LoCoMoDe · Logic and Combinatorics in Monadically Dependent Graph Classes

The (first-order) model checking problem is a fundamental problem in theoretical computer science, bridging the fields of structural graph theory, finite model theory, and parameterized complexity.Given as input a graph and a formula in first-order logic, the goal is to decide whether the graph satisfies the formula.Instead of solving a single algorithmic problem, model checking algorithms provide a uniform way to solve all first-order definable problems. Examples include parameterized versions of the dominating set, independent set, and subgraph isomorphism problems.While model checking is hard on the class of all graphs, the recently formulated Model Checking Conjecture predictsthat a hereditary graph class admits fixed-parameter tractable first-order model checking if and only if it is monadically dependent.Monadically dependent graph classes are very general.They include many sparse classes like planar graphs, bounded degree, bounded tree-width, excluded minor, and nowhere dense classes; and also dense classes like monadically stable, bounded clique-width, and bounded twin-width classes.While tractable model checking has been established for many restricted fragments of monadic dependence, the general case has so far remained elusive.The main obstacle is the lack of a combinatorial structure theory for monadically dependent classes.Originating in model theory, monadic dependence is defined in terms of logic and has predominantly been studied on infinite structures.In this fellowship, I will combine tools from combinatorics and logic to develop a combinatorial structure theory for monadically dependent classes of finite graphs.The goal is to resolve the Model Checking Conjecture and enable further progress in the algorithmic and combinatorial treatment of monadically dependent graph classes.