We have extended Maruyama's [5, 6, 7] constraint dependency grammar (CDG) to process a lattice or graph of sentence hypotheses instead of separate text strings. A post-processor to a speech recognizer producing N-best hypotheses generates the word graph representation, which is then augmented with information required for parsing. We will summarize the CDG parsing algorithm and then describe how the algorithm is extended to process a word graph on a single processor machine.