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

Heegaard surfaces for certain graphs in compressionbodies