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We study the existence of incompressible embeddings of surfaces into thegenus two handlebody. We show that for every compact surface with boundary,orientable or not, there is an incompressible embedding of the surface into thegenus two handlebody. In the orientable case the embedding can be eitherseparating or non-separating. We also consider the case in which the genus twohandlebody is replaced by an orientable 3-manifold with a compressible boundarycomponent of genus greater than or equal to two.

Incompressible surfaces in handlebodies and boundary reducible 3-manifolds