Saturated simple and $k$-simple topological graphs

TL;DR

This paper constructs saturated simple and $k$-simple topological graphs with linear edges, explores connectivity properties, and establishes bounds on crossings, advancing understanding of topological graph saturation and crossing limits.

Contribution

It provides the first constructions of saturated simple and $k$-simple topological graphs with linear edges and proves bounds on crossings between independent vertices.

Findings

Constructed saturated simple topological graphs with $O(n)$ edges.

Extended constructions to $k$-simple graphs for $k>1$.

Proved that any two independent vertices can be connected with a curve crossing each edge at most $2k$ times.

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.