Szymon Toruńczyk
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View article: First-order transducibility among classes of sparse graphs
First-order transducibility among classes of sparse graphs Open
We prove several negative results about first-order transducibility for classes of sparse graphs: - for every $t \in \mathbb{N}$, the class of graphs of treewidth at most $t+1$ is not transducible from the class of graphs of treewidth at m…
View article: Separability Properties of Monadically Dependent Graph Classes
Separability Properties of Monadically Dependent Graph Classes Open
A graph class $\mathcal C$ is monadically dependent if one cannot interpret all graphs in colored graphs from $\mathcal C$ using a fixed first-order interpretation. We prove that monadically dependent classes can be exactly characterized b…
View article: Merge-width and First-Order Model Checking
Merge-width and First-Order Model Checking Open
We introduce merge-width, a family of graph parameters that unifies several structural graph measures, including treewidth, degeneracy, twin-width, clique-width, and generalized coloring numbers. Our parameters are based on new decompositi…
View article: Flip-Breakability: A Combinatorial Dichotomy for Monadically Dependent Graph Classes
Flip-Breakability: A Combinatorial Dichotomy for Monadically Dependent Graph Classes Open
A conjecture in algorithmic model theory predicts that the model-checking problem for first-order logic is fixed-parameter tractable on a hereditary graph class if and only if the class is monadically dependent. Originating in model theory…
View article: Flip-Breakability: A Combinatorial Dichotomy for Monadically Dependent Graph Classes
Flip-Breakability: A Combinatorial Dichotomy for Monadically Dependent Graph Classes Open
A conjecture in algorithmic model theory predicts that the model-checking problem for first-order logic is fixed-parameter tractable on a hereditary graph class if and only if the class is monadically dependent. Originating in model theory…
View article: Twin-Width IV: Ordered Graphs and Matrices
Twin-Width IV: Ordered Graphs and Matrices Open
We establish a list of characterizations of bounded twin-width for hereditary classes of totally ordered graphs: as classes of at most exponential growth studied in enumerative combinatorics, as monadically NIP classes studied in model the…
View article: Elementary first-order model checking for sparse graphs
Elementary first-order model checking for sparse graphs Open
It is known that for subgraph-closed graph classes the first-order model checking problem is fixed-parameter tractable if and only if the class is nowhere dense [Grohe, Kreutzer, Siebertz, STOC 2014]. However, the dependency on the formula…
View article: First-Order Model Checking on Monadically Stable Graph Classes
First-Order Model Checking on Monadically Stable Graph Classes Open
A graph class $\mathscr{C}$ is called monadically stable if one cannot interpret, in first-order logic, arbitrary large linear orders in colored graphs from $\mathscr{C}$. We prove that the model checking problem for first-order logic is f…
View article: Flipper games for monadically stable graph classes
Flipper games for monadically stable graph classes Open
A class of graphs $\mathscr{C}$ is monadically stable if for any unary expansion $\widehat{\mathscr{C}}$ of $\mathscr{C}$, one cannot interpret, in first-order logic, arbitrarily long linear orders in graphs from $\widehat{\mathscr{C}}$. I…
View article: Canonical Decompositions in Monadically Stable and Bounded Shrubdepth Graph Classes
Canonical Decompositions in Monadically Stable and Bounded Shrubdepth Graph Classes Open
We use model-theoretic tools originating from stability theory to derive a result we call the Finitary Substitute Lemma, which intuitively says the following. Suppose we work in a stable graph class 𝒞, and using a first-order formula φ wit…
View article: Model Checking on Interpretations of Classes of Bounded Local Cliquewidth
Model Checking on Interpretations of Classes of Bounded Local Cliquewidth Open
International audience
View article: Indiscernibles and Flatness in Monadically Stable and Monadically NIP Classes
Indiscernibles and Flatness in Monadically Stable and Monadically NIP Classes Open
Monadically stable and monadically NIP classes of structures were initially studied in the context of model theory and defined in logical terms. They have recently attracted attention in the area of structural graph theory, as they general…
View article: Twin-width and types
Twin-width and types Open
We study problems connected to first-order logic in graphs of bounded twin-width. Inspired by the approach of Bonnet et al. [FOCS 2020], we introduce a robust methodology of local types and describe their behavior in contraction sequences …
View article: Twin-width IV: ordered graphs and matrices
Twin-width IV: ordered graphs and matrices Open
We establish a list of characterizations of bounded twin-width for\nhereditary, totally ordered binary structures. This has several consequences.\nFirst, it allows us to show that a (hereditary) class of matrices over a finite\nalphabet ei…
View article: Register Automata with Extrema Constraints, and an Application to Two-Variable Logic
Register Automata with Extrema Constraints, and an Application to Two-Variable Logic Open
