Minimum lethal sets in grids and tori under 3-neighbour bootstrap percolation

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Let r≥1 be any non negative integer and let G=(V,E) be any undirected graph in which a subset D⊆V of vertices are initially infected. We consider the process in which, at every step, each non-infected vertex with at least r infected neighbours becomes infected and an infected vertex never becomes non-infected. The problem consists in determining the minimum size sr(G) of an initially infected vertices set D that eventually infects the whole graph G. This problem is closely related to cellular automata, to percolation problems and to the Game of Life studied by John Conway. Note that s1(G)=1 for any connected graph G. The case when G is the n×n grid, Gn×n, and r=2 is well known and appears in many puzzle books, in particular due to the elegant proof that shows that s2(Gn×n)=n for all n∈N. We study the cases of square grids, Gn×n, and tori, Tn×n, when r∈{3,4}. We show that s3(Gn×n)=⌈n2+2n+43⌉ for every n even and that ⌈n2+2n3⌉≤s3(Gn×n)≤⌈n2+2n3⌉+1 for any n odd. When n is odd, we show that both bounds are reached, namely s3(Gn×n)=⌈n2+2n3⌉ if n≡5(mod6) or n=2p−1 for any p∈N∗, and s3(Gn×n)=⌈n2+2n3⌉+1 if n∈{9,13}. Finally, for all n∈N, we give the exact expression of s3(Tn×n).