K-cut on paths and some trees

TL;DR

This paper introduces the k-cut number for rooted graphs, analyzing its expectation, variance, and distribution for paths, and discusses extensions to trees and general graphs, providing new insights into network destruction models.

Contribution

It defines the k-cut number for graphs, derives its expectation, variance, and distribution for paths, and explores extensions to trees and general graphs.

Findings

First order expectation and variance of the k-cut number for paths are established.

Rescaled k-cut number converges in distribution to a complex limit .

Analytic results for k-cut numbers on trees and graphs are provided.

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 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.