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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)

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. 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.

Keywords

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

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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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