The Weisfeiler-Leman dimension of planar graphs is at most 3

Abstract

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 any arc-colored 3-connected graph belonging to this class.Then we prove that, apart from several exceptional graphs (which have WLdimension at most 2), the individualization of two correctly chosen vertices of a colored 3-connected planar graph followed by the 1-dimensional WL-algorithm produces the discrete vertex partition. This implies that the 3-dimensional WLalgorithm determines the orbits of a colored 3-connected planar graph.As a byproduct of the proof, we get a classification of the 3-connected planar graphs with fixing number 3. This is an extended version of the paper with the same title published in the

The Power of the Weisfeiler--Leman Algorithm to Decompose Graphs

“…More specifically, we prove that already the 2-dimensional WL algorithm distinguishes separating pairs, i.e., pairs of vertices that separate the given graph, from other vertex pairs. This improves on a result from [29], where an analogous statement was proved for the 3-dimensional WL algorithm. Using the decomposition techniques discussed there, we conclude that for the k-dimensional WL algorithm with k ≥ 2, to identify a graph, it suffices to determine vertex orbits on all arc-colored 3-connected components of it.…”

“…The expressive power of the k-dimensional algorithm corresponds to definability in the logic C k+1 , the extension of the (k + 1)-variable fragment of first-order logic by counting quantifiers [9,27]. Exploiting this correspondence, our results imply that for every n ∈ N, there is a formula ϕ n (x 1 , x 2 ) ∈ C 3 (first-order logic with counting quantifiers over three variables) such that for an n-vertex graph G, it holds that G |= ϕ n (v, w) if and only if {v, w} is a 2-separator in G. With only three variables at our disposal, it is not possible to take the route of [29] by comparing certain numbers of walks between different pairs of vertices. Instead, the formulas obtained from our proof are essentially a disjunction over all n-vertex graphs and subformulas for two distinct graphs may look completely different, exploiting specific structural properties of the graphs.…”

The graph isomorphism problem

“…Explicit bounds on the WL dimension are known for specific graph classes. Most notably, Kiefer, Ponomarenko, and Schweitzer [51] proved that the WL dimension of planar graphs is at most 3. The example of two triangles vs a cycle of length 6 shows that it is at least 2.…”

Canonisation and Definability for Graphs of Bounded Rank Width

“…Another way of stating Theorem 1.1 is that the Weisfeiler-Leman (WL) dimension [18] of graphs of rank width k is at most 3k + 4. While it is known that many natural graph classes have bounded WL dimension, among them the class of planar graphs [16,30], classes of bounded genus [17], bounded tree width [19], classes of graphs excluding some fixed graph as a minor [18], and interval graphs [31,33], all these except for the class of interval graphs are classes of sparse graphs (with an edge number linear in the number of vertices). Our result adds a rich family of classes that include dense graphs to the picture.…”