We define the (random) kappa-cut number of a rooted graph to model the difficulty of the destruction of a resilient network. The process is as the cut model of Meir and Moon [21] except now a node must be cut kappa times before it is destroyed. The first order terms of the expectation and variance of chi(n), the kappa-cut number of a path of length n, are proved. We also show that chi(n), after rescaling, converges in distribution to a limit B-kappa, which has a complicated representation. The paper then briefly discusses the kappa-cut number of some trees and general graphs. We conclude by some analytic results which may be of interest.