Characterizations of Monadic Second Order Definable Context-Free Sets of Graphs

We give a characterization of the sets of graphs that are both definable in Counting Monadic Second Order Logic (CMSO) and context-free, i.e., least solutions of Hyperedge-Replacement (HR) grammars introduced by Courcelle and Engelfriet. We prove the equivalence of these sets with: (a) recognizable sets (in the algebra of graphs with HR-operations) of bounded tree-width; we refine this condition further and show equivalence with recognizability in a finitely generated subalgebra of the HR-algebra of graphs; (b) parsable sets, for which there is an MSO-definable transduction from graphs to a set of derivation trees labelled by HR operations, such that the set of graphs is the image of the set of derivation trees under the canonical evaluation of the HR operations; (c) images of recognizable unranked sets of trees under an MSO-definable transduction, whose inverse is also MSO-definable. We rely on a novel connection between two seminal results, a logical characterization of context-free graph languages in terms of tree to graph MSO-definable transductions, by Courcelle and Engelfriet and a proof that an optimal-width tree decomposition of a graph can be built by an MSO-definable transduction, by Bojanczyk and Pilipczuk.