# The Maximum Time of 2-Neighbour Bootstrap Percolation: Complexity Results

• Thiago Marcilon
• Samuel Nascimento
• Rudini Sampaio
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8747)

## Abstract

In $$2$$-neighbourhood bootstrap percolation on a graph $$G$$, an infection spreads according to the following deterministic rule: infected vertices of $$G$$ remain infected forever and in consecutive rounds healthy vertices with at least $$2$$ already infected neighbours become infected. Percolation occurs if eventually every vertex is infected. The maximum time $$t(G)$$ is the maximum number of rounds needed to eventually infect the entire vertex set. In 2013, it was proved [7] that deciding if $$t(G)\ge k$$ is polynomial time solvable for $$k=2$$, but is NP-Complete for $$k=4$$ and is NP-Complete if the graph is bipartite and $$k=7$$. In this paper, we solve the open questions. Let $$n = |V(G)|$$ and $$m = |E(G)|$$. We obtain an $$\varTheta (m n^5)$$-time algorithm to decide if $$t(G)\ge 3$$ in general graphs. In bipartite graphs, we obtain an $$\varTheta (m n^3)$$-time algorithm to decide if $$t(G)\ge 3$$ and an $$O(m n^{13})$$-time algorithm to decide if $$t(G)\ge 4$$. We also prove that deciding if $$t(G)\ge 5$$ is NP-Complete in bipartite graphs.

## Keywords

2-Neighbour bootstrap percolation $$P_3$$-convexity Maximum time Infection on graphs

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© Springer International Publishing Switzerland 2014

## Authors and Affiliations

• Thiago Marcilon
• 1
Email author
• Samuel Nascimento
• 1
• Rudini Sampaio
• 1
1. 1.Dept. ComputaçãoUniversidade Federal Do CearáFortalezaBrazil