Disjoint edges in geometric graphs
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A geometric graph is a graph drawn in the plane so that its vertices and edges are represented by points in general position and straight line segments, respectively. A vertex of a geometric graph is called convex if it lies outside of the convex hull of its neighbors. We show that for a geometric graph with n vertices and e edges there are at least n/22e/n3 pairs of disjoint edges provided that 2e≥ n and all the vertices of the graph are convex. Besides, we prove that if any edge of a geometric graph with n vertices is disjoint from at most m edges, then the number of edges of this graph does not exceed n(√(1+8m)+3)/4 provided that n is sufficiently large. These two results are tight for an infinite family of graphs.
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