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Given a point set P and a class C of geometric objects, GC(P) is a geometric graph with vertex set P such that any two vertices p and q are adjacent if and only if there is some C ∈ C containing both p and q but no other points from P. We study G (P) graphs where is the class of downward equilateral triangles (ie. equilateral triangles with one of their(More)
Terrains are often modeled by triangulations. One of the criteria that a triangulation should have is “nice shape” triangles. Delaunay triangulation is a good way to formalize nice shape. Another criterion is reality of drainage in terrains. Natural terrains do not have many local minima and have drainage lines in the bottom of valleys. To achieve these(More)
Let P be a set of n points in general position in the plane which is partitioned into color classes. P is said to be color-balanced if the number of points of each color is at most n/2. Given a color-balanced point set P , a balanced cut is a line which partitions P into two color-balanced point sets, each of size at most 2n/3+1. A colored matching of P is(More)
Let R and B be two disjoint sets of points in the plane such that |B| |R|, and no three points of R ∪ B are collinear. We show that the geometric complete bipartite graph K(R, B) contains a non-crossing spanning tree whose maximum degree is at most max 3, |R|−1 |B| + 1 ; this is the best possible upper bound on the maximum degree. This solves an open(More)
A bottleneck plane perfect matching of a set of n points in R 2 is defined to be a perfect non-crossing matching that minimizes the length of the longest edge; the length of this longest edge is known as bottleneck. The problem of computing a bottleneck plane perfect matching has been proved to be NP-hard. We present an algorithm that computes a bottleneck(More)
Let S be a finite set of points in the interior of a simple polygon P. A geodesic graph, G P (S, E), is a graph with vertex set S and edge set E such that each edge (a, b) ∈ E is the shortest geodesic path between a and b inside P. G P is said to be plane if the edges in E do not cross. If the points in S are colored, then G P is said to be properly colored(More)
We consider an extension of the triangular-distance Delaunay graphs (TD-Delaunay) on a set P of points in general position in the plane. In TD-Delaunay, the convex distance is defined by a fixed-oriented equilateral triangle , and there is an edge between two points in P if and only if there is an empty homothet of having the two points on its boundary. We(More)
Given a set of n red points and n blue points in the plane, we are interested to match the red points with the blue points by straight line segments in such a way that the segments do not cross each other and the length of the longest segment is minimized. In general, this problem in NP-hard. We give exact solutions for some special cases of the input point(More)