A simple topological graph is a topological graph in which any two edges have at most one common point, which is either a 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 bounds of Kynčl, Pach, Radoičić, and Tóth (2013) and show that there are saturated simple graphs on n vertices with 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.
Last update: January 12, 2018.