  The degree of a based graph is the number of essential nonbasepoint vertices
after generic perturbation. Hatcher--Vogtmann's degree theorem states that the
subcomplex of Auter space of graphs of degree at most d is (d-1)-connected. We
extend the definition of degree to the simplicial closure of Auter space and
prove a version of Hatcher--Vogtmann's result in this context.
