Minimal obstructions for 1-immersions and hardness of 1-planarity testing

Abstract

A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by no more than one other edge. A non-1-planar graph G is minimal if the graph G∈-∈e is 1-planar for every edge e of G. We construct two infinite families of minimal non-1-planar graphs and show that for every integer n∈ ∈63, there are at least non-isomorphic minimal non-1-planar graphs of order n. It is also proved that testing 1-planarity is NP-complete. As an interesting consequence we obtain a new, geometric proof of NP-completeness of the crossing number problem, even when restricted to cubic graphs. This resolves a question of Hliněný. © 2009 Springer Berlin Heidelberg.