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Given a tree T on n vertices and a set P of n points in the plane in general position, it is known that T can be straight line embedded in P without crossings. The problem becomes more diicult if T is rooted and we want to root it at any particular point of P. The problem in this form was posed by Perles and partially solved by Pach and Torosick 5]. A(More)
Around 1958, Hill conjectured that the crossing number CRg(K<sub>n</sub>) of the complete graph KK<sub>n</sub> is Z(n):=1/4 &#8970; n/2 &#8971; &#8970;(n-1)/2&#8971; &#8970; (n-2)/2 &#8971; &#8970; (n-3)/2 &#8971; and provided drawings of K<sub>n</sub> with exactly Z(n) crossings. Towards the end of the century, substantially different drawings of(More)
The Harary-Hill Conjecture states that the number of crossings in any drawing of the complete graph K n in the plane is at least Z(n) := 1 4 n 2 n−1 2 n−2 2 n−3 2. In this paper, we settle the Harary-Hill conjecture for shellable drawings. We say that a drawing D of K n is s-shellable if there exist a subset S = {v 1 , v 2 ,. .. , v s } of the vertices and(More)
Given a set P of n points in the plane, the order-k Delaunay graph is a graph with vertex set P and an edge exists between two points p, q ∈ P when there is a circle through p and q with at most k other points of P in its interior. We provide upper and lower bounds on the number of edges in an order-k Delaunay graph. We study the combinatorial structure of(More)
In this paper we study the separability in the plane by two criteria: double-wedge separability and Θ-separability. We give an O(N log N)-time optimal algorithm for computing all the vertices of separating double wedges of two disjoint sets of objects (points, segments, polygons and circles) and an O((N/Θ 0) log N)-time algorithm for computing a(More)
Multi-VMap is a compact framework from which plane graphs representing geographic maps at different levels of detail can be extracted. Its main feature is that the scale of the extracted map can be variable through its domain, while each entity maintains consistent combinatorial relations with the rest of entities represented in the map. The model is based(More)
Given a set of points S = {pi,...,pn} in the euclidean d-dimensional space, we address the problem of computing the d-dimensional annulus of smallest width containing the set. We give a complete characterization of the centers of annuli which are locally minimal in arbitrary dimension and we show that, for d = 2, a locally minimal annulus has two points on(More)
We study the problem of converting triangulated domains to quadrangula-tions, under a variety of constraints. We obtain a variety of characterizations for when a triangulation (of some structure such as a polygon, set of points, line segments or planar subdivision) admits a quadrangulation without the use of Steiner points, or with a bounded number of(More)
In this paper we consider the tolerance of a geometric or combinatorial structure associated to a set of points as a measure of how much the set of points can be perturbed while leaving the (topological or combinatorial) structure essentially unchanged. We concentrate on studying the Delaunay triangulation and show that its tolerance can be computed in O(n)(More)
Given a polygonal object (simple polygon, geometric graph, wire-frame, skeleton or more generally a set of line segments) in three-dimensional Euclidean space, we consider the problem of computing a variety of " nice " parallel (orthographic) projections of the object. We show that given a general polygonal object consisting of n line segments in space,(More)