The Maximum Time of 2-Neighbour Bootstrap Percolation: Complexity Results

  • Thiago MarcilonEmail author
  • Samuel Nascimento
  • Rudini Sampaio
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8747)


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. The maximum time \(t(G)\) is the maximum number of rounds needed to eventually infect the entire vertex set. In 2013, it was proved [7] that deciding if \(t(G)\ge k\) is polynomial time solvable for \(k=2\), but is NP-Complete for \(k=4\) and is NP-Complete if the graph is bipartite and \(k=7\). In this paper, we solve the open questions. Let \(n = |V(G)|\) and \(m = |E(G)|\). We obtain an \(\varTheta (m n^5)\)-time algorithm to decide if \(t(G)\ge 3\) in general graphs. In bipartite graphs, we obtain an \(\varTheta (m n^3)\)-time algorithm to decide if \(t(G)\ge 3\) and an \(O(m n^{13})\)-time algorithm to decide if \(t(G)\ge 4\). We also prove that deciding if \(t(G)\ge 5\) is NP-Complete in bipartite graphs.


2-Neighbour bootstrap percolation \(P_3\)-convexity Maximum time Infection on graphs 


  1. 1.
    Amini, H.: Bootstrap percolation in living neural networks. J. Stat. Phys. 141(3), 459–475 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Balogh, J., Bollobás, B.: Bootstrap percolation on the hypercube. Probab. Theor. Relat. Fields 134(4), 624–648 (2006)CrossRefzbMATHGoogle Scholar
  3. 3.
    Balogh, J., Bollobás, B., Duminil-Copin, H., Morris, R.: The sharp threshold for bootstrap percolation in all dimensions. Trans. Amer. Math. Soc. 364(5), 2667–2701 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Balogh, J., Pete, G.: Random disease on the square grid. Random Struct. Algorithms 13, 409–422 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Balogh, J., Bollobás, B., Morris, R.: Bootstrap percolation in three dimensions. Ann. Probab. 37(4), 1329–1380 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Balogh, J., Bollobás, B., Morris, R.: Bootstrap percolation in high dimensions. Combin. Probab. Comput. 19(5–6), 643–692 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Benevides, F., Campos, V., Dourado, M.C., Sampaio, R.M., Silva, A.: The maximum time of 2-neighbour bootstrap percolation: algorithmic aspects. The Seventh European Conference on Combinatorics, Graph Theory and Applications. CRM Series, vol. 16, pp. 135–139. Scuola Normale Superiore, Pisa (2013)CrossRefGoogle Scholar
  8. 8.
    Benevides, F., Przykucki, M.: Maximal percolation time in two-dimensional bootstrap percolation (Submitted)Google Scholar
  9. 9.
    Benevides, F., Przykucki, M.: On slowly percolating sets of minimal size in bootstrap percolation. Electron. J. Comb. 20(2), P46 (2013)MathSciNetGoogle Scholar
  10. 10.
    Bollobás, B., Holmgren, C., Smith, P.J., Uzzell, A.J.: The time of bootstrap percolation with dense initial sets (Submitted)Google Scholar
  11. 11.
    Calder, J.: Some elementary properties of interval convexities. J. London Math. Soc. 3, 422–428 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Centeno, C., Dourado, M.C., Penso, L., Rautenbach, D., Szwarcfiter, J.L.: Irreversible conversion of graphs. Theor. Comput. Sci. 412, 3693–3700 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Chalupa, J., Leath, P.L., Reich, G.R.: Bootstrap percolation on a bethe lattice. J. Phys. C 12(1), 31–35 (1979)CrossRefGoogle Scholar
  14. 14.
    Chen, N.: On the approximability of influence in social networks. SIAM J. Discrete Math. 23(3), 1400–1415 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Dreyer, P.A., Roberts, F.S.: Irreversible k-threshold processes: graph-theoretical threshold models of the spread of disease and of opinion. Discrete Appl. Math. 157(7), 1615–1627 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Duchet, P.: Convex sets in graphs, II. minimal path convexity. J. Comb. Theor. B 44, 307–316 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Erdős, P., Fried, E., Hajnal, A., Milner, E.C.: Some remarks on simple tournaments. Algebra Univers. 2, 238–245 (1972)CrossRefGoogle Scholar
  18. 18.
    Farber, M., Jamison, R.E.: Convexity in graphs and hypergraphs. SIAM J. Algebraic Discrete Methods 7, 433–444 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Fey, A., Levine, L., Peres, Y.: Growth rates and explosions in sandpiles. J. Stat. Phys. 138, 143–159 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Holroyd, A.E.: Sharp metastability threshold for two-dimensional bootstrap percolation. Probab. Theor. Relat. Fields 125(2), 195–224 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Levi, F.W.: On Helly’s theorem and the axioms of convexity. J. Indian Math. Soc. 15, 65–76 (1951)zbMATHGoogle Scholar
  22. 22.
    Morris, R.: Minimal percolating sets in bootstrap percolation. Electron. J. Comb. 16(1), 20 (2009)Google Scholar
  23. 23.
    Przykucki, M.: Maximal percolation time in hypercubes under 2-bootstrap percolation. Electron. J. Comb. 19(2), 41 (2012)MathSciNetGoogle Scholar
  24. 24.
    Riedl, E.: Largest minimal percolating sets in hypercubes under 2-bootstrap percolation. Electron. J. Comb. 17(1), 13 (2010)MathSciNetGoogle Scholar
  25. 25.
    Van de Vel, M.L.J.: Theory of Convex Structures. North-Holland, Amsterdam (1993)zbMATHGoogle Scholar
  26. 26.
    Barbosa, R.M., Coelho, E.M.M., Dourado, M.C., Rautenbach, D., Szwarcfiter, J.L.: On the carathodory number for the convexity of paths of order three. SIAM J. Discrete Math. 26, 929–939 (2012)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Thiago Marcilon
    • 1
    Email author
  • Samuel Nascimento
    • 1
  • Rudini Sampaio
    • 1
  1. 1.Dept. ComputaçãoUniversidade Federal Do CearáFortalezaBrazil

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