The Weisfeiler--Leman Dimension of Planar Graphs Is at Most 3

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 variabl...

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Veröffentlicht in:Journal of the ACM 2019-12, Vol.66 (6), p.1-31, Article 44
Hauptverfasser: Kiefer, Sandra, Ponomarenko, Ilia, Schweitzer, Pascal
Format: Artikel
Sprache:eng
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Zusammenfassung: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.
ISSN:0004-5411
1557-735X
DOI:10.1145/3333003