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Fermionic extensions of generic 2d gravity theories obtained from the gradedPoisson-Sigma model (gPSM) approach show a large degree of ambiguity. Inaddition, obstructions may reduce the allowed range of fields as given by thebosonic theory, or even prohibit any extension in certain cases. In our presentwork we relate the finite W-algebras inherent in the gPSM algebra ofconstraints to algebras which can be interpreted as supergravities in the usualsense (Neuveu-Schwarz or Ramond algebras resp.), deformed by the presence ofthe dilaton field. With very straightforward and natural assumptions on them--like demanding rigid supersymmetry in a certain flat limit, or linking theanti-commutator of certain fermionic charges to the Hamiltonian constraint-- inthe ``genuine'' supergravity obtained in this way the ambiguities disappear, aswell as the obstructions referred to above. Thus all especially interestingbosonic models (spherically reduced gravity, the Jackiw-Teitelboim model etc.)\under these conditions possess a unique fermionic extension and are free fromnew singularities. The superspace supergravity model of Howe is found as aspecial case of this supergravity action. For this class of models the relationbetween bosonic potential and prepotential does not introduce obstructions aswell.Comment: 22 pages, LaTeX, JHEP class. v3: Final version, to appear in JHE

Graded Poisson-sigma models and dilaton-deformed 2D supergravity algebra