Abstract
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 [21] 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.
Citation
Xing Shi Cai. Cecilia Holmgren. Luc Devroye. Fiona Skerman. "$k$-cut on paths and some trees." Electron. J. Probab. 24 1 - 22, 2019. https://doi.org/10.1214/19-EJP318