Minimal Obstructions for 1-Immersions and Hardness of 1-Planarity Testing

Authors:
Vladimir P. Korzhik;Bojan Mohar
Affiliations:
National Academy of Science, National University of Chernivtsi and Institute of APMM, Lviv, Ukraine;Department of Mathematics, Simon Fraser University, Burnaby, Canada B.C. V5A 1S6
Venue:
Graph Drawing
Year:
2009

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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 e 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 Hlinný.