Contagious Sets in Dense Graphs

Abstract

We study the activation process in undirected graphs known as bootstrap percolation: a vertex is active either if it belongs to a set of initially activated vertices or if at some point it had at least r active neighbors, for a threshold r that is identical for all vertices. A contagious set is a vertex set whose activation results with the entire graph being active. Let m(G, r) be the size of a smallest contagious set in a graph G. We examine density conditions that ensure that a given n-vertex graph \(G=(V,E)\) has a small contagious set. With respect to the minimum degree, we prove that if G has minimum degree \(n{\slash }2\) then \(m(G,2)=2\). We also provide tight upper bounds on the number of rounds until all nodes are active.