Let be the bananas surface arising from the torus by contracting two different meridians of the torus to a simple point each. It was proved in [8] that there is not a finite Kuratowski theorem for .
A graph is outer-bananas-surface if it can be embedded in so that all its vertices lie on the same face. In this paper, we prove that the class of the outer- graphs is closed under minors. In fact, we give the complete set of minor-minimal non-outer- graphs and we also characterize these graphs by a finite list of forbidden topological minors.
We also extend outer embeddings to other pseudosurfaces. The pseudosurfaces treated are spheres joined by points in such a way that each sphere has two singular points. We give an excluded minor characterization of outer- graphs and we also give an explicit and finite list of forbidden topological minors for these pseudosurfaces.
