We show that the second homology of the configuration spaces of a planar graph is generated under the operations of embedding, disjoint union, and edge stabilization by three atomic graphs: the cycle graph with one edge, the star graph with three edges, and the theta graph with four edges. We give an example of a non-planar graph for which this statement is false.

中文翻译：

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平面图辫群的二次同调

我们证明了平面图的配置空间的第二同调是在三个原子图的嵌入、不相交和边稳定操作下生成的：一个边的循环图、三边的星形图和 theta具有四个边的图。我们给出一个非平面图的例子，这个陈述是错误的。