On Infinite Transition Graphs Having a Decidable Monadic Theory

We deene a family of graphs whose monadic theory is (in linear space) reducible to the monadic theory S2S of the complete ordered binary tree. This family contains strictly the context-free graphs investigated by Muller and Schupp, and also the equational graphs deened by Courcelle. Using words as representations of vertices, we give a complete set of representatives by preex rewriting of rational languages. This subset of possible representatives is a boolean algebra preserved by transitive closure of arcs and by rational restriction on vertices.