Convex drawings of the complete graph: topology meets geometry

@article{Arroyo2021ConvexDO,
  title={Convex drawings of the complete graph: topology meets geometry},
  author={Alan Arroyo and Dan McQuillan and R. Bruce Richter and Gelasio Salazar},
  journal={Ars Mathematica Contemporanea},
  year={2021}
}
In this work, we introduce and develop a theory of convex drawings of the complete graph $K_n$ in the sphere. A drawing $D$ of $K_n$ is convex if, for every 3-cycle $T$ of $K_n$, there is a closed disc $\Delta_T$ bounded by $D[T]$ such that, for any two vertices $u,v$ with $D[u]$ and $D[v]$ both in $\Delta_T$, the entire edge $D[uv]$ is also contained in $\Delta_T$. As one application of this perspective, we consider drawings containing a non-convex $K_5$ that has restrictions on its… Expand
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