Deterministic bootstrap percolation on trees

Abstract

In a graph, k-bootstrap percolation is a process by which an “infection” spreads from an initial set of infected vertices, according to the rule that on each iteration an uninfected vertex with k infected neighbors becomes infected. This process continues until either every vertex is infected or every uninfected vertex has fewer than k infected neighbors. We are particularly interested in the case where every vertex is eventually infected. The cardinality of a smallest set that results in this is the k-bootstrap percolation number of the graph. In this paper, we determine the k-bootstrap percolation number for trees of small diameter, spiders, complete N-ary trees, and caterpillars. For these graph families we also consider the smallest number of iterations needed for any smallest set to spread to the entire graph. Finally, we give an upper bound for the k-bootstrap percolation number for general trees which improves upon previous results.