Graph planarity, Duality and Related Topics

graph theory; it is part of the theory of graph embeddings
into surfaces of different types. Interconnections exist
e.g. with matroid theory, linear algebra, graph minors and
convex polytopes.

We describe some old and new results on graph planarity and duality.
We discuss the relationships between some known planarity criteria.
We also describe further developments and approaches in graph
planarity theory that use ideas of decomposition and lead to various
strengthenings of some known criteria. New results concern
strengthenings of some classical planarity criteria for quasi
4-connected graphs, for bipartite quasi 4-connected graphs,
and for cubic bipartite graphs.

Further we give generalizations of classical results
of Whitney and Kuratowski in this area and show some results
on Dirac's and Barnette's conjectures.