NDLI: Adding One Edge to Planar Graphs Makes Crossing Number and 1-Planarity Hard

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Cabello, Sergio Mohar, Bojan
Copyright Year	2013
Abstract	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 Hliněný.
Starting Page	1803
Ending Page	1829
Page Count	27
File Format	PDF
ISSN	00975397
DOI	10.1137/120872310
e-ISSN	10957111
Journal	SIAM Journal on Computing (SMJCAT)
Issue Number	5
Volume Number	42
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2013-09-19
Access Restriction	Subscribed
Subject Keyword	crossing number NP-hardness Planar graphs; geometric and topological aspects of graph theory Computational difficulty of problems 1-planarity topological graph theory Graph theory
Content Type	Text
Resource Type	Article
Subject	Mathematics Computer Science

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