  We consider the performance of the Depth First Search (DFS) algorithm on the
random graph $G\left(n,\frac{1+\epsilon}{n}\right)$, $\epsilon>0$ a small
constant. Recently, Enriquez, Faraud and M\'enard [2] proved that the stack $U$
of the DFS follows a specific scaling limit, reaching the maximal height of
$(1+o_{\epsilon}(1))\epsilon^2n$. Here we provide a simple analysis for the
typical length of a maximum path discovered by the DFS.
