  We define the (random) $k$-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 except now a node must be cut $k$ times before it is
destroyed. The first order terms of the expectation and variance of
$\mathcal{X}_{n}$, the $k$-cut number of a path of length $n$, are proved. We
also show that $\mathcal{X}_{n}$, after rescaling, converges in distribution to
a limit $\mathcal{B}_{k}$, which has a complicated representation. The paper
then briefly discusses the $k$-cut number of some trees and general graphs. We
conclude by some analytic results which may be of interest.