We introduce a model of register automata over infinite trees with extrema constraints. Such an automaton can store elements of a linearly ordered domain in its registers, and can compare those values to the suprema and infima of register …
View article: Model Checking on Interpretations of Classes of Bounded Local Cliquewidth
Model Checking on Interpretations of Classes of Bounded Local Cliquewidth Open
We present a fixed-parameter tractable algorithm for first-order model checking on interpretations of graph classes with bounded local cliquewidth. Notably, this includes interpretations of planar graphs, and more generally, of classes of …
View article: Combinatorial and Algorithmic Aspects of Monadic Stability
Combinatorial and Algorithmic Aspects of Monadic Stability Open
Nowhere dense classes of graphs are classes of sparse graphs with rich structural and algorithmic properties, however, they fail to capture even simple classes of dense graphs. Monadically stable classes, originating from model theory, gen…
View article: Algorithms and Data Structures for First-Order Logic with Connectivity Under Vertex Failures
Algorithms and Data Structures for First-Order Logic with Connectivity Under Vertex Failures Open
We introduce a new data structure for answering connectivity queries in undirected graphs subject to batched vertex failures. Precisely, given any graph G and integer parameter k, we can in fixed-parameter time construct a data structure t…
View article: Algorithms and data structures for first-order logic with connectivity under vertex failures
Algorithms and data structures for first-order logic with connectivity under vertex failures Open
We introduce a new data structure for answering connectivity queries in undirected graphs subject to batched vertex failures. Precisely, given any graph G and integer k, we can in fixed-parameter time construct a data structure that can la…
View article: Stable graphs of bounded twin-width
Stable graphs of bounded twin-width Open
We prove that every class of graphs $\mathscr C$ that is monadically stable and has bounded twin-width can be transduced from some class with bounded sparse twin-width. This generalizes analogous results for classes of bounded linear cliqu…
View article: Ordered graphs of bounded twin-width
Ordered graphs of bounded twin-width Open
We consider hereditary classes of graphs equipped with a total order. We provide multiple equivalent characterisations of those classes which have bounded twin-width. In particular, we prove a grid theorem for classes of ordered graphs whi…
View article: Twin-width IV: ordered graphs and matrices
Twin-width IV: ordered graphs and matrices Open
We establish a list of characterizations of bounded twin-width for hereditary, totally ordered binary structures. This has several consequences. First, it allows us to show that a (hereditary) class of matrices over a finite alphabet eithe…
View article: Register Automata with Extrema Constraints, and an Application to Two-Variable Logic
Register Automata with Extrema Constraints, and an Application to Two-Variable Logic Open
We introduce a model of register automata over infinite trees with extrema constraints. Such an automaton can store elements of a linearly ordered domain in its registers, and can compare those values to the suprema and infima of register …
View article: First-Order Interpretations of Bounded Expansion Classes
First-Order Interpretations of Bounded Expansion Classes Open
The notion of bounded expansion captures uniform sparsity of graph classes and renders various algorithmic problems that are hard in general tractable. In particular, the model-checking problem for first-order logic is fixed-parameter trac…
View article: Aggregate Queries on Sparse Databases
Aggregate Queries on Sparse Databases Open
We propose an algebraic framework for studying efficient algorithms for query evaluation, aggregation, enumeration, and maintenance under updates, on sparse databases. Our framework allows to treat those problems in a unified way, by consi…
View article: Aggregate Queries on Sparse Databases
Aggregate Queries on Sparse Databases Open
We propose an algebraic framework for studying efficient algorithms for query evaluation, aggregation, enumeration, and maintenance under updates, on sparse databases. Our framework allows to treat those problems in a unified way, by consi…
View article: Definable isomorphism problem
Definable isomorphism problem Open
We investigate the isomorphism problem in the setting of definable sets (equivalent to sets with atoms): given two definable relational structures, are they related by a definable isomorphism? Under mild assumptions on the underlying struc…
View article: Definable isomorphism problem
Definable isomorphism problem Open
We investigate the isomorphism problem in the setting of definable sets (equivalent to sets with atoms): given two definable relational structures, are they related by a definable isomorphism? Under mild assumptions on the underlying struc…
View article: Progressive Algorithms for Domination and Independence
Progressive Algorithms for Domination and Independence Open
We consider a generic algorithmic paradigm that we call progressive exploration, which can be used to develop simple and efficient parameterized graph algorithms. We identify two model-theoretic properties that lead to efficient progressiv…