Visibility Graphs, Dismantlability, and the Cops and Robbers Game

We study versions of cop and robber pursuit-evasion games on the visibility graphs of polygons, and inside polygons with straight and curved sides. Each player has full information about the other player's location, players take turns, and the robber is captured when the cop arrives at the same point as the robber. In visibility graphs we show the cop can always win because visibility graphs are dismantlable, which is interesting as one of the few results relating visibility graphs to other known graph classes. We extend this to show that the cop wins games in which players move along straight line segments inside any polygon and, more generally, inside any simply connected planar region with a reasonable boundary. Essentially, our problem is a type of pursuit-evasion using the link metric rather than the Euclidean metric, and our result provides an interesting class of infinite cop-win graphs.Comment: 23 page

Maximum independent set for intervals by divide and conquer with pruning

As an easy example of divide, prune, and conquer, we give an output-sensitive O(n log k)-time algorithm to compute, for n intervals, a maximum independent set of size k

Computing contour trees in all dimensions

AbstractWe show that contour trees can be computed in all dimensions by a simple algorithm that merges two trees. Our algorithm extends, simplifies, and improves work of Tarasov and Vyalyi and of van Kreveld et al

Visibility graphs, dismantlability, and the cops and robbers game

We study versions of cop and robber pursuit–evasion games on the visibility graphs of polygons, and inside polygons with straight and curved sides. Each player has full information about the other player's location, players take turns, and the robber is captured when the cop arrives at the same point as the robber. In visibility graphs we show the cop can always win because visibility graphs are , which is interesting as one of the few results relating visibility graphs to other known graph classes. We extend this to show that the cop wins games in which players move along straight line segments inside any polygon and, more generally, inside any simply connected planar region with a reasonable boundary. Essentially, our problem is a type of pursuit–evasion using the link metric rather than the Euclidean metric, and our result provides an interesting class of infinite cop-win graphs

A 2-chain can interlock with a k-chain

One of the open problems posed in [3] is: what is the minimal number k such that an open, flexible k-chain can interlock with a flexible 2-chain? In this paper, we establish the assumption behind this problem, that there is indeed some k that achieves interlocking. We prove that a flexible 2-chain can interlock with a flexible, open 16-chain.Comment: 10 pages, 6 figure

Reconstructing polygons from scanner data

A range-finding scanner can collect information about the shape of an (unknown) polygonal room in which it is placed. Suppose that a set of scanners returns not only a set of points, but also additional information, such as the normal to the plane when a scan beam detects a wall. We consider the problem of reconstructing the floor plan of a room from different types of scan data. In particular, we present algorithmic and hardness results for reconstructing two-dimensional polygons from point-wall pairs, point-normal pairs, and visibility polygons. The polygons may have restrictions on topology (e.g., to be simply connected) or geometry (e.g., to be orthogonal). We show that this reconstruction problem is NP-hard under most models, but that some restrictive assumptions do allow polynomial-time reconstruction algorithms

Ununfoldable Polyhedra with Convex Faces

Unfolding a convex polyhedron into a simple planar polygon is a well-studied problem. In this paper, we study the limits of unfoldability by studying nonconvex polyhedra with the same combinatorial structure as convex polyhedra. In particular, we give two examples of polyhedra, one with 24 convex faces and one with 36 triangular faces, that cannot be unfolded by cutting along edges. We further show that such a polyhedron can indeed be unfolded if cuts are allowed to cross faces. Finally, we prove that ``open'' polyhedra with triangular faces may not be unfoldable no matter how they are cut.Comment: 14 pages, 9 figures, LaTeX 2e. To appear in Computational Geometry: Theory and Applications. Major revision with two new authors, solving the open problem about triangular face

The problem of managing a strategic reserve

AbstractWe develop a method of managing strategic reserves, in this case, the U.S. Strategic Cobalt Reserve. A rationale for the stockpiling of cobalt is presented, followed by a method for bringing the stockpiled amount from any level to a desired goal, in this case an amount determined by the Federal Emergency Management Agency to last through three years of conventional warfare. The method involves solving a stochastic programming problem in order to balance the expected values of the social benefit and the social cost of building the stockpile—Social benefit is accrued by decreasing the impact and the probability of a war or a major supply disruption occuring before the stockpile goal is realized; social cost is determined from the additional amount U.S. cobalt consumers must pay due to the increase in world demand brought about by stockpiling. The management of the filled stockpile is then discussed, introducing the idea of using the stockpile to assure stability in the world price of cobalt and of defraying maintenance costs by market speculation. Least-squares fitting is used to determine whether prices are high or low and how much to sell or buy, respectively, to bring prices back into line. Then the conditions under which the stockpile should be drawn down are considered, with the proper rate and total amount of released stockpile material determined for two cases, that of a major supply disruption and that of actual warfare. Finally, generalization of the method to cover other strategic stockpiles is discussed