Geometric graphs with no self-intersecting path of length three

Authors:
János Pach;Rom Pinchasi;Gábor Tardos;Géza Tóth
Affiliations:
City College, CUNY and Courant Institute of Mathematical Sciences, New York University, New York, NY;Department of Mathematics Massachusetts Institute of Technology, Cambridge, MA;Rényi Institute of the Hungarian Academy of Sciences, H-1364 Budapest, P.O.B. 127, Hungary;Rényi Institute of the Hungarian Academy of Sciences, H-1364 Budapest, P.O.B. 127, Hungary
Venue:
European Journal of Combinatorics - Special issue: Topological graph theory
Year:
2004

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Abstract

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 longer forbidden paths and for graphs drawn by not necessarily straight-line edges are also considered.