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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, this notion is defined in terms of logic, and encompasses nowhere dense classes, monadically stable classes, and classes of bounded twin-width. Working towards this conjecture, we provide the first two combinatorial characterizations of monadically dependent graph classes. This yields the following dichotomy. On the structure side, we characterize monadic dependence by a Ramsey-theoretic property called flip-breakability. This notion generalizes the notions of uniform quasi-wideness, flip-flatness, and bounded grid rank, which characterize nowhere denseness, monadic stability, and bounded twin-width, respectively, and played a key role in their respective model checking algorithms. Natural restrictions of flip-breakability additionally characterize bounded treewidth and cliquewidth and bounded treedepth and shrubdepth. On the non-structure side, we characterize monadic dependence by explicitly listing few families of forbidden induced subgraphs. This result is analogous to the characterization of nowhere denseness via forbidden subdivided cliques, and allows us to resolve one half of the motivating conjecture: First-order model checking is AW[$*$]-hard on every hereditary graph class that is monadically independent. The result moreover implies that hereditary graph classes which are small, have almost bounded twin-width, or have almost bounded flip-width, are monadically dependent. Lastly, we lift our result to also obtain a combinatorial dichotomy in the more general setting of monadically dependent classes of binary structures., Comment: v2: added section "Conclusions and Future Work"

A class of graphs is structurally nowhere dense if it can be constructed from a nowhere dense class by a first-order transduction. Structurally nowhere dense classes vastly generalize nowhere dense classes and constitute important examples of monadically stable classes. We show that the first-order model checking problem is fixed-parameter tractable on every structurally nowhere dense class of graphs. Our result builds on a recently developed game-theoretic characterization of monadically stable graph classes. As a second key ingredient of independent interest, we provide a polynomial-time algorithm for approximating weak neighborhood covers (on general graphs). We combine the two tools into a recursive locality-based model checking algorithm. This algorithm is efficient on every monadically stable graph class admitting flip-closed sparse weak neighborhood covers, where flip-closure is a mild additional assumption. Thereby, establishing efficient first-order model checking on monadically stable classes is reduced to proving the existence of flip-closed sparse weak neighborhood covers on these classes - a purely combinatorial problem. We complete the picture by proving the existence of the desired covers for structurally nowhere dense classes: we show that every structurally nowhere dense class can be sparsified by contracting local sets of vertices, enabling us to lift the existence of covers from sparse classes.

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 $\widehat{\mathscr{C}}$. It is known that nowhere dense graph classes are monadically stable; these encompass most of the studied concepts of sparsity in graphs, including graph classes that exclude a fixed topological minor. On the other hand, monadic stability is a property expressed in purely model-theoretic terms and hence it is also suited for capturing structure in dense graphs. For several years, it has been suspected that one can create a structure theory for monadically stable graph classes that mirrors the theory of nowhere dense graph classes in the dense setting. In this work we provide a step in this direction by giving a characterization of monadic stability through the Flipper game: a game on a graph played by Flipper, who in each round can complement the edge relation between any pair of vertex subsets, and Connector, who in each round localizes the game to a ball of bounded radius. This is an analog of the Splitter game, which characterizes nowhere dense classes of graphs (Grohe, Kreutzer, and Siebertz, J.ACM'17). We give two different proofs of our main result. The first proof uses tools from model theory, and it exposes an additional property of monadically stable graph classes that is close in spirit to definability of types. Also, as a byproduct, we give an alternative proof of the recent result of Braunfeld and Laskowski (arXiv 2209.05120) that monadic stability for graph classes coincides with existential monadic stability. The second proof relies on the recently introduced notion of flip-wideness (Dreier, M\"ahlmann, Siebertz, and Toru\'nczyk, arXiv 2206.13765) and provides an efficient algorithm to compute Flipper's moves in a winning strategy.

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, generalize nowhere dense classes and close them under transductions, i.e. transformations defined by colorings and simple first-order interpretations. In this work we aim to extend some combinatorial and algorithmic properties of nowhere dense classes to monadically stable classes of finite graphs. We prove the following results. - In monadically stable classes the Ramsey numbers $R(s,t)$ are bounded from above by $\mathcal{O}(t^{s-1-\delta})$ for some $\delta>0$, improving the bound $R(s,t)\in \mathcal{O}(t^{s-1}/(\log t)^{s-1})$ known for general graphs and the bounds known for $k$-stable graphs when $s\leq k$. - For every monadically stable class $\mathcal{C}$ and every integer $r$, there exists $\delta > 0$ such that every graph $G \in \mathcal{C}$ that contains an $r$-subdivision of the biclique $K_{t,t}$ as a subgraph also contains $K_{t^\delta,t^\delta}$ as a subgraph. This generalizes earlier results for nowhere dense graph classes. - We obtain a stronger regularity lemma for monadically stable classes of graphs. - Finally, we show that we can compute polynomial kernels for the independent set and dominating set problems in powers of nowhere dense classes. Formerly, only fixed-parameter tractable algorithms were known for these problems on powers of nowhere dense classes.

An indiscernible sequence $(\bar a_i)_{1\leq i\leq n}$ in a structure is an ordered sequence of tuples of elements which is very homogeneous in the sense that any two finite subsequences of the same length satisfy the same first-order formulas. We provide new characterizations of monadically stable and monadically NIP classes of structures in terms of indiscernible sequences by showing that they impose a strong structure on their neighborhoods. In particular, we show that every formula~$\phi(x,\bar y)$, where $x$ is a single free variable, has alternation rank at most $2$ over every sufficiently long indiscernible sequence in a monadically NIP class. We provide a second new characterization of monadically stable classes of graphs in terms of a new notion called flip-wideness. Flip-wideness generalizes the notion of uniform quasi-wideness, which characterizes nowhere dense classes and had a key impact on the combinatorial and algorithmic treatment of nowhere dense classes. All our proofs are constructive and yield efficient algorithms.

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 bounded genus. To obtain this result we develop a new tool which works in a very general setting of dependent classes and which we believe can be an important ingredient in achieving similar results in the future., Comment: 28 pages, 5 figures

A strong backdoor in a formula $\phi$ of propositional logic to a tractable class $\mathcal{C}$ of formulas is a set $B$ of variables of $\phi$ such that every assignment of the variables in $B$ results in a formula from $\mathcal{C}$. Strong backdoors of small size or with a good structure, e.g. with small backdoor treewidth, lead to efficient solutions for the propositional satisfiability problem SAT. In this paper we propose the new notion of recursive backdoors, which is inspired by the observation that in order to solve SAT we can independently recurse into the components that are created by partial assignments of variables. The quality of a recursive backdoor is measured by its recursive backdoor depth. Similar to the concept of backdoor treewidth, recursive backdoors of bounded depth include backdoors of unbounded size that have a certain treelike structure. However, the two concepts are incomparable and our results yield new tractability results for SAT.