  The Fortuin-Kasteleyn (FK) random cluster model, which can be exactly mapped
from the $q$-state Potts spin model, is a correlated bond percolation model. By
extensive Monte Carlo simulations, we study the FK bond representation of the
critical Ising model ($q=2$) on a finite complete graph, i.e. the mean-field
Ising model. We provide strong numerical evidence that the configuration space
for $q=2$ contains an asymptotically vanishing sector in which quantities
exhibit the same finite-size scaling as in the critical uncorrelated bond
percolation ($q=1$) on the complete graph. Moreover, we observe that in the
full configuration space, the power-law behaviour of the cluster-size
distribution for the FK Ising clusters except the largest one is governed by a
Fisher exponent taking the value for $q=1$ instead of $q=2$. This demonstrates
the percolation effects in the FK Ising model on the complete graph.
