  We propose a modification to the random destruction of graphs: Given a finite
network with a distinguished set of sources and targets, remove (cut) vertices
at random, discarding components that do not contain a source node. We
investigate the number of cuts required until all targets are removed, and the
size of the remaining graph. This model interpolates between the random cutting
model going back to Meir and Moon and site percolation. We prove several
general results, including that the size of the remaining graph is a tight
family of random variables for compatible sequences of expander-type graphs,
and determine limiting distributions for binary caterpillar trees and complete
binary trees.
