  Lattice animals provide a discretized model for the theta transition
displayed by branched polymers in solvent. Exact graph enumeration studies have
given some indications that the phase diagram of such lattice animals may
contain two collapsed phases as well as an extended phase. This has not been
confirmed by studies using other means. We use the exact correspondence between
the q --> 1 limit of an extended Potts model and lattice animals to investigate
the phase diagram of lattice animals on phi-cubed random graphs of arbitrary
topology (``thin'' random graphs). We find that only a two phase structure
exists -- there is no sign of a second collapsed phase.
  The random graph model is solved in the thermodynamic limit by saddle point
methods. We observe that the ratio of these saddle point equations give
precisely the fixed points of the recursion relations that appear in the
solution of the model on the Bethe lattice by Henkel and Seno. This explains
the equality of non-universal quantities such as the critical lines for the
Bethe lattice and random graph ensembles.
