Rankwidth meets stability

TL;DR

This paper explores the model-theoretic properties of graph classes, linking stability concepts with structural graph parameters like rankwidth, and establishes characterizations and algorithmic implications for these classes.

Contribution

It characterizes classes of graphs with bounded rankwidth and stable edge relations as first-order transductions of bounded treewidth classes, and connects monadic dependence with unstable edge relations.

Findings

Classes with bounded rankwidth excluding half-graphs are linearly χ-bounded.

A class is a first-order transduction of bounded treewidth iff it has bounded rankwidth and stable edges.

If monadically dependent but not stable, then the class has an unstable edge relation.

Abstract

We study two notions of being well-structured for classes of graphs that are inspired by classic model theory. A class of graphs $C$ is monadically stable if it is impossible to define arbitrarily long linear orders in vertex-colored graphs from $C$ using a fixed first-order formula. Similarly, monadic dependence corresponds to the impossibility of defining all graphs in this way. Examples of monadically stable graph classes are nowhere dense classes, which provide a robust theory of sparsity. Examples of monadically dependent classes are classes of bounded rankwidth (or equivalently, bounded cliquewidth), which can be seen as a dense analog of classes of bounded treewidth. Thus, monadic stability and monadic dependence extend classical structural notions for graphs by viewing them in a wider, model-theoretical context. We explore this emerging theory by proving the following: - A class of graphs $C$ is a first-order transduction of a class with bounded treewidth if and only if $C$ has bounded rankwidth and a stable edge relation (i.e. graphs from $C$ exclude some half-graph as a semi-induced subgraph). - If a class of graphs $C$ is monadically dependent and not monadically stable, then $C$ has in fact an unstable edge relation. As a consequence, we show that classes with bounded rankwidth excluding some half-graph as a semi-induced subgraph are linearly $\chi$-bounded. Our proofs are effective and lead to polynomial time algorithms.