关。

在本文中，我们对寻找可以使平面，空间和球面上的最小刚度图的实际嵌入数最大化的边长感兴趣。我们使用代数公式来提供上限。为了找到导致具有大量实际实现的图（可能达到（代数）上限）的参数值，我们使用了一些标准试探法，并且还开发了一种受耦合器曲线启发的新方法。我们应用这种新方法来获取嵌入${\mathbb{[R}}^3$。它的主要新颖之处之一是它允许我们通过在每次迭代中仅选择一部分参数来从大量参数中进行有效采样。

我们的结果包括7个顶点图的完全分类，具体取决于在嵌入情况下它们的最大实际嵌入数。 ${\mathbb{[R}}^2$ 和 ${\mathbb{[R}}^3$，而在 ${\mathrm{小号}}^2$我们为所有6个顶点图实现了这种分类。另外，通过增加选定图的嵌入数量，我们改善了最大实现数上的先前已知的渐近下界。有关空间嵌入的方法和结果是ISSAC 2018程序的一部分（Bartzos et al。，2018）。