Abstract
A simple topological graph G is a graph drawn in the plane so that any pair of edges have at most one point in common, which is either an endpoint or a proper crossing. G is called saturated if no further edge can be added without violating this condition.
We construct saturated simple topological graphs with n vertices and O(n) edges. For every k>1, we give similar constructions for k-simple topological graphs, that is, for graphs drawn in the plane so that any two edges have at most k points in common.
We show that in any k-simple topological graph, any two independent vertices can be connected by a curve that crosses each of the original edges at most 2k times. Another construction shows that the bound 2k cannot be improved.
Several other related problems are also considered.