Minimal obstructions for 1-immersions and hardness of 1-planarity testing
Korzhik, Vladimir P. ; Mohar, Bojan, 1956-
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 ▫$n \ge 63$▫, there are at least ▫$2^{\frac{n}{4}-\frac{54}{4}}$▫ 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.
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