The maximum time of 2-neighbor bootstrap percolation: Complexity 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 2 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 [7] that deciding whether t(G) ≥ k is polynomial time solvable for k = 2, but is NP-Complete for k = 4 and, if the problem is restricted to bipartite graphs, it is NP-Complete for k = 7. In this paper, we solve the open questions. We obtain an O(mn 5 )time algorithm to decide whether t(G) ≥ 3. For bipartite graphs, we obtain an O(mn 3 )-time algorithm to decide whether t(G) ≥ 3, an O(m 2 n 9 )-time algorithm to decide whether t(G) ≥ 4 and we prove that t(G) ≥ 5 is NP-Complete.Keywords: 2-neighbor bootstrap percolation, P 3 -Convexity, maximum time, infection on graphs t r (G, S, v) = ∞ if there is no such t). We say that a set S (0) infects G (or that S (0) is a percolating set of G) if eventually every vertex of G becomes infected, that is, there exists t such that S (t) = V (G). If S is a percolating set of G, then we define t r (G, S) as the minimum t such that S (t) = V (G). Also, define the percolation time of G as t r (G) = max{t r (G, S) : S infects G}. In this paper, we shall focus on the case where r = 2 and in such case we omit the subscript of the functions t r (G, S, v), t r (G, S) and t r (G).Bootstrap percolation was introduced by Chalupa, Leath and Reich [15] as a model for certain interacting particle systems in physics. Since then it has found applications in clustering phenomena, sandpiles [22], and many other areas of statistical physics, as well as in neural networks [1] and computer science [18].There are two broad classes of questions one can ask about bootstrap percolation. The first, and the most extensively studied, is what happens when the initial configuration S (0) is chosen randomly under some probability distribution? For example, vertices are included in S (0) independently with some fixed probability p. One would like to know how likely percolation is to occur, and if it does occur, how long it takes.The answer to the first of these questions is now well understood for various graphs. An interesting case is the one of the lattice graph [n] d , in which d is fixed and n tends to infinity, since the probability of percolation under the r-neighbor model displays a sharp threshold between no percolation with high probability and percolation with high probability. The existence of thresholds in the strong sense just described first appeared in papers by Holroyd, Balogh, Bollobás, Duminil-Copin and Morris [24,5,3]. Sharp thresholds have also been proved for the hypercube (Balogh and Bollobás [2], and Balogh, Bollobás and Morris [6]). There are also very recent results due to Bollobás, Holmgren, Smith and Uzzell [11], about the time perco...

On the $P_3$-Hull Number of Kneser Graphs

“…The zero forcing number of G is the size of a smallest set S of initially infected vertices that forces the whole graph to become infected. Another infection problem is the bootstrap percolation on a graph (see for example, [4,3,23,24,26,27] and references therein): an infection spreads over the vertices of a connected graph G following a deterministic spreading rule in such a way that an infected vertex will remain infected forever. Given a set S ⊆ V (G) of initially infected vertices, we can build a sequence S 0 = S, S 1 , S 2 , .…”

“…The 2-neighbor bootstrap percolation problem has been studied by several authors. For example, the maximum percolation time of the 2-neighbor bootstrap percolation problem has been studied by Benevides et al [4], Marcilon et al [23] and Przykucki [26]. The smallest or largest size of a percolating set with a given property has been studied by Benevides et al [3] and Morris [24].…”

3-Neighbor bootstrap percolation on grids

“…A minimum r-percolating set in G is an r-percolating set S of G satisfying m(G, r) = |S|. Bootstrap percolation is very well studied in graphs, see, for example, [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][23][24][25][26][27].…”

On the $P_3$-hull number and infecting times of generalized Petersen graphs

“…As introduced in [4] and [7], the P 3 -hull number of a simple connected graph is the minimum cardinality of a set U of initially infected vertices that will eventually infect the entire graph where an uninfected node becomes infected if two or more of its neighbors are infected. There has been much work on formulas for the P 3 -hull numbers of various types of graphs, [5,6,8,10], as well as with the closely related notion of the 2-neighbor bootstrap percolation problem, [3,12,13].…”