  We introduce an algorithm that constructs a random uniform graph with
prescribed degree sequence together with a depth first exploration of it. In
the so-called supercritical regime where the graph contains a giant component,
we prove that the renormalized contour process of the Depth First Search Tree
has a deterministic limiting profile that we identify. The proof goes through a
detailed analysis of the evolution of the empirical degree distribution of
unexplored vertices. This evolution is driven by an infinite system of
differential equations which has a unique and explicit solution. As a
byproduct, we deduce the existence of a macroscopic simple path and get a lower
bound on its length.
