Colour Refinement is a combinatorial algorithm that computes a vertex colouring for an input graph to reveal its structural properties. Each iteration of the algorithm refines the current colouring by assessing local information. More precisely, the new colour of a vertex is determined by its current colour and the multiset of colours in its neighbourhood. This refinement procedure continues until it reaches a stable partition of the vertex set into colour classes. On the practical side, the algorithm admits fast implementations. Because the final colouring is isomorphism-invariant, executing the algorithm on two graphs in parallel can be used to demonstrate that they are not isomorphic. From a theoretical perspective, the algorithm is arguably the most straightforward combinatorial approach to detecting asymmetries - specifically for distinguishing vertices that do not belong to the same orbit of the automorphism group of the graph. Its numerous connections to other areas in computer science stand as evidence of its robustness and naturalness and make it a fascinating object of research. Among the most elegant connections is the link to counting logic. Colour Refinement assigns distinct final colours to two vertices in a graph if and only if there is a formula in the two-variable fragment C² of the logic C that distinguishes them, meaning that the formula holds for precisely one of the two vertices. In fact, the vertex colours translate directly into logical formulas with one free variable. As a consequence, Colour Refinement distinguishes two graphs if and only if there is a C²-sentence that distinguishes them. This correspondence extends to higher dimensions: the k-variable fragment C^k of C corresponds to the (k-1)-dimensional extension of Colour Refinement, the (k-1)-dimensional Weisfeiler-Leman algorithm. This algorithm computes a unique colouring for a graph G if and only if G is definable in C^k, i.e. there is a sentence in C^k whose only models are G and its isomorphic copies. As a matter of fact, the link to the logic C goes even deeper: the number of Colour Refinement iterations required to compute distinct colours corresponds exactly to the quantifier depth of a distinguishing formula. Since the iterations induce a sequence of strictly nested vertex partitions, the process must terminate after at most n-1 rounds, where n is the number of vertices. Consequently, the value n-1 serves as a trivial upper bound on both the number of iterations and the quantifier depth required to distinguish any two (distinguishable) vertices in C². My talk provides an introduction to the link between the Colour Refinement procedure and the logic C². We revisit a simple characterisation of their expressivity on graphs and on general relational structures. The characterisation implies that the definability of a graph in C² can be checked very efficiently. We then discuss tight lower bounds on the quantifier depth of C²-formulas required to distinguish vertices. Through a thorough analysis of computational data from Colour Refinement executions, we constructed infinite families of graphs that witness those bounds. We finish with a presentation of a recent purely theoretical reverse-engineering approach to finding long-refinement graphs and a classification of all such graphs with small (or, equivalently, large) degrees. The talk is based on the collaborations[Sandra Kiefer and T. Devini de Mel, 2026; Kiefer and McKay, 2020; Sandra Kiefer et al., 2022] and unpublished work.

The Colour Refinement algorithm is a classical procedure to detect symmetries in graphs, whose most prominent application is in graph-isomorphism tests. The algorithm and its generalisation, the Weisfeiler–Leman algorithm, evaluate local information to compute a colouring for the vertices in an iterative fashion. Different final colours of two vertices certify that no isomorphism can map one onto the other.

We propose a scalable framework for deciding, proving, and explaining (in-)equivalence of context-free grammars. We present an implementation of the framework and evaluate it on large data sets collected within educational support systems. Even though the equivalence problem for context-free languages is undecidable in general, the framework is able to handle a large portion of these datasets. It introduces and combines techniques from several areas, such as an abstract grammar transformation language to identify equivalent grammars as well as sufficiently similar inequivalent grammars, theory-based comparison algorithms for a large class of context-free languages, and a graph-theory-inspired grammar canonization that allows to efficiently identify isomorphic grammars.

Population protocols are a well-studied model of distributed computation in which a group of anonymous finite-state agents communicates via pairwise interactions. Together they decide whether their initial configuration, i. e., the initial distribution of agents in the states, satisfies a property. As an extension in order to express properties of multisets over an infinite data domain, Blondin and Ladouceur (ICALP'23) introduced population protocols with unordered data (PPUD). In PPUD, each agent carries a fixed data value, and the interactions between agents depend on whether their data are equal or not. Blondin and Ladouceur also identified the interesting subclass of immediate observation PPUD (IOPPUD), where in every transition one of the two agents remains passive and does not move, and they characterised its expressive power. We study the decidability and complexity of formally verifying these protocols. The main verification problem for population protocols is well-specification, that is, checking whether the given PPUD computes some function. We show that well-specification is undecidable in general. By contrast, for IOPPUD, we exhibit a large yet natural class of problems, which includes well-specification among other classic problems, and establish that these problems are in ExpSpace. We also provide a lower complexity bound, namely coNExpTime-hardness.

