Ahmad Biniaz

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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)
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)
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)
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)
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)
Let P and S be two disjoint sets of n and m points in the plane, respectively. We consider the problem of computing a Steiner tree whose Steiner vertices belong to S, in which each point of P is a leaf, and whose longest edge length is minimum. We present an algorithm that computes such a tree in O((n + m) log m) time, improving the previously best result(More)