Saturated simple and 2-simple topological graphs with few edges

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 2-simple 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 [4] and show that there are saturated simple graphs on n vertices with only 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.

Saturated $k$-Plane Drawings with Few Edges

“…Stashing into the free cells yields a family of drawings on n vertices obtaining the desired bounds. For k = 3 the above construction gives an edge-vertex ratio of 6 5 , but if instead of stashing isolated vertices we stash isolated edges we can improve the ratio to 7 6 ; see Fig. 6c.…”

“…They presented bounds on the minimum number of edges in saturated -simple drawings. Recently the bounds for simple and 2-simple drawings were improved by Hajnal et al [7]. Saturated drawings with few edges have also been studied by Aichholzer et al [2] in the context of thrackles, that are drawings in which every pair of edges intersects exactly once, either at a proper crossing or at a common endpoint.…”

Extending Simple Drawings

“…Note that the term augmentation has also been used in the literature for the similar problem of inserting edges and/or vertices to a graph [10]. Extensions of simple drawings have been previously considered in the context of saturated drawings, that is, drawings where no edge can be inserted [12,16].…”

On the Edge-Vertex Ratio of Maximal Thrackles

“…A thrackle is maximal if no edge between already existing vertices can be inserted such that the resulting drawing remains a thrackle. Our work is partially motivated by the results of Hajnal et al [10] on saturated k-simple graphs. A graph is k-simple if every pair of edges has at most k common points, either proper crossings and/or a common endpoint.…”

“…A k-simple graph is saturated if no further edge can be added while maintaining th k-simple property. In [10], saturated simple graphs on n vertices with only 7n edges are constructed, as well as saturated 2-simple graphs on n vertices with 14.5n edges.…”