A Modification of the Random Cutting Model

TL;DR

This paper introduces a modified graph destruction model that randomly removes vertices while preserving source-connected components, analyzing the number of cuts needed to eliminate targets and the residual graph size.

Contribution

It presents a novel modification to the random cutting model, providing theoretical results on the remaining graph size and limiting distributions for specific tree structures.

Findings

Remaining graph size is tight for expander graphs.

Derived limiting distributions for binary caterpillar and complete binary trees.

Model interpolates between existing random cutting and percolation models.

Abstract

We propose a modification to the random destruction of graphs: Given a finite network with a distinguished set of sources and targets, remove (cut) vertices at random, discarding components that do not contain a source node. We investigate the number of cuts required until all targets are removed, and the size of the remaining graph. This model interpolates between the random cutting model going back to Meir and Moon and site percolation. We prove several general results, including that the size of the remaining graph is a tight family of random variables for compatible sequences of expander-type graphs, and determine limiting distributions for binary caterpillar trees and complete binary trees.