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.

For $n \ge k \ge r$, we denote by M(n, k, r) the maximum number of edges in an n-vertex graph G satisfying $m(G,r)>k$. We determine the precise value of M(n, k, 2) and M(n, k, k) assuming that n is sufficiently large compared to k.

D. Freund—Supported in part by U.S. Army Research Office grant W911NF-14-1-0477.

M. Poloczek—Supported by the Alexander von Humboldt Foundation within the Feodor Lynen program, and in part by NSF grant CCF-1115256.

D. Reichman—Supported in part by NSF grants IIS-0911036 and CCF-1214844, AFOSR grant FA9550-08-1-0266, and ARO grant W911NF-14-1-0017.