The Weisfeiler-Leman (WL) dimension is an established measure for the inherent descriptive complexity of graphs and relational structures. It corresponds to the number of variables that are needed and sufficient to define the object of interest in a counting version of first-order logic (FO). These bounded-variable counting logics were even candidates to capture graph isomorphism, until a celebrated construction due to Cai, Fürer, and Immerman [Combinatorica 1992] showed that Ω(n) variables are required to distinguish all non-isomorphic n-vertex graphs.

The fixed-point logic LREC=was developed by Grohe et al. (CSL 2011) in the quest for a logic to capture all problems decidable in logarithmic space. It extends FO+C, first-order logic with counting, by an operator that formalises a limited form of recursion. We show that for every LREC=-definable property on relational structures, there is a constant k such that the k-variable fragment of first-order logic with counting quantifiers expresses the property via formulae of logarithmic quantifier depth. This yields that any pair of graphs separable by the property can be distinguished with the k-dimensional Weisfeiler–Leman algorithm in a logarithmic number of iterations. In particular, it implies that a constant dimension of the algorithm identifies every interval graph and every chordal claw-free graph in logarithmically many iterations, since every such graph admits LREC=-definable canonisation.

The Weisfeiler-Leman (WL) algorithm is a combinatorial procedure that computes colorings on graphs, which can often be used to detect their (non-)isomorphism. Particularly the 1- and 2-dimensional versions 1-WL and 2-WL have received much attention, due to their numerous links to other areas of computer science. Knowing the expressive power of a certain dimension of the algorithm usually amounts to understanding the computed colorings. An increase in the dimension leads to finer computed colorings and, thus, more graphs can be distinguished. For example, on the class of planar graphs, 3-WL solves the isomorphism problem. However, the expressive power of 2-WL on the class is poorly understood (and, in particular, it may even well be that it decides isomorphism). In this paper, we investigate the colorings computed by 2-WL on planar graphs. Towards this end, we analyze the graphs induced by edge color classes in the graph. Based on the obtained classification, we show that for every 3-connected planar graph, it holds that: a) after coloring all pairs with their 2-WL color, the graph has fixing number 1 with respect to 1-WL, or b) there is a 2-WL-definable matching that can be used to transform the graph into a smaller one, or c) 2-WL detects a connected subgraph that is essentially the graph of a Platonic or Archimedean solid, a prism, a cycle, or a bipartite graph K_{2,\ell}. In particular, the graphs from case (a) are identified by 2-WL.

While message-passing graph neural networks have clear limitations in approximating permutation-equivariant functions over graphs or general relational data, more expressive, higher-order graph neural networks do not scale to large graphs. They either operate on $k$-order tensors or consider all $k$-node subgraphs, implying an exponential dependence on $k$ in memory requirements, and do not adapt to the sparsity of the graph. By introducing new heuristics for the graph isomorphism problem, we devise a class of universal, permutation-equivariant graph networks, which, unlike previous architectures, offer a fine-grained control between expressivity and scalability and adapt to the sparsity of the graph. These architectures lead to vastly reduced computation times compared to standard higher-order graph networks in the supervised node- and graph-level classification and regression regime while significantly improving standard graph neural network and graph kernel architectures in terms of predictive performance.

We give new decomposition theorems for classes of graphs that can be transduced in first-order logic from classes of sparse graphs — more precisely, from classes of bounded expansion and nowhere dense classes. In both cases, the decomposition takes the form of a single colored rooted tree of bounded depth where, in addition, there can be links between nodes that are not related in the tree. The constraint is that the structure formed by the tree and the links has to be sparse. Using the decomposition theorem for transductions of nowhere dense classes, we show that they admit low-shrubdepth covers of size , where n is the vertex count and ε > 0 is any fixed real. This solves an open problem posed by Gajarský et al. (ACM TOCL ’20) and also by Briański et al. (SIDMA ’21).

