Saturated simple and 2-simple topological graphs with few edges

Abstract: 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 Kyn\v{c}l, Pach, Radoi\v{c}i\'c, and T\'oth 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.