Number of Crossing-Free Geometric Graphs vs. Triangulations

We show that there is a constant > 0 such that, for any set P of n 5 points in general position in the plane, a crossing-free geometric graph on P that is chosen uniformly at random contains, in expectation, at least (1 2 + )M edges, where M denotes the number of edges in any triangulation of P. From this we derive (to our knowledge) the first non-trivial upper bound of the form cn

• Type: Journal
• Year: 2008
• Where: ENDM
• Authors: Andreas Razen, Jack Snoeyink, Emo Welzl
• Keywords: Crossing-free Geometric Graphs, ENDM 2008, Non-trivial Upper Bound, Upper Bound