Rankwidth meets stability

We study two notions of being well-structured for classes of graphs that are inspired by classic model theory. A class of graphs is monadically stable if it is impossible to define arbitrarily long linear orders in vertex-colored graphs from 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 is a first-order transduction of a class with bounded treewidth if and only if has bounded rankwidth and a stable edge relation (i. e. graphs from exclude some half-graph as a semi-induced subgraph). 2) If a class of graphs is monadically dependent and not monadically stable, then 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.

Authors

• Jaroslav Nesetril
• Patrice Ossona de Mendez
• Michal Pilipczuk
• Roman Rabinovich 0001
• Sebastian Siebertz

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