An Introduction to Framed 4-Graph Minor Theory

The well-known Pontrjagin-Kuratowski Theorem says that a graph is non-planar if it does not contain $K_5$ and $K_{3,3}$ (in the modern formulation we can say does not contain as a minor). In the talk we deal with regular 4-graphs with an additional structure of opposite edges at each vertex (we call them framed 4-graphs) . A theorem due to the speaker (conjectured by V.A.Vassiliev) says that such a graph is non-planar if it does not contain two cycles with no common edges having exactly one transverse intersection. The equivalence of the Pontrjagin-Kuratowski Theorem and Vassiliev's conjecture was proved by I.M.Nikonov.

In the talk we prove that for framed 4-graphs (with source-sink structure) there is a unique graph which plays the role of planarity abstraction as well as intrinsic linkedness obstruction as well as obstruction of crossing number no more than two.