  Let M be a compressionbody containing a graph T (with at least one edge) such
that \boundary_+ M is parallel to the union of T and \boundary_- M. We extend
methods of Hayashi and Shimokawa to classify bridge surfaces for T. The results
of this paper are used in later work to show that if a bridge surface for a
graph in a 3-manifold is c-weakly reducible then either a degenerate situation
occurs or the exterior of the graph contains an essential meridional surface.
