Percolating sets in bootstrap percolation on the Hamming graphs and triangular graphs

Bidgoli, M.R.; Mohammadian, Abdolmajid; Tayfeh-Rezaie, B.

doi:10.1016/j.ejc.2020.103256

Cited by 2 publications

(1 citation statement)

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“…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].…”

Section: Introductionmentioning

confidence: 99%

3-Neighbor bootstrap percolation on grids

Hedžet,

Henning

2023

Discuss. Math. Graph Theory

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Given a graph G and assuming that some vertices of G are infected, the r-neighbor bootstrap percolation rule makes an uninfected vertex v infected if v has at least r infected neighbors. The r-percolation number, m(G, r), of G is the minimum cardinality of a set of initially infected vertices in G such that after continuously performing the r-neighbor bootstrap percolation rule each vertex of G eventually becomes infected. In this paper, we consider the 3-bootstrap percolation number of grids with fixed widths. If G is the Cartesian product P 3 P m of two paths of orders 3 and m, we prove that m(G, 3) = 3 2 (m + 1) − 1, when m is odd, and m(G, 3) = 3 2 m + 1, when m is even. Moreover, if G is the Cartesian product P 5 P m , we prove that m(G, 3) = 2m + 2, when m is odd, and m(G, 3) = 2m + 3, when m is even. If G is the Cartesian product P 4 P m , we prove that m(G, 3) takes on one of two possible values, namely m(G, 3) = 5(m+1) 3

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