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

Authors

  • Vladimir P. Korzhik,

    1. NATIONAL UNIVERSITY OF CHERNIVTSI CHERNIVTSI UKRAINE AND INSTITUTE OF APPLIED PROBLEMS OF MECHANICS AND MATHEMATICS OF NATIONAL ACADEMY OF SCIENCE OF UKRAINE, LVIV, UKRAINE
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    • This paper was done while the author visited Simon Fraser University.

  • Bojan Mohar

    1. DEPARTMENT OF MATHEMATICS, SIMON FRASER UNIVERSITY, BURNABY, CANADA
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    • Contract grant sponsors: ARRS (Slovenia); NSERC Discovery Grant (Canada); Canada Research Chair program; Contract grant number: P1–0297.

    • On leave from IMFM, and FMF, Department of Mathematics, University of Ljubljana, Ljubljana, Slovenia.


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 math formula 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 math formula, there are at least math formula nonisomorphic minimal non-1-planar graphs of order n. It is also proved that testing 1-planarity is NP-complete.

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