As already noted, planarity is a very mild relaxation of the better known projectivity constraint, the only difference being that planarity allows graphs with covered roots (see Definition 2), and studies of natural language treebanks have shown the vast majority of non-projective structures to be non-planar as well (Kuhlmann and Nivre 2006;Havelka 2007).
the parser by Goldberg and Elhadad (2010) can be seen as an implementation of this transition system, which uses a training algorithm that assigns a weight to each of the ATTACH-RIGHT(i) EF and ATTACH-LEFT(i) EF transitions in such a way that "easier" (more reliable) attachments are performed first.