Convex drawings of the complete graph: topology meets geometry
Alan Arroyo, Dan McQuillan, R. Bruce Richter, Gelasio Salazar

TL;DR
This paper develops a theory of convex drawings of complete graphs on a sphere, characterizes convexity by excluding certain subdrawings, and explores implications for optimal graph drawings and their subclasses.
Contribution
It introduces convex, h-convex, and f-convex drawings, linking convexity to pseudolinear and pseudospherical drawings, and provides new conditions for suboptimality in graph drawings.
Findings
Convex drawings characterized by excluding two K5 subdrawings.
Convexity relates to pseudolinear and pseudospherical drawings.
Local conditions can determine suboptimality without counting crossings.
Abstract
In this work, we introduce and develop a theory of convex drawings of the complete graph in the sphere. A drawing of is convex if, for every 3-cycle of , there is a closed disc bounded by such that, for any two vertices with and both in , the entire edge is also contained in . As one application of this perspective, we consider drawings containing a non-convex that has restrictions on its extensions to drawings of . For each such drawing, we use convexity to produce a new drawing with fewer crossings. This is the first example of local considerations providing sufficient conditions for suboptimality. In particular, we do not compare the number of crossings {with the number of crossings in} any known drawings. This result sheds light on Aichholzer's computer proof (personal…
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