Minimal Representations of Order Types by Geometric Graphs
Oswin Aichholzer, Martin Balko, Michael M. Hoffmann, Jan Kynčl, Wolfgang Mulzer, Irene Parada, Alexander Pilz, Manfred Scheucher, Pável Valtr, Birgit Vogtenhuber, Emo Welzl · Journal of Graph Algorithms and Applications · 2020
In order to have a compact visualization of the order type of a given point set $S$, we are interested in geometric graphs on $S$ with few edges that unambiguously display the order type of $S$. We introduce the concept of exit edges, which prevent the order type from changing under continuous motion of vertices. That is, in the geometric graph on $S$ whose edges are the exit edges, in order to change the order type of $S$, at least one vertex needs to move across an exit edge. Exit edges have a natural dual characterization, which allows us to efficiently compute them and to bound their number.