Eduardo Rivera-Campo

Spanning trees with bounded degrees

Lets andk be positive integers. We prove that ifG is ak-connected graph containing no independent set withks+2 vertices thenG has a spanning tree with maximum degree at mosts+1. Moreover ifs≥3 and the independence number α(G) is such that α(G)≤1+k(s−1)+c for some0≤c≤k thenG has a spanning tree with no more thanc vertices of degrees+1.

Hamilton cycles in the path graph of a set of points in convex position

Abstract Agarwal, P.K. and M. Sharir, Off-line dynamic maintenance of the width of a planar point set, Computational Geometry: Theory and Applications 1 (1990) 65-78. In this paper we present an efficient algorithm for the off-line dynamic maintenance of the width of a planar point set in the following restricted case: We are given a real parameter W and a sequence Σ=(σ1,...,σn) of n insert and delete operations on a set S of points in R 2, initially consisting of n points, and we want to determine whether there is an i such that the width of S the ith operation is less than or equal to W. Our algorithm runs in time O(nlog3n) and uses O(n) space.

Guarding rectangular art galleries

Abstract Consider a rectangular art gallery divide into n rectangular rooms, such that any two rooms sharing a wall in common have a door connecting them. How many guards need to be stationed in the gallery so as to protect all of the rooms in our gallery? Note that of a guard is stationed at a door, he will be able to guard two rooms. Our main aim in this paper is to show that ⌈ n /2⌉ guards are always sufficient to protect all rooms in a rectangular art gallery. Extensions of our result are obtained for non-rectangular galleries and for 3-dimensional art galleries.

Illuminating rectangles and triangles in the plane

Abstract A set S of light sources, idealized as points, illuminates a collection F of convex sets if each point in the boundary of the sets of F is visible from at least one point in S. For any n disjoint plane isothetic rectangles, ⌊(4n + 4)/3⌋ lights are sufficient to illuminate their boundaries. If, in addition, the rectangles have equal width, then n + 1 lights always suffice and n − 1 are occasionally necessary. For any family of n plane triangles, ⌊(4n + 4)/3⌋ light sources are sufficient and n − 1 are occasionally necessary.

Separating convex sets in the plane

Given a setA inR2 and a collectionS of plane sets, we say that a lineL separatesA fromS ifA is contained in one of the closed half-planes defined byL, while every set inS is contained in the complementary closed half-plane.We prove that, for any collectionF ofn disjoint disks inR2, there is a lineL that separates a disk inF from a subcollection ofF with at least ⌌(n−7)/4⌍ disks. We produce configurationsHn andGn, withn and 2n disks, respectively, such that no pair of disks inHn can be simultaneously separated from any set with more than one disk ofHn, and no disk inGn can be separated from any subset ofGn with more thann disks.We also present a setJm with 3m line segments inR2, such that no segment inJm can be separated from a subset ofJm with more thanm+1 elements. This disproves a conjecture by N. Alonet al. Finally we show that ifF is a set ofn disjoint line segments in the plane such that they can be extended to be disjoint semilines, then there is a lineL that separates one of the segments from at least ⌌n/3⌍+1 elements ofF.

Graphs of Triangulations and Perfect Matchings

Given a set P of points in general position in the plane, the graph of triangulations of P has a vertex for every triangulation of P, and two of them are adjacent if they differ by a single edge exchange. We prove that the subgraph of , consisting of all triangulations of P that admit a perfect matching, is connected. A main tool in our proof is a result of independent interest, namely that the graph that has as vertices the non-crossing perfect matchings of P and two of them are adjacent if their symmetric difference is a single non-crossing cycle, is also connected.

Radial Perfect Partitions of Convex Sets in the Plane

In this paper we study the following problem: how to divide a cake among the children attending a birthday party such that all the children get the same amount of cake and the same amount of icing. This leads us to the study of the following. A perfect k-partitioning of a convex set S is a partitioning of S into k convex pieces such that each piece has the same area and \(\frac{1}{k}\) of the perimeter of S . We show that for any k, any convex set admits a perfect k-partitioning. Perfect partitionings with additional constraints are also studied.

On circumscribing polygons for line segments

Abstract Any family of k3 + 1 pairwise disjoint line segments in the Euclidean plane E2, such that no three of their endpoints are collinear, has k + 1 members admitting a circumscribing polygon. That is, one can find a simple polygon P with 2k + 2 vertices such that each of these segments is either an edge or an internal diagonal of P.

Edge guarding polyhedral terrains

Abstract In this paper we show that [ n 3 ] edge guards are always sufficient to cover a triangulated polyhedral terrain on n vertices. We prove this by showing that [ n 3 ] edges are always sufficient to cover all of the faces of a plane triangulation on n vertices. We also show that [ 2n 5 ] edges are sufficient to cover all of the faces of an arbitrary plane graph on n vertices.

A Note on Convex Decompositions of a Set of Points in the Plane

Abstract.For any set P of n points in general position in the plane there is a convex decomposition of P with at most elements. Moreover, any minimal convex decomposition of such a set P has at most elements, where k is the number of points in the boundary of the convex hull of P.