  We extend the Achlioptas model for the delay of criticality in the
percolation problem. Instead of having a completely random connectivity
pattern, we generalize the idea of the two-site probe in the Achlioptas model
for connecting smaller clusters, by introducing two models: the first one by
allowing any number k of probe sites to be investigated, k being a parameter,
and the second one independent of any specific number of probe sites, but with
a probabilistic character which depends on the size of the resulting clusters.
We find numerically the complete spectrum of critical points and our results
indicate that the value of the critical point behaves linearly with k after the
value of k = 3. The range k = 2-3 is not linear but parabolic. The more general
model of generating clusters with probability inversely proportional to the
size of the resulting cluster produces a critical point which is equivalent to
the value of k being in the range k = 5-7.
