Largest and Smallest Minimal Percolating Sets in Trees

Bootstrap percolation is the process on a graph where, given an initial infected set, vertices with at least r infected neighbors are infected until no new vertices can be infected. A set percolates if it infects all the vertices of the graph, and a percolating set is minimal if no proper subset percolates. We consider the question for trees. We describe an O(n) algorithm for computing the largest and smallest minimal percolating sets and show that if A is a minimal percolating set on a tree T with n vertices and ` leaves, (r−1)n+1 r ≤ |A| ≤ rn+(r−1)` r+1 . Moreover, we show that the difference between the sizes of a largest and smallest minimal percolating set divided by the number of vertices is less than r−1 r2 .