Bootstrap percolation, connectivity, and graph distance

Bootstrap percolation is a process defined on a graph which begins with an initial set of infected vertices. In each subsequent round, an uninfected vertex becomes infected if it is adjacent to at least r previously infected vertices. If an initially infected set of vertices, A0, begins a process in which every vertex of the graph eventually becomes infected, then we say that A0 percolates. In this paper we investigate bootstrap percolation as it relates to graph distance and connectivity. We find a sufficient condition for the existence of cardinality 2 percolating sets in diameter 2 graphs when r = 2. We also investigate connections between connectivity and bootstrap percolation and lower and upper bounds on the number of rounds to percolation in terms of invariants related to graph distance.