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We investigate the number of plane geometric, i.e., straight-line, graphs, a set <i>S</i> of <i>n</i> points in the plane admits. We show that the number of plane graphs and connected plane graphs as well as the number of cycle-free plane graphs is minimized when <i>S</i> is in convex position. Moreover, these results hold for all these graphs with an… (More)

Let T S be the set of all crossing-free straight line spanning trees of a planar n-point set S. Consider the graph T S where two members T and T of T S are adjacent if T intersects T only in points of S or in common edges. We prove that the diameter of T S is O(log k), where k denotes the number of convex layers of S. Based on this result, we show that the… (More)

A geometric graph is ;I g~.aph G = (V, E) drawn in the plane, such that V is a point set in general position and E is a set of straight-line segments whose endpoints belong to I,'. We study the following extremal problem for geometric graphs: How many arbitrary edges can be removed from a complete geometric graph with n vertices such that the remaining… (More)

This paper studies non-crossing geometric perfect matchings. Two such perfect matchings are compatible if they have the same vertex set and their union is also non-crossing. Our first result states that for any two perfect match-ings M and M of the same set of n points, for some k ∈ O(log n), there is a sequence of perfect matchings M = M 0 , M 1 ,. .. , M… (More)

We consider a variation of a problem stated by Erd˝ os and Szekeres in 1935 about the existence of a number f ES (k) such that any set S of at least f ES (k) points in general position in the plane has a subset of k points that are the vertices of a convex k-gon. In our setting the points of S are colored, and we say that a (not necessarily convex) spanned… (More)

We propose a novel subdivision of the plane that consists of both convex polygons and pseudo-triangles. This pseudo-convex decomposition is significantly sparser than either convex decompositions or pseudo-triangulations for planar point sets and simple polygons. We also introduce pseudo-convex partitions and coverings. We establish some basic properties… (More)