Characterizations of monadically dependent tree-ordered weakly sparse structures

TL;DR

This paper characterizes monadically dependent classes of tree-ordered weakly sparse structures, linking their properties to sparsification and graph minors, and explores implications for model checking and structural graph theory.

Contribution

It provides new characterizations of monadically dependent classes using various graph constructions and establishes connections with sparsity and minor-exclusion.

Findings

A class is monadically dependent iff its sparsification is nowhere-dense.

Sparsification transduces bounded clique-width into bounded tree-width.

Model checking is not fixed parameter tractable on certain classes assuming PT PT.

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.