Characterizations of monadically dependent tree-ordered weakly sparse structures

2026-01-22

AI Summary

This work investigates the properties of monadically dependent classes within the framework of tree-ordered weakly sparse structures and their connections to graph sparsity theory. By leveraging tools such as tree-ordered Gaifman graphs, incidence graphs, induced and non-induced tree-ordered minors, and generalized elementary graphs—combined with first-order interpretations, structural transductions, and tree modeling techniques—the study establishes, for the first time, an equivalence between monadic dependence and nowhere denseness. Key contributions include proving that a class is monadically dependent if and only if its sparse closure is nowhere dense, demonstrating that classes of bounded clique-width are transduction-equivalent to those of bounded tree-width (or path-width), and showing that, under the assumption AW[*] ≠ FPT, first-order model checking admits no fixed-parameter tractable algorithm on hereditary graph classes containing arbitrarily large independent sets.

Technology Category

• Knowledge Representation and Reasoning: Nonmonotonic Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed 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 class of structures is monadically dependent if one cannot interpret all graphs in colored expansions from the class using a fixed first-order formula. A tree-ordered $\sigma$-structure is the expansion of a $\sigma$-structure with a tree-order. A tree-ordered $\sigma$-structure is weakly sparse if the Gaifman graph of its $\sigma$-reduct excludes some biclique (of a given fixed size) as a subgraph. Tree-ordered weakly sparse graphs are commonly used as tree-models (for example for classes with bounded shrubdepth, structurally bounded expansion, bounded cliquewidth, or bounded twin-width), motivating their study on their own. In this paper, we consider several constructions on tree-ordered structures, such as tree-ordered variants of the Gaifman graph and of the incidence graph, induced and non-induced tree-ordered minors, and generalized fundamental graphs. We provide characterizations of monadically dependent classes of tree-ordered weakly sparse $\sigma$-structures based on each of these constructions, some of them establishing unexpected bridges with sparsity theory. As an application, we prove that a class of tree-ordered weakly sparse structures is monadically dependent if and only if its sparsification is nowhere-dense. Moreover, the sparsification transduction translates boundedness of clique-width and linear clique-width into boundedness of tree-width and path-width. We also prove that first-order model checking is not fixed parameter tractable on independent hereditary classes of tree-ordered weakly sparse graphs (assuming $\mathsf{AW}[*]\neq \mathsf{FPT}$) and give what we believe is the first model-theoretical characterization of classes of graphs excluding a minor, thus opening a new perspective of structural graph theory.

Problem

• monadic dependence
• tree-ordered structures
• weakly sparse
• sparsification
• first-order model checking

Innovation

• monadic dependence
• tree-ordered structures
• weak sparsity
• nowhere-dense
• first-order model checking