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In 2-neighborhood bootstrap percolation on a graph G, an infection spreadsaccording to the following deterministic rule: infected vertices of G remaininfected forever and in consecutive rounds healthy vertices with at least 2already infected neighbors become infected. Percolation occurs if eventuallyevery vertex is infected. The maximum time t(G) is the maximum number of roundsneeded to eventually infect the entire vertex set. In 2013, it was proved\cite{eurocomb13} that deciding whether $t(G)\geq k$ is polynomial timesolvable for k=2, but is NP-Complete for k=4 and, if the problem is restrictedto bipartite graphs, it is NP-Complete for k=7. In this paper, we solve theopen questions. We obtain an $O(mn^5)$-time algorithm to decide whether$t(G)\geq 3$. For bipartite graphs, we obtain an $O(mn^3)$-time algorithm todecide whether $t(G)\geq 3$, an $O(m^2n^9)$-time algorithm to decide whether$t(G)\geq 4$ and we prove that $t(G)\geq 5$ is NP-Complete.

The maximum time of 2-neighbor bootstrap percolation: complexity results