Amalgamations along surfaces with boundary in a handlebody

Let M be a connected orientable 3-manifold, and F a compact connected orientable surface properly embedded in M. If F cuts M into two connected 3-manifolds X and Y, that is, M=X \cup_F Y, we say that M is an amalgamation of X and Y along F; and if F cuts M into a connected 3-manifold X, we say that M is a self-amalgamation of X along F. A characterization of an amalgamation of two handlebodies along a surface, incompressible in both, to be a handlebody was obtained by Lei, Liu, Li, and Vesnin. The case of amalgamation of two handelbodies along a compressional surface was studdied by Xu, Fang, and Lei. In the present paper, a characterization of an amalgamation and self-amalgamation of a handlebody to be a handlebody is given.