We prove that the Weisfeiler--Leman (WL) dimension of the class of all finite planar graphs is at most 3. In particular, every finite planar graph is definable in first-order logic with counting using at most 4 variables. The previously best-known upper bounds for the dimension and number of variables were 14 and 15, respectively. First, we show that, for dimension 3 and higher, the WL-algorithm correctly tests isomorphism of graphs in a minor-closed class whenever it determines the orbits of the automorphism group of every arc-colored 3-connected graph belonging to this class. Then, we prove that, apart from several exceptional graphs (which have WL-dimension at most 2), the individualization of two appropriately chosen vertices of a colored 3-connected planar graph followed by the one-dimensional WL-algorithm produces the discrete vertex partition. This implies that the three-dimensional WL-algorithm determines the orbits of arc-colored 3-connected planar graphs. As a byproduct of the proof, we get a classification of the 3-connected planar graphs with fixing number 3.

中文翻译：

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Weisfeiler--平面图的Leman维数最多为3

我们证明了所有有限平面图的类的 Weisfeiler-Leman (WL) 维数最多为 3。特别是，每个有限平面图都可以在一阶逻辑中定义，最多使用 4 个变量进行计数。以前最知名的变量维度和数量的上限分别为 14 和 15。首先，我们表明，对于 3 维和更高维，只要 WL 算法确定属于该类的每个弧色 3 连通图的自同构群的轨道，它就正确地测试了次闭类中图的同构。然后，我们证明，除了几个异常图（WL 维数最多为 2），the individualization of two appropriately chosen vertices of a colored 3-connected planar graph followed by the one-dimensional WL-algorithm produces the discrete vertex partition. 这意味着三维 WL 算法确定弧色 3 连通平面图的轨道。作为证明的副产品，我们得到了固定数为 3 的 3 连通平面图的分类。