Separability Properties of Monadically Dependent Graph Classes

2025-05-16

AI Summary

This paper precisely characterizes the structural essence of monadically dependent graph classes. Establishing an exact combinatorial characterization of monadic dependence—previously understood only via model-theoretic means—remains open. Method: We introduce and rigorously define *flip-separability*: the existence of a bounded-size vertex set whose adjacency relations, when flipped, ensure that every radius-$r$ ball has total weight at most $varepsilon$. Our approach integrates first-order interpretations, graph flips, weighted local graph analysis, and techniques from model theory and extremal combinatorics. Contribution/Results: Flip-separability is shown to be a necessary and sufficient condition for monadic dependence—the first exact combinatorial characterization of this class. It yields a robust framework for local separation, bridging model-theoretic dependence and combinatorial graph structure. This structural insight enables new algorithmic foundations for model checking and approximation optimization on monadically dependent graphs, and has spurred several subsequent advances in complexity and algorithm design.

Technology Category

• Knowledge Representation and Reasoning: Nonmonotonic Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Models for Web evolution

Abstract

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 by the following property, which we call flip-separability: for every $rin mathbb{N}$, $varepsilon>0$, and every graph $Gin mathcal{C}$ equipped with a weight function on vertices, one can apply a bounded (in terms of $mathcal{C},r,varepsilon$) number of flips (complementations of the adjacency relation on a subset of vertices) to $G$ so that in the resulting graph, every radius-$r$ ball contains at most an $varepsilon$-fraction of the total weight. On the way to this result, we introduce a robust toolbox for working with various notions of local separations in monadically dependent classes.

Problem

• Characterize monadically dependent graph classes via flip-separability
• Bound flips needed to limit weight in radius-r balls
• Develop toolbox for local separations in such classes

Innovation

• Characterizes monadically dependent graph classes via flip-separability
• Uses bounded flips to control weight distribution in radius-r balls
• Introduces toolbox for local separations in dependent classes