Saturated simple and k-simple topological graphs

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)O(n) edges. For every k>1k>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.