My latest work has generalised Fary's wellknown theorem (1948) on straightline representations of
"planar" graphs to certain classes of nonplanar graphs. One of the most common graph drawing
conventions is the straightline drawing convention, that edges are straightline segments. Existing
straightline drawing algorithms have a strong assumption that the input graph is planar. In practice,
however, most graphs arising from realworld applications are nonplanar.
My innovation was to relax the assumption of planarity, to sparse nonplanar graph classes such as
"kplanar graphs" (i.e., each edge has at most k crossings) and "kskewness graphs" (i.e. graphs
whose removal of k edges result in planar graphs). These optimal algorithms can construct straightline
drawings of such nonplanar graphs in linear time. 
P. Eades, S. Hong, G. Liotta, S. Poon, "Fary's Theorem for 1planar Graphs", TR ITIVG201201, School of IT, University of Sydney, 2012.
Fary'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 Fary's theorem to nonplanar graphs. More specifically,
we study the problem of drawing 1plane graphs with straightline edges.
A 1plane graph is a graph embedded in a plane with at most one crossing per
edge.We give a characterisation of those 1plane graphs that admit a straightline
drawing. The proof of the characterisation consists of a linear time testing algorithm
and a drawing algorithm. We also show that there are 1plane graphs for
which every straightline drawing has exponential area. To our best knowledge,
this is the first result to extend Fary's theorem to nonplanar graphs.
