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.
This research was supported by Capes, CNPq, FUNCAP and FAPERJ
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Benevides, F., Campos, V., Dourado, M.C., Sampaio, R.M., Silva, A. (2013). The maximum time of 2-neighbour bootstrap percolation: algorithmic aspects. In: Nešetřil, J., Pellegrini, M. (eds) The Seventh European Conference on Combinatorics, Graph Theory and Applications. CRM Series, vol 16. Edizioni della Normale, Pisa. https://doi.org/10.1007/978-88-7642-475-5_22
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DOI: https://doi.org/10.1007/978-88-7642-475-5_22
Publisher Name: Edizioni della Normale, Pisa
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