Electronic Journal of Probability

$k$-cut on paths and some trees

Xing Shi Cai, Cecilia Holmgren, Luc Devroye, and Fiona Skerman

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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

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

Received: 9 April 2018
Accepted: 7 May 2019
First available in Project Euclid: 5 June 2019

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Primary: 60C05: Combinatorial probability

cutting k-cut random trees

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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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