(2) Centro de Investigacion y de Estudios Avanzados del Instituto Politécnico Nacional
(3) Departament de Matemàtica Aplicada II
(4) Unidad Académica de Matemáticas [Mexico]
(5) Department of Combinatorics and Optimization
(6) Instituto de Fisica [Mexico]
(7) Departamento de Sistemas [Azcapotzalco]
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.
