The Maximum Number of Edges in Geometric Graphs with Pairwise Virtually Avoiding Edges

The Maximum Number of Edges in Geometric Graphs with Pairwise Virtually Avoiding Edges

Let G be a geometric graph on n vertices that are not necessarily in general position. Assume that no line passing through one edge of G meets the relative interior of another edge. We show that in this case the number of edges in G is at most 2n− 3.

The Maximum Number of Edges in a Strongly Multiplicative Graph

On Geometric Graphs with No k Pairwise Parallel Edges

A geometric graph is a graph G = (V;E) drawn in the plane so that the vertex set V consists of points in general position and the edge set E consists of straight line segments between points of V . Two edges of a geometric graph are said to be parallel , if they are opposite sides of a convex quadrilateral. In this paper we show that, for any xed k 3, any geometric graph on n vertices with no k...

the maximum number of edges in a strongly multiplicative graph

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On the maximum number of edges in quasi-planar graphs

A topological graph is quasi-planar, if it does not contain three pairwise crossing edges. Agarwal et al. [2] proved that these graphs have a linear number of edges. We give a simple proof for this fact that yields the better upper bound of 8n edges for n vertices. Our best construction with 7n−O(1) edges comes very close to this bound. Moreover, we show matching upper and lower bounds for seve...