Straight-line Drawability of a Planar Graph Plus an Edge

TL;DR

This paper characterizes almost-planar graphs that can be drawn with straight lines, provides efficient algorithms for testing and constructing such drawings, and discusses area requirements.

Contribution

It offers a linear-time characterization and algorithms for straight-line drawings of almost-planar graphs, advancing understanding of their geometric representations.

Findings

Linear-time testing algorithm for straight-line drawability

Linear-time construction algorithm for such drawings

Some graphs require exponential area for straight-line drawings

Abstract

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.