Fáry’s Theorem for 1-Planar Graphs
- Cite this paper as:
- Hong SH., Eades P., Liotta G., Poon SH. (2012) Fáry’s Theorem for 1-Planar Graphs. In: Gudmundsson J., Mestre J., Viglas T. (eds) Computing and Combinatorics. COCOON 2012. Lecture Notes in Computer Science, vol 7434. Springer, Berlin, Heidelberg
A plane graph is a graph embedded in a plane without edge crossings. Fáry’s theorem states that every plane graph can be drawn as a straight-line drawing, preserving the embedding of the plane graph. In this paper, we extend Fáry’s theorem to a class of non-planar graphs. More specifically, we study the problem of drawing 1-plane graphs with straight-line edges. A 1-plane graph is a graph embedded in a plane with at most one crossing per edge. We give a characterisation of those 1-plane graphs that admit a straight-line drawing. The proof of the characterisation consists of a linear time testing algorithm and a drawing algorithm. Further, we show that there are 1-plane graphs for which every straight-line drawing has exponential area. To the best of our knowledge, this is the first result to extend Fáry’s theorem to non-planar graphs.
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