A Linear Upper Bound on the Weisfeiler-Leman Dimension of Graphs of Bounded Genus

The Weisfeiler-Leman (WL) dimension of a graph is a measure for the inherent descriptive complexity of the graph. While originally derived from a combinatorial graph isomorphism test called the Weisfeiler-Leman algorithm, the WL dimension can also be characterised in terms of the number of variables that is required to describe the graph up to isomorphism in first-order logic with counting quantifiers. It is known that the WL dimension is upper-bounded for all graphs that exclude some fixed graph as a minor [M. Grohe, 2017]. However, the bounds that can be derived from this general result are astronomic. Only recently, it was proved that the WL dimension of planar graphs is at most 3 [S. Kiefer et al., 2017]. In this paper, we prove that the WL dimension of graphs embeddable in a surface of Euler genus g is at most 4g+3. For the WL dimension of graphs embeddable in an orientable surface of Euler genus g, our approach yields an upper bound of 2g + 3

The Power of the Weisfeiler-Leman Algorithm to Decompose Graphs

The Weisfeiler-Leman procedure is a widely-used approach for graph isomorphism testing that works by iteratively computing an isomorphism-invariant coloring of vertex tuples. Meanwhile, a fundamental tool in structural graph theory, which is often exploited in approaches to tackle the graph isomorphism problem, is the decomposition into 2- and 3-connected components. We prove that the 2-dimensional Weisfeiler-Leman algorithm implicitly computes the decomposition of a graph into its 3-connected components. Thus, the dimension of the algorithm needed to distinguish two given graphs is at most the dimension required to distinguish the corresponding decompositions into 3-connected components (assuming it is at least 2). This result implies that for k >= 2, the k-dimensional algorithm distinguishes k-separators, i.e., k-tuples of vertices that separate the graph, from other vertex k-tuples. As a byproduct, we also obtain insights about the connectivity of constituent graphs of association schemes. In an application of the results, we show the new upper bound of k on the Weisfeiler-Leman dimension of graphs of treewidth at most k. Using a construction by Cai, F\"urer, and Immerman, we also provide a new lower bound that is asymptotically tight up to a factor of 2.Comment: 30 pages, 4 figures, full version of a paper accepted at MFCS 201

Canonisation and Definability for Graphs of Bounded Rank Width

We prove that the combinatorial Weisfeiler-Leman algorithm of dimension $(3k+4)$ is a complete isomorphism test for the class of all graphs of rank width at most $k$. Rank width is a graph invariant that, similarly to tree width, measures the width of a certain style of hierarchical decomposition of graphs; it is equivalent to clique width. It was known that isomorphism of graphs of rank width $k$ is decidable in polynomial time (Grohe and Schweitzer, FOCS 2015), but the best previously known algorithm has a running time $n^{f(k)}$ for a non-elementary function $f$. Our result yields an isomorphism test for graphs of rank width $k$ running in time $n^{O(k)}$. Another consequence of our result is the first polynomial time canonisation algorithm for graphs of bounded rank width. Our second main result is that fixed-point logic with counting captures polynomial time on all graph classes of bounded rank width.Comment: 32 page

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 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 WL-dimension 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 WL-algorithm 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.Comment: 34 pages, 3 figures, extended version of LICS 2017 pape

Logarithmic Weisfeiler--Leman and Treewidth

In this paper, we show that the $(3k+4)$-dimensional Weisfeiler--Leman algorithm can identify graphs of treewidth $k$ in $O(\log n)$ rounds. This improves the result of Grohe & Verbitsky (ICALP 2006), who previously established the analogous result for $(4k+3)$-dimensional Weisfeiler--Leman. In light of the equivalence between Weisfeiler--Leman and the logic $\textsf{FO} + \textsf{C}$ (Cai, F\"urer, & Immerman, Combinatorica 1992), we obtain an improvement in the descriptive complexity for graphs of treewidth $k$. Precisely, if $G$ is a graph of treewidth $k$, then there exists a $(3k+5)$-variable formula $\varphi$ in $\textsf{FO} + \textsf{C}$ with quantifier depth $O(\log n)$ that identifies $G$ up to isomorphism

