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 on n vertices. We examine density conditions that ensure m(G,r) = r for all r >= 2. With respect to the minimum degree, we prove that such a smallest possible contagious set is guaranteed to exist if and only if G has minimum degree at least (k1)/k * n. Moreover, we study the speed with which the activation spreads and provide tight upper bounds on the number of rounds it takes until all nodes are activated in such a graph. We also investigate what average degree asserts the existence of small contagious sets. For n >= k >= r, we denote by M(n,k,r) the maximum number of edges in an nvertex 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.
 Publication:

arXiv eprints
 Pub Date:
 February 2015
 arXiv:
 arXiv:1503.00158
 Bibcode:
 2015arXiv150300158F
 Keywords:

 Computer Science  Discrete Mathematics;
 Mathematics  Combinatorics;
 05C35;
 68R10;
 G.2.2
 EPrint:
 Extended version of the IWOCA'15 paper that generalizes the results on the minimum degree condition and the speed of the activation process to arbitrary values for the threshold parameter r