International Workshop on Combinatorial Algorithms

Combinatorial Algorithms pp 185-196

# Contagious Sets in Dense Graphs

• Daniel Freund
• Matthias Poloczek
• Daniel Reichman
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9538)

## 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(Gr) 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(nkr) the maximum number of edges in an n-vertex graph G satisfying $$m(G,r)>k$$. We determine the precise value of M(nk, 2) and M(nkk) assuming that n is sufficiently large compared to k.

### Keywords

Bootstrap percolation Target set selection Extremal graph theory

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© Springer International Publishing Switzerland 2016

## Authors and Affiliations

• Daniel Freund
• 1
• Matthias Poloczek
• 2
• Daniel Reichman
• 3
1. 1.Center for Applied MathematicsCornell UniversityIthacaUSA
2. 2.School of Operations Research and Information EngineeringCornell UniversityIthacaUSA
3. 3.Department of Computer ScienceCornell UniversityIthacaUSA