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Let G be a connected plane geometric graph with n vertices. In this paper, we study bounds on the number of edges required to be added to G to obtain 2-vertex or 2-edge connected plane geometric graphs. In particular, we show that for G to become 2-edge connected, 2n 3 additional edges are required in some cases and that 6n 7 additional edges are always… (More)

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)

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)

We consider combinatorial and computational issues that are related to the problem of moving coins from one configuration to another. Coins are defined as non-overlapping discs, and moves are defined as collision free translations, all in the Euclidean plane. We obtain combinatorial bounds on the number of moves that are necessary and/or sufficient to move… (More)

We are given a transportation line where displacements happen at a bigger speed than in the rest of the plane. A shortest time path is a path between two points which takes less than or equal time to any other. We consider the time to follow a shortest time path to be the time distance between the two points. In this paper, we give a simple algorithm for… (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)

- Manuel Abellanas, Ferran Hurtado, Christian Icking, Rolf Klein, Elmar Langetepe, Lihong Ma +2 others
- ESA
- 2001

Motivated by questions in location planning, we show for a set of colored point sites in the plane how to compute the smallest (by perimeter or area) axis-parallel rectangle, the narrowest strip, and other smallest objects enclosing at least one site of each color.

In this work we study the order-k Delaunay graph, which is formed by edges pq having a circle through p and q and containing no more than k sites. We study the combinatorial structure of the set of trian-gulations that can be constructed with edges of this graph and show that it is connected under the flip operation if k ≤ 1 and for every k if points are in… (More)

A known result in combinatorial geometry states that any collection P n of points on the plane contains two such that any circle containing them contains n/c elements of P n , c a constant. We prove: Let be a family of n noncrossing compact convex sets on the plane, and let S be a strictly convex compact set. Then there are two elements S i , S j of such… (More)