Fáry's Theorem for 1-Planar Graphs

Fáry’s theorem states that every plane graph can be drawn as a straightline drawing. A plane graph is a graph embedded in a plane without edge crossings. In this paper, we extend Fáry’s theorem to 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. We also show that there are 1-plane 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.