In this contribution, we deal with graph parsing, i.e., mapping input strings to graph-structured output representations, using tree approximations. We experiment with the data from the SemEval 2014 Semantic Dependency Parsing (SDP) task. We define various tree approximation schemes for graphs, and make twofold use of them. First, we statically analyze the semantic dependency graphs, seeking to unscover which linguistic phenomena in particular require the additional annotation expressivity provided by moving from trees to graphs. We focus on undirected base cycles in the SDP graphs, and discover strong connections to grammatical control and coordination. Second, we make use of the approximations in a statistical parsing scenario. In it, we convert the training set graphs to dependency trees, and use the resulting treebanks to build standard dependency tree parsers. We perform lossy graph reconstructions on parser outputs, and evaluate our models as dependency graph parsers. Our system outperforms the baselines by a large margin, and evaluates as the best non-voting tree approximation–based parser on the SemEval 2014 data, scoring at just over 81% in labeled F1.