  We show that the chronological removal of cycles from a walk on a graph,
known as Lawler's loop-erasing procedure, generates a preLie co-algebra on the
vector space spanned by the walks. In addition, we prove that the tensor and
symmetric algebras of graph walks are graded Hopf algebras, provide their
antipodes explicitly and recover the preLie co-algebra from a brace coalgebra
on the tensor algebra of graph walks. Finally we exhibit sub-Hopf algebras
associated to particular types of walks.
