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The Erdős-Sós conjecture for geometric graphs

Abstract : Let f(n,k) be the minimum number of edges that must be removed from some complete geometric graph G on n points, so that there exists a tree on k vertices that is no longer a planar subgraph of G. In this paper we show that ( 1 / 2 )n2 / k-1-n / 2≤f(n,k) ≤2 n(n-2) / k-2. For the case when k=n, we show that 2 ≤f(n,n) ≤3. For the case when k=n and G is a geometric graph on a set of points in convex position, we completely solve the problem and prove that at least three edges must be removed.
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Luis Barba, Ruy Fabila-Monroy, Dolores Lara, Jesús Leaños, Cynthia Rodrıguez, et al.. The Erdős-Sós conjecture for geometric graphs. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2013, Vol. 15 no. 1 (1), pp.93--100. ⟨hal-00990608⟩

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