Rankwidth meets stability

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 $χ$-bounded. Our proofs are effective and lead to polynomial time algorithms.