Prosenjit Bose

Guarding polyhedral terrains

Abstract We prove that ⌊ n 2 ⌋ vertex guards are always sufficient and sometimes necessary to guard the surface of an n-vertex polyhedral terrain. We also show that ⌊ (4n − 4) 13 ⌋ edge guards are sometimes necessary to guard the surface of an n-vertex polyhedral terrain. The upper bound on the number of edge guards is ⌊ n 3 ⌋ (Everett and Rivera-Campo, 1994). Since both upper bounds are based on the four color theorem, no practical polynomial time algorithm achieving these bounds seems to exist, but we present a linear time algorithm for placing ⌊ 3n 5 ⌋ vertex guards for covering a polyhedral terrain and a linear time algorithm for placing ⌊ 2n 5 ⌋ edge guards.

Characterizing Proximity Trees

Complete characterizations are given for those trees that can be drawn as either the relative neighborhood graph, relatively closest graph, Gabriel graph, or modified Gabriel graph of a set of points in the plane. The characterizations give rise to linear-time algorithms for determining whether a tree has such a drawing; if such a drawing exists one can be constructed in linear time in the real RAM model. The characterization of Gabriel graphs settles several conjectures of Matula and Sokal [17].

On Rectangle Visibility Graphs

We study the problem of drawing a graph in the plane so that the vertices of the graph are rectangles that are aligned with the axes, and the edges of the graph are horizontal or vertical lines-of-sight. Such a drawing is useful, for example, when the vertices of the graph contain information that we wish displayed on the drawing; it is natural to write this information inside the rectangle corresponding to the vertex. We call a graph that can be drawn in this fashion a rectangle-visibility graph, or RVG. Our goal is to find classes of graphs that are RVGs. We obtain several results: n n1. n nFor 1 ≤ k ≤ 4, k-trees are RVGs. n n n n n2. n nAny graph that can be decomposed into two caterpillar forests is an RVG. n n n n n3. n nAny graph whose vertices of degree four or more form a distance-two independent set is an RVG. n n n n n4. n nAny graph with maximum degree four is an RVG. Our proofs are constructive and yield linear-time layout algorithms.

On plane geometric spanners: A survey and open problems

Abstract Given a weighted graph G = ( V , E ) and a real number t ⩾ 1 , a t-spanner of G is a spanning subgraph G ′ with the property that for every edge xy in G, there exists a path between x and y in G ′ whose weight is no more than t times the weight of the edge xy. We review results and present open problems on different variants of the problem of constructing plane geometric t-spanners.

Making triangulations 4-connected using flips

We show that any combinatorial triangulation on n vertices can be transformed into a 4-connected one using at most @?(3n-9)/[emailxa0protected]? edge flips. We also give an example of an infinite family of triangulations that requires this many flips to be made 4-connected, showing that our bound is tight. In addition, for n>=19, we improve the upper bound on the number of flips required to transform any 4-connected triangulation into the canonical triangulation (the triangulation with two dominant vertices), matching the known lower bound of 2n-15. Our results imply a new upper bound on the diameter of the flip graph of 5.2n-33.6, improving on the previous best known bound of 6n-30.

A Linear-Time Algorithm for the Geodesic Center of a Simple Polygon

Let P be a closed simple polygon with n vertices. For any two points in P, the geodesic distance between them is the length of the shortest path that connects them among all paths contained in P. The geodesic center of P is the unique point in P that minimizes the largest geodesic distance to all other points of P. In 1989, Pollack et al. (Discrete Comput Geom 4(1): 611–626, 1989) showed an

Gabriel Triangulations and Angle-Monotone Graphs: Local Routing and Recognition

O(nlog n)

All convex polyhedra can be clamped with parallel jaw grippers

O(nlogn)-time algorithm that computes the geodesic center of P. Since then, a longstanding question has been whether this running time can be improved. In this paper we affirmatively answer this question and present a deterministic linear-time algorithm to solve this problem.