Largest Minimal Percolating Sets in Hypercubes under $2$-Bootstrap Percolation

Abstract

Consider the following process, known as $r$-bootstrap percolation, on a graph $G$. Designate some initial infected set $A$ and infect any vertex with at least $r$ infected neighbors, continuing until no new vertices can be infected. We say $A$ percolates if it eventually infects the entire graph. We say $A$ is a minimal percolating set if $A$ percolates, but no proper subset percolates. We compute the size of a largest minimal percolating set for $r=2$ in the $n$-dimensional hypercube.