Geometric Graphs with No Self-intersecting Path of Length Three
Let G be a geometric graph with n vertices, i.e., a graph drawn in the plane with straight-line edges. It is shown that if G has no self-intersecting path of length 3, then its number of edges is O(n log n). This result is asymptotically tight. Analogous questions for curvilinear drawings and for longer paths are also considered.
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