A minor-closed class of graphs is addable if each excluded minor is 2-connected. We see that such a class A of labelled graphs has smooth growth; and, for the random graph R n sampled uniformly from the n-vertex graphs in A, the fragment not in the giant component asymptotically has a simple 'Boltzmann Poisson distribution'. In particular, as n → ∞ the probability that R n is connected tends to 1/A(ρ), where A(x) is the exponential generating ...