  We consider the fractal dimensions of critical clusters occurring in
configurations of a q-state Potts model coupled to the planar random graphs of
the dynamical triangulations formulation of Euclidean quantum gravity in two
dimensions. For regular lattices, it is well-established that at criticality
the properties of Fortuin-Kasteleyn clusters are directly related to the
conventional critical exponents, whereas the corresponding properties of the
geometric clusters of like spins are not. Recently it has been observed that
the latter are related to the critical properties of a tricritical Potts model
with the same central charge. We apply the KPZ formalism to develop a related
prediction for the case of Potts models coupled to quantum gravity and employ
numerical simulation methods to confirm it for the Ising case q=2.
