Dismantling sparse random graphs

TL;DR

This paper investigates the minimal vertex removal needed to break down large components in sparse random and regular graphs, revealing a universal behavior as the graph size grows.

Contribution

It demonstrates that the vertex removal threshold is asymptotically the same across all large enough component sizes in sparse and regular graphs.

Findings

Threshold is consistent across different k values tending to infinity.

Removals are asymptotically optimal for large sparse graphs.

Universal behavior observed in vertex removal across graph types.

Abstract

We consider the number of vertices that must be removed from a graph G in order that the remaining subgraph has no component with more than k vertices. Our principal observation is that, if G is a sparse random graph or a random regular graph on n vertices with n tending to infinity, then the number in question is essentially the same for all values of k such that k tends to infinity but k=o(n).