Thomas C. Shermer

Recent results in art galleries (geometry)

Two points in a polygon are called if the straight line between them lies entirely inside the polygon. The art gallery problem for a polygon P is to find a minimum set of points G in P such that every point in P is visible from some point of G. The author provides an introduction to art gallery theorems and surveys the recent results of the field. The emphasis is on the results rather than the techniques. Several new problems that have the same geometric flavor as art gallery problems are also examined. >

Additive graph spanners

A spanning subgraph S = (V, E′) of a connected simple graph G = (V, E) is a f(x)-spanner if for any pair of nodes u and v, dS(u, v) ≦ f(dG(u, v)), where dG and dS are the usual distance functions in graphs G and S, respectively. We are primarily interested in (t + x)-spanners, which we refer to as additive spanners. We construct low-degree additive spanners for X-trees, pyramids, and multidimensional grids. We prove, for arbitrary t > 0, that to determine whether a given graph G has an additive spanner with no more than m edges is NP-complete.

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.

On representations of some thickness-two graphs

This paper considers representations of graphs as rectanglevisibility graphs and as doubly linear graphs. These are, respectively, graphs whose vertices are isothetic rectangles in the plane with adjacency determined by horizontal and vertical visibility, and graphs that can be drawn as the union of two straight-edged planar graphs. We prove that these graphs have, with n vertices, at most 6n−20 (resp., 6n−18) edges, and we provide examples of these graphs with 6n−20 edges for each n≥8.

A Visibility Representation for Graphs in Three Dimensions

This paper proposes a 3-dimensional visibility representation of graphs G =( V;E) in which vertices are mapped to rectangles floating in R 3 parallel to the x;y-plane, with edges represented by vertical lines of sight. We apply an extension of the Erd} os-Szekeres Theorem in a geometric setting to obtain an upper bound of n = 56 for the largest representable complete graph Kn. On the other hand, we show by construction that n 22. These are the best existing bounds. We also note that planar graphs and complete bipartite graphs Km;n are representable, but that the family of representable graphs is not closed under graph minors.

Graph Folding: Extending Detail and Context Viewing into a Tool for Subgraph Comparisons

It is a difficult problem to display large, complex graphs in a manner which furthers comprehension. A useful approach is to expand selected sections (foci) of the graph revealing details of subgraphs. If this expansion is maintained within the context of the entire graph, information is provided about how subgraphs are embedded in the overall structure. Often it is also desirable to realign these foci in order to facilitate the visual comparison of subgraphs. We have introduced a distortion-based viewing tool, three-dimensional pliable surface (3DPS) [1], which allows for multiple arbitrarily-shaped foci on a surface that can be manipulated by the viewer to control the level of detail contained within each region. This paper extends 3DPS to include the repositioning of foci so as to bring together spatially separated regions for the purpose of comparison while retaining the effect of detail in context viewing. The significance of this approach is that it utilizes precognitive perceptual cues about the three-dimensional surface to make the distortions comprehensible, and allows the user to interactively control the location, shape, and extent of the distortion in very large graphs.

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.

Generalized guarding and partitioning for rectilinear polygons

Abstract A Tk-guard G in a rectilinear polygon P is a tree of diameter k completely contained in P. The guard G is said to cover a point x if x is visible from some point contained in G. We investigate the function r(n,h,k), which is the largest number of Tk-guards necessary to cover any rectilinear polygon with h holes and n vertices. The aim of this paper is to prove new lower and upper bounds on parts of this function. In particular, we show the following upper bounds: 1. r(n,0,k)⩽uec02 n k+4 uec01 , with equality for even k. 2. r(n,h,1)=uec02 n+ 4h 3 + 4 3 4+ 4 3 uec01 3. (n,h,2)⩾uec02 n 6 These bounds, along with other lower bounds that we establish, suggest that the presence of holes reduces the number of guards required, if k > 1. In the course of proving the upper bounds, new results on partitioning are obtained which also have efficient algorithmic versions.

ADDITIVE SPANNERS FOR HYPERCUBES

A t-spanner of a network is a subnetwork in which every two nodes that were connected by an edge in the original network are connected by a path of at most t edges in the subnetwork. This guarantees that if nodes u and v are at distance d(u,v) in the original network, they are at distance no more than t·d(u,v) in the t-spanner. We introduce a more general definition of spanners. A special case of this more general definition is the additive t-spanner: a subnetwork in which every two nodes u and v that were separated by distance d(u,v) in the original network are at distance no more than t+d(u,v) in the subnetwork. We construct low-degree additive t-spanners for hypercubes.

On the sectional area of convex polytopes

A function j : %3 -+ ‘R is unimodal if it increases to a maximum value and then decreases. It is strictly unimodal if the increase and decrease are strict. To be precise, ~ is strictly unimodal iff for all reals z < y, the value o = min{~(z), j(y)} is either the global minimum or maximum of ~ or v < ~(z) for all z G (z, y). Unimodality is important for the design of efficient search algorithms because it permits prune-and-search strategies. E.g., Chazelle and Dobkin [2] showed that the perpendicular distance from a line 1 to an n-vertex convex polygon Q is bimodal. From this, unimodal functions can be constructed and the farthest point from t can be computed in O(log n) time. Unimodality can also simplify proofs. It was shown in 1973 that if A(z) is the length of the intersection of a convex polygon Q with the vertical line through Z, then A(x) is a strictly unimodal function [4]. We note that this generalizes to higher dimensions: the area in 7?3 (or volume in 7?~) of the intersection of a convex polytope K and a hyperplane h(z) = {(x,23, ..., Zd) I VZi G ‘R} is a strictly unimodal function A(x). (If the plane is defined by rotation instead of translation, then there are convex polytopes for which sectional area is not unimodal.) Prune-andsearch can be used to compute the intersection with maximum area (volume) in time proportional to the size of K, if K is stored with its complete facial lattice. For %33 our algorithm has an application to shape matching: Given convex polygons P and Q and a direction in which to translate P, one can find the translation