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

Authors:
Vladimir P. Korzhik;Bojan Mohar
Affiliations:
(This paper was done while the author visited Simon Fraser University) (Contract grant sponsors: ARRS (Slovenia)/ NSERC Discovery Grant (Canada)/ Canada Research Chair program/ Contract grant numb ...;(On leave from IMFM, and FMF, Department of Mathematics, University of Ljubljana, Ljubljana, Slovenia.) DEPARTMENT OF MATHEMATICS, SIMON FRASER UNIVERSITY, BURNABY, B.C. V5A 1S6, CANADA
Venue:
Journal of Graph Theory
Year:
2013

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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 (and any pair of crossing edges cross only once). A non-1-planar graph G is minimal if the graph 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 , there are at least nonisomorphic minimal non-1-planar graphs of order n. It is also proved that testing 1-planarity is NP-complete. © 2013 Wiley Periodicals, Inc.