## Electronic Journal of Probability

### $k$-cut on paths and some trees

#### 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.

#### Article information

Source
Electron. J. Probab., Volume 24 (2019), paper no. 53, 22 pp.

Dates
Accepted: 7 May 2019
First available in Project Euclid: 5 June 2019

https://projecteuclid.org/euclid.ejp/1559700303

Digital Object Identifier
doi:10.1214/19-EJP318

Zentralblatt MATH identifier
07068784

Subjects
Primary: 60C05: Combinatorial probability

Keywords
cutting k-cut random trees

#### Citation

Cai, Xing Shi; Holmgren, Cecilia; Devroye, Luc; Skerman, Fiona. $k$-cut on paths and some trees. Electron. J. Probab. 24 (2019), paper no. 53, 22 pp. doi:10.1214/19-EJP318. https://projecteuclid.org/euclid.ejp/1559700303

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