  In this note, we consider the width of a supercritical random graph according
to some commonly studied width measures. We give short, direct proofs of
results of Lee, Lee and Oum, and of Perarnau and Serra, on the rank- and
tree-width of the random graph $G(n,p)$ when $p= \frac{1+\epsilon}{n}$ for
$\epsilon > 0$ constant. Our proofs avoid the use, as a black box, of a result
of Benjamini, Kozma and Wormald on the expansion properties of the giant
component in this regime, and so as a further benefit we obtain explicit bounds
on the dependence of these results on $\epsilon$. Finally, we also consider the
width of the random graph in the weakly supercritical regime, where $\epsilon =
o(1)$ and $\epsilon^3n \to \infty$. In this regime, we determine, up to a
constant multiplicative factor, the rank- and tree-width of $G(n,p)$ as a
function of $n$ and $\epsilon$.