The Weisfeiler--Leman procedure is a widely used technique 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 two-dimensional Weisfeiler--Leman algorithm implicitly computes the decomposition of a graph into its 3-connected components. This implies that the dimension of the algorithm needed to distinguish two given nonisomorphic graphs is at most the dimension required to distinguish nonisomorphic 3-connected components of the graphs (assuming dimension at least 2). To obtain our decomposition result, we show 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 the class of graphs of treewidth at most 𝑘. Using a construction by Cai, Fürer, and Immerman, we also provide a new lower bound that is asymptotically tight up to a factor of 2. (A corrected version is attached.)

We classify graphs and, more generally, finite relational structures that are identified by C^2 , that is, two-variable first-order logic with counting. Using this classification, we show that it can be decided in almost linear time whether a structure is identified by C^2 . Our classification implies that for every graph identified by this logic, all vertex-colored versions of it are also identified. A similar statement is true for finite relational structures. We provide constructions that solve the inversion problem for finite relational structures in linear time. By a result due to Otto, this problem has been known to be polynomial-time solvable. For graphs, we conclude that every C^2 -equivalence class contains a representative whose orbits are exactly the classes of the C^2 -partition of its vertex set and which has a single automorphism witnessing this fact. We show that such statements are not true for general k by providing examples of graphs of order linear in k which are identified by C^3 , but for which the orbit partition is strictly finer than the C^k -partition. We also construct identified graphs which have vertex-colored versions that are not identified by C^k .

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 Colour Refinement procedure and its generalisation to higher dimensions, the Weisfeiler-Leman algorithm, are central subroutines in approaches to the graph isomorphism problem. In an iterative fashion, Colour Refinement computes a colouring of the vertices of its input graph. A trivial upper bound on the iteration number of Colour Refinement on graphs of order n is n-1. We show that this bound is tight. More precisely, we prove via explicit constructions that there are infinitely many graphs G on which Colour Refinement takes |G|-1 iterations to stabilise. Modifying the infinite families that we present, we show that for every natural number n >= 10, there are graphs on n vertices on which Colour Refinement requires at least n-2 iterations to reach stabilisation.

String-to-string MSO interpretations are like Courcelle's MSO transductions, except that a single output position can be represented using a tuple of input positions instead of just a single input position. In particular, the output length is polynomial in the input length, as opposed to MSO transductions, which have output of linear length. We show that string-to-string MSO interpretations are exactly the polyregular functions. The latter class has various characterizations, one of which is that it consists of the string-to-string functions recognized by pebble transducers. Our main result implies the surprising fact that string-to-string MSO interpretations are closed under composition.

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 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 bi- and triconnected components. We prove that the 2-dimensional Weisfeiler-Leman algorithm implicitly computes the decomposition of a graph into its triconnected 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 dimension 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, Fürer, and Immerman, we also provide a new lower bound that is asymptotically tight up to a factor of 2.

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.

We show that on graphs with n vertices, the 2-dimensional Weisfeiler-Leman algorithm requires at most O(n^2/log(n)) iterations to reach stabilization. This in particular shows that the previously best, trivial upper bound of O(n^2) is asymptotically not tight. In the logic setting, this translates to the statement that if two graphs of size n can be distinguished by a formula in first-order logic with counting with 3 variables (i.e., in C3), then they can also be distinguished by a C3-formula that has quantifier depth at most O(n^2/log(n)). To prove the result we define a game between two players that enables us to decouple the causal dependencies between the processes happening simultaneously over several iterations of the algorithm. This allows us to treat large color classes and small color classes separately. As part of our proof we show that for graphs with bounded color class size, the number of iterations until stabilization is at most linear in the number of vertices. This also yields a corresponding statement in first-order logic with counting. Similar results can be obtained for the respective logic without counting quantifiers, i.e., for the logic L3.

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.

We show that on graphs with n vertices the 2-dimensional Weisfeiler-Leman algorithm requires at most O(n2 / log(n)) iterations to reach stabilization. This in particular shows that the previously best, trivial upper bound of O(n2) is asymptotically not tight. In the logic setting this translates to the statement that if two graphs of size n can be distinguished by a formula in first order logic with counting with 3 variables (i.e., in C3) then they can also be distinguished by a C3-formula that has quantifier depth at most O(n2 / log(n)).