Extensions of the Caucal Hierarchy?

The Caucal hierarchy contains graphs that can be obtained from finite graphs by alternately applying the unfolding operation and inverse rational mappings. The goal of this work is to check whether the hierarchy is closed under interpretations in logics extending the monadic second-order logic by the unbounding quantifier \(\mathsf U\). We prove that by applying interpretations described in the MSO+\(\mathsf {U^{fin}}\) logic (hence also in its fragment WMSO+\(\mathsf U\)) to graphs of the Caucal hierarchy we can only obtain graphs on the same level of the hierarchy. Conversely, interpretations described in the more powerful MSO+\(\mathsf U\) logic can give us graphs with undecidable MSO theory, hence outside of the Caucal hierarchy.