We investigate straight-line drawings of topological graphs that consist of a planar graph plus one edge, also called almost-planar graphs. We present a characterization of such graphs that admit a straight-line drawing. The characterization enables a linear-time testing algorithm to determine whether an almost-planar graph admits a straight-line drawing, and a linear-time drawing algorithm that constructs such a drawing, if it exists. We also show that some almost-planar graphs require exponential area for a straight-line drawing.
@article{arxiv.1504.06540,
title = {Straight-line Drawability of a Planar Graph Plus an Edge},
author = {Peter Eades and Seok-Hee Hong and Giuseppe Liotta and Naoki Katoh and Sheung-Hung Poon},
journal= {arXiv preprint arXiv:1504.06540},
year = {2015}
}