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 \(2^{\frac{n}{4}-\frac{54}{4}}\) 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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Korzhik, V.P., Mohar, B. (2009). Minimal Obstructions for 1-Immersions and Hardness of 1-Planarity Testing. In: Tollis, I.G., Patrignani, M. (eds) Graph Drawing. GD 2008. Lecture Notes in Computer Science, vol 5417. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-00219-9_29
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DOI: https://doi.org/10.1007/978-3-642-00219-9_29
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