Incorporating Weisfeiler-Leman into algorithms for group isomorphism

In this paper we combine many of the standard and more recent algebraic techniques for testing isomorphism of finite groups (GpI) with combinatorial techniques that have typically b

Improving Expressivity of Graph Neural Networks using Localization

In this paper, we propose localized versions of Weisfeiler-Leman (WL) algorithms in an effort to both increase the expressivity, as well as decrease the computational overhead. We focus on the specific problem of subgraph counting and give localized versions of $k-$WL for any $k$. We analyze the power of Local $k-$WL and prove that it is more expressive than $k-$WL and at most as expressive as $(k+1)-$WL. We give a characterization of patterns whose count as a subgraph and induced subgraph are invariant if two graphs are Local $k-$WL equivalent. We also introduce two variants of $k-$WL: Layer $k-$WL and recursive $k-$WL. These methods are more time and space efficient than applying $k-$WL on the whole graph. We also propose a fragmentation technique that guarantees the exact count of all induced subgraphs of size at most 4 using just $1-$WL. The same idea can be extended further for larger patterns using $k>1$. We also compare the expressive power of Local $k-$WL with other GNN hierarchies and show that given a bound on the time-complexity, our methods are more expressive than the ones mentioned in Papp and Wattenhofer[2022a]

Logarithmic Weisfeiler-Leman Identifies All Planar Graphs

The Weisfeiler-Leman (WL) algorithm is a well-known combinatorial procedure for detecting symmetries in graphs and it is widely used in graph-isomorphism tests. It proceeds by iteratively refining a colouring of vertex tuples. The number of iterations needed to obtain the final output is crucial for the parallelisability of the algorithm. We show that there is a constant k such that every planar graph can be identified (that is, distinguished from every non-isomorphic graph) by the k-dimensional WL algorithm within a logarithmic number of iterations. This generalises a result due to Verbitsky (STACS 2007), who proved the same for 3-connected planar graphs. The number of iterations needed by the k-dimensional WL algorithm to identify a graph corresponds to the quantifier depth of a sentence that defines the graph in the (k+1)-variable fragment C^{k+1} of first-order logic with counting quantifiers. Thus, our result implies that every planar graph is definable with a C^{k+1}-sentence of logarithmic quantifier depth

The Iteration Number of the Weisfeiler-Leman Algorithm

We prove new upper and lower bounds on the number of iterations the $k$-dimensional Weisfeiler-Leman algorithm ($k$-WL) requires until stabilization. For $k \geq 3$, we show that $k$-WL stabilizes after at most $O(kn^{k-1}\log n)$ iterations (where $n$ denotes the number of vertices of the input structures), obtaining the first improvement over the trivial upper bound of $n^{k}-1$ and extending a previous upper bound of $O(n \log n)$ for $k=2$ [Lichter et al., LICS 2019]. We complement our upper bounds by constructing $k$-ary relational structures on which $k$-WL requires at least $n^{\Omega(k)}$ iterations to stabilize. This improves over a previous lower bound of $n^{\Omega(k / \log k)}$ [Berkholz, Nordstr\"{o}m, LICS 2016]. We also investigate tradeoffs between the dimension and the iteration number of WL, and show that $d$-WL, where $d = \lceil\frac{3(k+1)}{2}\rceil$, can simulate the $k$-WL algorithm using only $O(k^2 \cdot n^{\lfloor k/2\rfloor + 1} \log n)$ many iterations, but still requires at least $n^{\Omega(k)}$ iterations for any $d$ (that is sufficiently smaller than $n$). The number of iterations required by $k$-WL to distinguish two structures corresponds to the quantifier rank of a sentence distinguishing them in the $(k + 1)$-variable fragment $C_{k+1}$ of first-order logic with counting quantifiers. Hence, our results also imply new upper and lower bounds on the quantifier rank required in the logic $C_{k+1}$, as well as tradeoffs between variable number and quantifier rank.Comment: 30 pages, 1 figure, full version of a paper accepted at LICS 2023; second version improves the presentation of the result