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We define 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 defined by Courcelle. Using words as representations of vertices, we give a complete set of representatives by prefix 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.