Maximal induced paths and minimal percolating sets in hypercubes

For a graph G, the r-bootstrap percolation process can be described as follows: Start with an initial set A of infected” vertices. Infect any vertex with at least r infected neighbours, and continue this process until no new vertices can be infected. A is said to percolate in G if eventually all the vertices of G are infected. A is a minimal percolating set in G if A percolates in G and no proper subset of A percolates in G. An induced path, P, in a hypercube Qn is maximal if no induced path in Qn properly contains P. Induced paths in hypercubes are also called snakes. We study the relationship between maximal snakes and minimal percolating sets (under 2-bootstrap percolation) in hypercubes. In particular, we show that every maximal snake contains a minimal percolating set, and that every minimal percolating set is contained in a maximal snake.