  We show that the domination number of the binomial random graph G_{n,p} with
edge-probability p is concentrated on two values for p \ge n^{-2/3+\eps}, and
not concentrated on two values for general p \le n^{-2/3}. This refutes a
conjecture of Glebov, Liebenau and Szabo, who showed two-point concentration
for p \ge n^{-1/2+\eps}, and conjectured that two-point concentration fails for
p \ll n^{-1/2}.
  The proof of our main result requires a Poisson type approximation for the
probability that a random bipartite graph has no isolated vertices, in a regime
where standard tools are unavailable (as the expected number of isolated
vertices is relatively large). We achieve this approximation by adapting the
proof of Janson's inequality to this situation, and this adaptation may be of
broader interest.
