  We show that the profile of the tree constructed by the Depth First Search
Algorithm in the giant component of an Erd\H{o}s-R\'enyi graph with $N$
vertices and connection probability $c/N$ converges to an explicit
deterministic shape. This makes it possible to exhibit a long non-intersecting
path of length $\left( \rho_c - \frac{\mathrm{Li}_2(\rho_c)}{c} \right) \times
N$, where $\rho_c$ is the density of the giant component.
