## Péter Hajnal, Alexander Igamberdiev, Günter Rote, and André
Schulz:

# Saturated simple and 2-simple topological graphs with few edges

In:
*
41st International Workshop on Graph-Theoretic Concepts in
Computer Science—WG 2015*, Garching, Germany, June 2015, Revised
Papers. Editor: Ernst Mayr, Lecture Notes in Computer Science,
9224, Springer-Verlag, 2016, pp. 391–405.
doi:10.1007/978-3-662-53174-7_28*,
arXiv:1503.01386 [cs.CG].
*
### Abstract

A *simple topological graph* is a topological graph in which any two
edges have at most one common point, which is either a 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 bounds of Kyncl, Pach, Radoicic, and
Tóth (2013) and show that there are saturated simple graphs
on `n` vertices with 7`n` edges and saturated 2-simple
graphs on `n` vertices with 14.5`n` edges. As a
consequence, 14.5`n` 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.

Last update: December 8, 2016.