  Sentences with gapping, such as Paul likes coffee and Mary tea, lack an overt
predicate to indicate the relation between two or more arguments. Surface
syntax representations of such sentences are often produced poorly by parsers,
and even if correct, not well suited to downstream natural language
understanding tasks such as relation extraction that are typically designed to
extract information from sentences with canonical clause structure. In this
paper, we present two methods for parsing to a Universal Dependencies graph
representation that explicitly encodes the elided material with additional
nodes and edges. We find that both methods can reconstruct elided material from
dependency trees with high accuracy when the parser correctly predicts the
existence of a gap. We further demonstrate that one of our methods can be
applied to other languages based on a case study on Swedish.
