Abstract
The \(k\)-cut number of rooted graphs was introduced by Cai et al. as a generalization of the classical cutting model by Meir and Moon. In this paper, we show that all moments of the \(k\)-cut number of conditioned Galton-Watson trees converge after proper rescaling, which implies convergence in distribution to the same limit law regardless of the offspring distribution of the trees. This extends the result of Janson. Using the same method, we also show that the \(k\)-cut number of various random or deterministic trees of logarithmic height converges in probability to a constant after rescaling, such as random split-trees, uniform random recursive trees, and scale-free random trees.
Highlights
In order to measure the difficulty for the destruction of a resilient network Cai et al [12] introduced a generalization of the cut model of Meir and Moon [30] where each vertex needs to be cut k ∈ N times before it is destroyed
Let Tn be a Galton-Watson tree conditioned on its number of vertices being n ∈ N with offspring distribution ξ satisfying (1)
The class of random split trees was first introduced by Devroye [13] to encompass many families of trees that are frequently used in algorithm analysis, e.g., binary search trees and tries
Summary
In order to measure the difficulty for the destruction of a resilient network Cai et al [12] introduced a generalization of the cut model of Meir and Moon [30] where each vertex (or edge) needs to be cut k ∈ N times (instead of only once) before it is destroyed. The appearance of the Brownian CRT in this framework should not come as a surprise since it is well-known that if we assign length n−1/2 to each edge of the Galton-Watson tree Tn, the latter converges weakly the electronic journal of combinatorics 28(1) (2021), #P1.25 to a Brownian CRT as n → ∞ We believe that this connection can be exploited even more than the one used in this work in order to obtain the precise distribution of ZCRT.
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