Let \(B_2\) 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 \(B_2\).
A graph is outer-bananas-surface if it can be embedded in \(B_2\) so that all its vertices lie on the same face. In this paper, we prove that the class of the outer-\(B_2\) graphs is closed under minors. In fact, we give the complete set of \(38\) minor-minimal non-outer-\(B_2\) graphs and we also characterize these graphs by a finite list of forbidden topological minors.
We also extend outer embeddings to other pseudosurfaces. The \(S\) 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-\(S\) graphs and we also give an explicit and finite list of forbidden topological minors for these pseudosurfaces.
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