Consequences of some outerplanarity extensions

In this expository paper we revise some extensions of Kuratowski planarity criterion, providing a link between the embeddings of infinite graphs without accumulation points and the embeddings of finite graphs with some distinguished vertices in only one face. This link is valid for any surface and for some pseudosurfaces. On the one hand, we present some key ideas that are not easily accessible. On the other hand, we state the relevance of infinite, locally finite graphs in practice and suggest some ideas for future research

The obstructions for toroidal graphs with no K3,3K_{3,3}'s

Forbidden minors and subdivisions for toroidal graphs are numerous. We consider the toroidal graphs with no $K_{3,3}$-subdivisions that coincide with the toroidal graphs with no $K_{3,3}$-minors. These graphs admit a unique decomposition into planar components and have short lists of obstructions. We provide the complete lists of four forbidden minors and eleven forbidden subdivisions for the toroidal graphs with no $K_{3,3}$'s and prove that the lists are sufficient.Comment: 10 pages, 7 figures, revised version with additional detail