Adding one edge to planar graphs makes crossing number and 1-planarity hard

Details Adding one edge to planar graphs makes crossing number and 1-planarity hard Adding one edge to planar graphs makes crossing number and 1-planarity hard

2012-03-27

arxiv.org/abs/1203.5944v1

Computer Science

Computational Geometry

27 pages, 10 figures. Part of the results appeared in Proceedings of the 26th Annual Symposium on Computational Geometry (SoCG

Scientific paper

A graph is near-planar if it can be obtained from a planar graph by adding an edge. We show the surprising fact that it is NP-hard to compute the crossing number of near-planar graphs. A graph is 1-planar if it has a drawing where every edge is crossed by at most one other edge. We show that it is NP-hard to decide whether a given near-planar graph is 1-planar. The main idea in both reductions is to consider the problem of simultaneously drawing two planar graphs inside a disk, with some of its vertices fixed at the boundary of the disk. This leads to the concept of anchored embedding, which is of independent interest. 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 Hlin\v{e}n\'y.

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