Abstract
We define a family of graphs whose the monadic theory is linearly reducible to the monadic theory S2S of the complete deterministic 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 for vertices, we give a complete set of representatives by prefix rewriting of rational languages. This subset is a boolean algebra preserved by transitive closure of arcs and by rational restriction on vertices.
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