Convex drawings of the complete graph: topology meets geometry


Recommended citation: Arroyo, A., McQuillan, D., Richter, R.B. and Salazar, G., 2017. Convex drawings of the complete graph: topology meets geometry. arXiv preprint arXiv:1712.06380.

A pseudoline is a homeomorphic image of the real line in the plane so that its complement is disconnected. An arrangement of pseudolines is a set of pseudolines in which every two cross exactly once. A drawing of a graph is pseudolinear if the edges can be extended to an arrangement of pseudolines. In the recent study of crossing numbers, pseudolinear drawings have played an important role as they are a natural combinatorial extension of rectilinear drawings. A characterization of the pseudolinear drawings of Kn was found recently. We extend this characterization to all graphs, by describing the set of minimal forbidden subdrawings for pseudolinear drawings. Our characterization also leads to a polynomial-time algorithm to recognize pseudolinear drawings and construct the pseudolines when it is possible.