K-cut on paths and some trees

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.