One of the goals of this in progress work made in collaboration with Foissy, Giscard and Ronco is to describe algebraically, thanks to Hopf algebras, the reconstruction of any walk of a given graph from simple cycles and self-avoiding walks. In this talk, we will first remind the definition of a combinatorial Hopf algebra. Then, we will present the Lawler's loop erasing procedure, detail the construction of a co-pre-Lie coproduct on walks and and explain how make walks into Hopf algebras. Finally, we will explain how we can calculate an algebraic reconstruction of the identity by using the dual algebraic structure.