The $k$-cut model in deterministic and random trees

Abstract

The k-cut number of rooted graphs was introduced by Cai et al. [12] as a generalization of the classical cutting model by Meir and Moon [30]. In this paper, we show that all moments of the k-cut number of conditioned Galton-Watson trees converges 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 [25]. 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.

$k$-cut model for the Brownian Continuum Random Tree

“…In the case k = 1, it is also known that Z 1 can be explicitly written as a functional of the so-called Aldous-Pitman fragmentation process, thanks to the works of Addario-Berry, Broutin & Holmgren [2], Bertoin & Miermont [5], Abraham & Delmas [1]. In this work, we extend this construction of Z 1 to the general setting of k ≥ 1, thus answering a question in [6] on the construction of Z k . To that end, let us start with a brief introduction to the Aldous-Pitman fragmentation process.…”

“…jointly with the convergences in (7). Combined with (6), this shows the convergence of both marginals in (13). To get to the joint convergence, it suffices to note that the law of (T , X k (T )) is the unique limit point of those on the left-hand side, which follows from the joint convergence in ( 16) and the fact that the family (R p ) p≥1 uniquely determines the law of (T , d, µ).…”

“…For n ≥ 1, let T n be a Galton-Watson tree with offspring distribution ξ conditioned on having n vertices. Berzunza, Cai and Holmgren show in [6] that…”

“…We'll give two proofs to the theorem. In Section 2, we give a first proof by identifying the conditional moments of X k (T ) given T with those of Z k , which were computed in [6]. In Section 3, we give a second proof via weak convergence arguments.…”

A Modification of the Random Cutting Model

“…In recent years, modifications of the original cutting model have been discussed: Kuba and Panholzer regarded the case of isolating a leaf, a general node, or multiple nodes simultaneously (instead of isolating the root) in [KP08a,KP08b,KP13], and Cai, Holmgren et al proposed and investigated the k-cut model in [CHDS19, CH19,BCH19], where a node is only removed after it has been cut for the k-th time.…”