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

In 2-neighborhood 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 neighbors 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 \cite{eurocomb13} that deciding whether $t(G)\geq k$ is polynomial time solvable for k=2, but is NP-Complete for k=4 and, if the problem is restricted to bipartite graphs, it is NP-Complete for k=7. In this paper, we solve the open 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 to decide 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.