ABSTRACT Computing the number of realizations of a minimally rigid graph is a notoriously difficult problem. Toward this goal, for graphs that are minimally rigid in the plane, we take advantage of a recently published algorithm, which is the fastest available method, although its complexity is still exponential. Combining computational results with the theory of constructing new rigid graphs by gluing, we give a new lower bound on the maximal possible number of (complex) realizations for graphs with a given number of vertices. We extend these ideas to rigid graphs in three dimensions and we derive similar lower bounds, by exploiting data from extensive Gröbner basis computations.

中文翻译：

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刚性图实现数量的下限

摘要 计算最小刚性图的实现数量是一个众所周知的难题。为了实现这个目标，对于平面中最小刚性的图，我们利用了最近发布的算法，这是最快的可用方法，尽管其复杂性仍然是指数级的。将计算结果与通过粘合构造新刚性图的理论相结合，我们给出了具有给定顶点数的图的最大可能（复杂）实现数的新下界。我们将这些想法扩展到三个维度的刚性图，并通过利用来自广泛 Gröbner 基础计算的数据得出类似的下界。