Maximal induced paths and minimal percolating sets in hypercubes

Abstract

For a graph $G$, the \emph{$r$-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 \emph{percolate in $G$} if eventually all the vertices of $G$ are infected. $A$ is a \emph{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 $Q_n$ is maximal if no induced path in $Q_n$ 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.