Limiting shape of the Depth First Search tree in an Erd\H{o}s-R\'enyi graph

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 $c\mathrm{/}N$ converges to an explicit deterministic shape. This makes it possible to exhibit a long non-intersecting path of length $\left( \rho_c - \frac{$\mathrm{L}\mathrm{i}$_2(\rho_c)}{c} \right) \times N$,where$\rho_c$ is the density of the giant component.