The Union of Matroids and the Rigidity of Frameworks
The Union of Matroids and the Rigidity of Frameworks
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拟阵并集与框架的刚性
DOI：
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发表时间：
1988
影响因子：
0.8
通讯作者：
W. Whiteley
中科院分区：
文献类型：
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作者：
W. Whiteley
从图的刚性矩阵的形式出发，我们证明了一个图(或多重图)上的k-框架具有图的k个圈拟阵的并的拟阵结构。该矩阵模式被应用于关于框架刚度的三个中心结果。这个拟阵并的一个直接推论是n-空间中刚性杆和刚体框架的刻画(泰氏定理)。这进一步专门刻画了n维空间中躯干和铰链结构的独立性和刚性(一个新的定理)。两个框架或图形拟阵的两个副本的并集被截断以产生平面杆和关节框架，该框架给出了平面上极小无限刚性杆和关节框架的特征(拉曼定理)。最后，利用图的圈拟阵和自行车拟阵的拟阵并，用这些技巧刻画了平面环面、圆柱面、圆锥面等平面上无限刚架的图。
From the pattern of its rigidity matrix, we show that a k-frame on a graph (or multigraph) has the matroid structure of the union of k copies of the cycle matroid of the graph. This matrix pattern is applied to three central results about the rigidity of frameworks. An immediate corollary of this matroid union is a characterization of rigid bar and body frameworks in n-space (Tay’s Theorem). This is further specialized to characterize the independence and the rigidity of body and hinge structures in n-space (a new theorem). The two-frame, or union of two copies of the graphic matroid, is truncated to produce plane bar and joint frameworks giving a characterization of minimal infinitesimally rigid bar and joint frameworks in the plane (Laman’s Theorem). Finally, these techniques are used to characterize the graphs of infinitesimally rigid frameworks on other surfaces, such as the flat torus, the cylinder, cones, etc., using matroid unions of cycle and bicycle matroids of the graph.