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

Abstract

Let $r\ge 1$ be any non negative integer and let $G=(V,E)$ be any undirected graph in which a subset $D\subseteq V$ 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 $s_r\left(G\right)$ 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 $s_1\left(G\right)=1$ for any connected graph $G$. The case when $G$ is the $n\times n$ grid, $G_{n\times n}$, and $r=2$ is well known and appears in many puzzle books, in particular due to the elegant proof that shows that $s_2\left(G_{n\times n}\right)=n$ for all $n\in N$. We study the cases of square grids, $G_{n\times n}$, and tori, $T_{n\times n}$, when $r\in \{3,4\}$. We show that $s_3\left(G_{n\times n}\right)=\lceil \frac{n^2+2n+4}{3}\rceil$ for every $n$ even and that $\lceil \frac{n^2+2n}{3}\rceil \le s_3\left(G_{n\times n}\right)\le \lceil \frac{n^2+2n}{3}\rceil +1$ for any $n$ odd. When $n$ is odd, we show that both bounds are reached, namely $s_3\left(G_{n\times n}\right)=\lceil \frac{n^2+2n}{3}\rceil$ if $n\equiv 5(mod6)$ or $n=2^p-1$ for any $p\in N^{\ast }$, and $s_3\left(G_{n\times n}\right)=\lceil \frac{n^2+2n}{3}\rceil +1$ if $n\in \{9,13\}$. Finally, for all $n\in N$, we give the exact expression of $s_3\left(T_{n\times n}\right)$.