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 \(\...

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Bibliographic Details
Published inarXiv.org
Main Authors Xing Shi Cai, Devroye, Luc, Holmgren, Cecilia, Skerman, Fiona
Format Paper Journal Article
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 30.01.2019
Subjects
Mathematics - Combinatorics
Mathematics - Probability
Rescaling
Trees
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Summary: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.
ISSN:2331-8422
DOI:10.48550/arxiv.1804.03069