Saturated Simple and 2-simple Topological Graphs with Few Edges
A simple topological graph is a topological graph in which any two edges have at most one common point, which is either their common endpoint or a proper crossing. More generally, in a k-simple topological graph, every pair of edges has at most k common points of this kind. We construct saturated simple and 2-simple graphs with few edges. These are k-simple graphs in which no further edge can be added. We improve the previous upper bounds of Kynčl, Pach, Radoičić, and Tóth [Comput. Geom., 48, 2015] and show that there are saturated simple graphs on n vertices with only 7n edges and saturated 2-simple graphs on n vertices with 14.5n edges. As a consequence, 14.5n edges is also a new upper bound for k-simple graphs (considering all values of k). We also construct saturated simple and 2-simple graphs that have some vertices with low degree.
KeywordsTopological Graph Geometric Graph Large Face Black Edge Local Saturation
The first author thanks Géza Tóth for presenting their inspiring results  in Szeged and for the encouragement during his investigation. This research was partially initiated at the EuroGIGA Workshop on Geometric Graphs in Münster, Germany, 2014, supported by the European Science Foundation (ESF).
- 3.Hajnal, P., Igamberdiev, A., Rote, G., Schulz, A.: Saturated simple and 2-simple topological graphs with few edges (2015). http://arxiv.org/abs/1503.01386