The maximum time of 2-neighbour bootstrap percolation: algorithmic aspects

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. In this paper, we are interested in calculating the maximum time t (G) the process can take, in terms of the number of rounds needed to eventually infect the entire vertex set. We prove that the problem of deciding if t (G) ≥ k is NP-complete for: (a) fixed k ≥ 4; (b) bipartite graphs with fixed k ≥ 7; and (c) planar bipartite graphs. Moreover, we obtain polynomial time algorithms for (a) k ≥ 2, (b) chordal graphs and (c) (q, q - 4)-graphs, for every fixed q.KeywordsBipartite GraphPolynomial Time AlgorithmChordal GraphAlgorithmic AspectPercolation TimeThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.