The Maximum Time of 2-neighbour Bootstrap Percolation in Grid Graphs and Parametrized Results

Abstract

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 two 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 by Benevides et al. [10] that t(G) is NP-hard for planar graphs and that deciding whether $t(G)\ge k$ is polynomial time solvable for $k\le 2$, but is NP-complete for $k\ge 4$. They left two open problems about the complexity for $k=3$ and for planar bipartite graphs. In 2014, we solved the first problem [24]. In this paper, we solve the second one by proving that t(G) is NP-complete even in grid graphs with maximum degree 3. We also prove that t(G) is polynomial time solvable for solid grid graphs with maximum degree 3. Moreover, we prove that the percolation time problem is fixed parameter tractable with respect to the parameter treewidth$\,+\,k$ and maximum degree$\,+\,k$. Finally, we obtain polynomial time algorithms for several graphs with few $P_4$’s, as cographs and $P_4$-sparse graphs.