Random point sets in the plane allow covering all edges of the complete geometric graph with O(n log n) crossing-free paths whp, while some configurations require Omega(n^2) monotone paths.
II , Discrete and Computational Geometry 20 (1998), 375--388
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Covering Complete Geometric Graphs by Monotone Paths
Random point sets in the plane allow covering all edges of the complete geometric graph with O(n log n) crossing-free paths whp, while some configurations require Omega(n^2) monotone paths.