The 2-planar parser can be restricted to graphs satisfying the SINGLE-HEAD and ACYCLICITY constraints in exactly the same way as the planar parser, and the proofs follow the same line of reasoning.
a version of the 2-planar parser that is sound and complete for the set of 2-planar dependency forests (that is, 2-planar dependency graphs without cycles and with each node having at most one head) can be defined as follows: only if ¬∃k | (k, i) ∈ A (single-head) and ¬i ↔ * j ∈ A (acyclicity) RIGHT-ARC 2P−ShAc = RIGHT-ARC H β (C)∩{(σ 1 |i,σ 2 ,j|β,A)∈C|¬(i↔ * j∈A)} only if ¬∃k|(k, j) ∈ A (single-head) and ¬i ↔ * j ∈ A (acyclicity) Because structures in dependency treebanks are typically restricted to forests, this is the version of the 2-planar parser that we use in the experimental evaluation in Section 5.2.