Straight-line Drawings of 1-planar Graphs

Abstract

The classical Fáry’s theorem from the 1930s states that every planar graph can be drawn as a straight-line drawing. In this paper, we extend Fáry’s theorem to non-planar graphs. More specifically, we study the problem of drawing 1-planar graphs with straight-line edges. A 1-planar graph is a sparse non-planar graph with at most one crossing per edge. We give a characterisation of those 1planar graphs that admit a straight-line drawing. The proof of the characterisation consists of a linear time testing algorithm and a drawing algorithm. We also show that there are 1-planar graphs for which every straight-line drawing has exponential area. To our best knowledge, this is the first result to extend Fáry’s theorem to non-planar graphs.

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Cite this paper

@inproceedings{Eades2011StraightlineDO, title={Straight-line Drawings of 1-planar Graphs}, author={Peter Eades and Seok-Hee Hong and Giuseppe Liotta and Sheung-Hung Poon}, year={2011} }