  We demonstrate that the invaded cluster algorithm, recently introduced by
Machta et al, is a fast and reliable tool for determining the critical
temperature and the magnetic critical exponent of periodic and aperiodic
ferromagnetic Ising models in two dimensions. The algorithm is shown to
reproduce the known values of the critical temperature on various periodic and
quasiperiodic graphs with an accuracy of more than three significant digits. On
two quasiperiodic graphs which were not investigated in this respect before,
the twelvefold symmetric square-triangle tiling and the tenfold symmetric
T\"ubingen triangle tiling, we determine the critical temperature. Furthermore,
a generalization of the algorithm to non-identical coupling strengths is
presented and applied to a class of Ising models on the Labyrinth tiling. For
generic cases in which the heuristic Harris-Luck criterion predicts deviations
from the Onsager universality class, we find a magnetic critical exponent
different from the Onsager value. But also notable exceptions to the criterion
are found which consist not only of the exactly solvable cases, in agreement
with a recent exact result, but also of the self-dual ones and maybe more.
