  Spreading of either information or matter can often be treated as a network
problem. It can be of great importance to be able to estimate the likelihood
that spreading through a network reaches essentially the entire network while
still not reaching certain sub-classes of the network. We show that excluding
nodes and edges from the network has a subtle effect on the percolation. We
study two specific examples of degree distributions (exponential and scale
free) for which analytical solutions can be obtained. The two cases exhibit
qualitatively different behavior.
