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 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, there is a $k$-simple graph (for a general $k$), which can be saturated using $14.5n$ edges, while previous upper bounds suggested $17.5n$ edges. We also construct saturated simple and 2-simple graphs that have some vertices with low degree.