Deep dependency parsing can be cast as the search for maximum acyclic subgraphs in weighted digraphs. Because this search problem is intractable in the general case, we consider its restriction to the class of 1-endpoint-crossing (1ec) graphs, which has high coverage on standard data sets. Our main contribution is a characterization of 1ec graphs as a subclass of the graphs with pagenumber at most 3. Building on this we show how to extend an existing parsing algorithm for 1-endpoint-crossing trees to the full class. While the runtime complexity of the extended algorithm is polynomial in the length of the input sentence, it features a large constant, which poses a challenge for practical implementations.