Flip-Breakability: A Combinatorial Dichotomy for Monadically Dependent Graph Classes

TL;DR

The paper advances a central open question in algorithmic model theory by giving the first two purely combinatorial characterizations of monadically dependent graph classes. It introduces flip-breakability as a Ramsey-style, distance-based separability property that exactly captures monadic dependence on graphs, and a forbidden-pattern perspective that yields AW[*]-hardness for monadically independent hereditary classes. The work unifies sparsity, twin-width, and stability themes by showing these are facets of a single tractable boundary and extends the results to binary structures and ordered graph classes. The combination of insulators, prepatterns, and transformers provides a constructive toolkit with potential algorithmic implications for FO model checking on monadically dependent classes and suggests future game-theoretic characterizations and quantitative refinements analogous to existing sparsity theories.

Abstract

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.

Figures (21)

• Figure 1: Hierarchy of selected properties of graph classes. Classes in green boxes are known do admit fpt model checking algorithms (with additional assumptions required in the case of bounded twin-width). Monadically dependent classes are known to generalize all these notions. It is conjectured that a hereditary class is monadically dependent if and only if its model checking problem is tractable.
• Figure 2: A half-graph of order 4.
• Figure 3: The sets $A$ and $B$ are far away after a $5$-flip shown on the right side. We flip between the blue squares and the blue circles, and between the orange squares and the orange circles.
• Figure 4: (i) star 4-crossing of order 4. (ii) clique 4-crossing of order 4. (iii) half-graph 4-crossing of order 4. (iv) comparability grid of order 4. In (i), (ii), (iii), the roots are adjacent to all vertices in their respective colorful strip.
• Figure 5: Left: An insulator surrounding the set $W_\star$. The shown stacked half-graph layout is easily described by a bounded number of colors and the column order. A bounded number of flips are sufficient to ensure the highlighted subsets $A \subseteq W_\star$ and $B \subseteq W_\star$ have distance larger than $4$. The shades of red and blue highlight upper bounds on the distance of other vertices to $A$ and $B$. In particular, all unshaded vertices have distance larger than two to these sets.

Theorems & Definitions (333)

• Conjecture 1.1: e.g., warwick-problemstwwIVssmc
• Definition 1.1: Flip-Breakability
• Theorem 1.2
• Theorem 1.3: twwIV
• Definition 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Theorem 1.9