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: 1) 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). 2) 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.