Let H m(n) be a random graph on n vertices, grown by adding vertices one at a time, joining each new vertex to a uniformly chosen set of m earlier vertices. If edges of H m(n) are deleted independently, each being retained with probability p, then there is a "phase transition". There is a certain critical value p c of p such that, with high probability, a component of order θ(n) remains as n → ∞ if and only if p > p c. Among other results, we obtain the exact value of p c, which depends on...