The fact that both the head and the dependent are kept in either λ 2 or β makes it possible to construct non-projective dependency graphs, because the NO-ARC n transition allows a node to be passed from λ 1 to λ 2 even if it does not (yet) have a head.
an arc can only be added between two nodes i and j if the dependent end of the arc is not the artificial root 0 and does not already have a head, which would violate ROOT and SINGLE-HEAD, respectively, and if there is no path connecting the dependent to the head, which would cause a violation of ACYCLICITY.