Connectivity of Graphs Induced by Directional Antennas

This paper addresses the problem of finding an orientation and a minimum radius for directional antennas of a fixed angle placed at the points of a planar set S, that induce a strongly connected communication graph. We consider problem instances in which antenna angles are fixed at 90 and 180 degrees, and establish upper and lower bounds for the minimum radius necessary to guarantee strong connectivity. In the case of 90-degree angles, we establish a lower bound of 2 and an upper bound of 7. In the case of 180-degree angles, we establish a lower bound of sqrt(3) and an upper bound of 1+sqrt(3). Underlying our results is the assumption that the unit disk graph for S is connected.Comment: 8 pages, 10 figure

An Infinite Class of Sparse-Yao Spanners

We show that, for any integer k > 5, the Sparse-Yao graph YY_{6k} (also known as Yao-Yao) is a spanner with stretch factor 11.67. The stretch factor drops down to 4.75 for k > 7.Comment: 17 pages, 12 figure

Partitioning Regular Polygons into Circular Pieces II:Nonconvex Partitions

We explore optimal circular nonconvex partitions of regular k-gons. The circularity of a polygon is measured by its aspect ratio: the ratio of the radii of the smallest circumscribing circle to the largest inscribed disk. An optimal circular partition minimizes the maximum ratio over all pieces in the partition. We show that the equilateral triangle has an optimal 4-piece nonconvex partition, the square an optimal 13-piece nonconvex partition, and the pentagon has an optimal nonconvex partition with more than 20 thousand pieces. For hexagons and beyond, we provide a general algorithm that approaches optimality, but does not achieve it.Comment: 13 pages, 11 figure

Undirected Connectivity of Sparse Yao Graphs

Given a finite set S of points in the plane and a real value d > 0, the d-radius disk graph G^d contains all edges connecting pairs of points in S that are within distance d of each other. For a given graph G with vertex set S, the Yao subgraph Y_k[G] with integer parameter k > 0 contains, for each point p in S, a shortest edge pq from G (if any) in each of the k sectors defined by k equally-spaced rays with origin p. Motivated by communication issues in mobile networks with directional antennas, we study the connectivity properties of Y_k[G^d], for small values of k and d. In particular, we derive lower and upper bounds on the minimum radius d that renders Y_k[G^d] connected, relative to the unit radius assumed to render G^d connected. We show that d=sqrt(2) is necessary and sufficient for the connectivity of Y_4[G^d]. We also show that, for d = 2/sqrt(3), Y_3[G^d] is always connected. Finally, we show that Y_2[G^d] can be disconnected, for any d >= 1.Comment: 7 pages, 11 figure

Partitioning Regular Polygons into Circular Pieces I: Convex Partitions

We explore an instance of the question of partitioning a polygon into pieces, each of which is as ``circular'' as possible, in the sense of having an aspect ratio close to 1. The aspect ratio of a polygon is the ratio of the diameters of the smallest circumscribing circle to the largest inscribed disk. The problem is rich even for partitioning regular polygons into convex pieces, the focus of this paper. We show that the optimal (most circular) partition for an equilateral triangle has an infinite number of pieces, with the lower bound approachable to any accuracy desired by a particular finite partition. For pentagons and all regular k-gons, k > 5, the unpartitioned polygon is already optimal. The square presents an interesting intermediate case. Here the one-piece partition is not optimal, but nor is the trivial lower bound approachable. We narrow the optimal ratio to an aspect-ratio gap of 0.01082 with several somewhat intricate partitions.Comment: 21 pages, 25 figure

Unfolding Orthogrids with Constant Refinement

We define a new class of orthogonal polyhedra, called orthogrids, that can be unfolded without overlap with constant refinement of the gridded surface.Comment: 19 pages, 12 figure

Unfolding Manhattan Towers

We provide an algorithm for unfolding the surface of any orthogonal polyhedron that falls into a particular shape class we call Manhattan Towers, to a nonoverlapping planar orthogonal polygon. The algorithm cuts along edges of a 4x5x1 refinement of the vertex grid.Comment: Full version of abstract that appeared in: Proc. 17th Canad. Conf. Comput. Geom., 2005, pp. 204--20