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

Saturated simple and 2-simple topological graphs with few edges

1. 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].  →BibTeX
2. Journal of Graph Algorithms and Applications 22, no. 1 (2017), pp. 117–138, special issue on "Graph Drawing Beyond Planarity". doi:10.7155/jgaa.00460  →BibTeX

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 Kynčl, Pach, Radoičić, and Tóth (2013) and show that there are saturated simple graphs on n vertices with 7n edges and saturated 2-simple graphs on n vertices with 14.5n edges. As a consequence, 14.5n 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.