Characterizations of monadically dependent tree-ordered weakly sparse structures

TL;DR

The paper addresses how monadic dependence can be characterized for tree-ordered weakly sparse structures, linking logical tameness to sparsity via a suite of tree-ordered constructions. It introduces a dichotomy for hereditary TOWS classes and a main equivalence that ties monadic dependence to nowhere denseness of associated minor closures, with broader implications across sparsification, width parameters, and model-checking complexity. Key contributions include the tree-ordered minor framework, the incidences and generalized fundamental graphs, and a sparsification operation that preserves core width-boundedness phenomena, plus a model-theoretic characterization for minor-excluding graph classes. The results forge a bridge between sparsity theory and model theory, yielding new perspectives on the tractability frontier for FO model checking and offering structural characterizations of minor-excluding graph families, including a special case where monadic dependence coincides with bounded twin-width when the tree-order cover graph has bounded degree.

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 $σ$-structure is the expansion of a $σ$-structure with a tree-order. A tree-ordered $σ$-structure is weakly sparse if the Gaifman graph of its $σ$-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 $σ$-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.

Figures (3)

• Figure 1: A core; red arrows are the cover relations of $\prec$, blue edges are the $E$-relations.
• Figure 2: On the right, the tree-ordered minor of the tree-ordered structure $\mathbf M$ obtained by contracting covers $d,i,j,k,n$ and deleting the triple $\beta$.
• Figure 3: Constructions derived from a tree-ordered $\sigma$-structure. Top left, a tree-ordered $\sigma$-structure $\mathbf M$ is composed of a tree-order (Hasse diagram in purple fat) and a weakly sparse set of triples (green, with arrows indicating the triple order). Top right: the generalized fundamental graph $\Lambda(\mathbf M)$ of $\mathbf M$. (If $k$ is the maximum arity of the relations in $\sigma$, then $\Lambda(\mathbf M)$ has $k-1$ types of edges). Bottom left: the tree-ordered Gaifman graph of $\mathbf M$. Bottom right: the tree-ordered incidence graph of $\mathbf M$.

Theorems & Definitions (6)

• Example 2.1
• proof
• theorem 1: DVORAK2018143
• theorem 2: store=thm:Adler,note=Adler2013
• proof
• proof