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Special Issue on Graph Drawing Beyond Planarity
DOI: 10.7155/jgaa.00460
Saturated simple and 2simple topological graphs with few edges
Péter Hajnal,
Alexander Igamberdiev,
Günter Rote, and
André Schulz
Vol. 22, no. 1, pp. 117138, 2018. Regular paper.
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 2simple 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 2simple 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 2simple graphs that have some vertices
with low degree.

Submitted: February 2017.
Reviewed: June 2017.
Revised: June 2017.
Accepted: July 2017.
Final: December 2017.
Published: January 2018.
