Hazel Everett

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.

The homogeneous set sandwich problem

Abstract The graph sandwich problem for property Φ is defined as follows: Given two graphs G 1 = ( V , E 1 ) and G 2 = ( V , E 2 ) such that E 1 ⊆ E 2 , is there a graph G = ( V , E ) such that E 1 ⊆ E ⊆ E 2 which satisfies property Φ? We present a polynomialtime algorithm for solving the graph sandwich problem, when property Φ is “to contain a homogeneous set”. The algorithm presented also provides the graph G and a homogeneous set in G in case it exists.

The voronoi diagram of three lines

We give a complete description of the Voronoi diagram of three lines in R3. In particular, we show that the topology of the Voronoi diagram is invariant for three lines in general position, that is, that are pairwise skew and not all parallel to a common plane. The trisector consists of four unbounded branches of either a non-singular quartic or of a cubic and line that do not intersect in real space. Each cell of dimension two consists of two connected components on a hyperbolic paraboloid that are bounded, respectively, by three and one of the branches of the trisector. The proof technique, which relies heavily upon modern tools of computer algebra, is of interest in its own right. This characterization yields some fundamental properties of the Voronoi diagram of three lines. In particular, we present linear semi-algebraic tests for separating the two connected components of each two-dimensional Voronoi cell and for separating the four connected components of the trisector. This enables us to answer queries of the form, given a point, determine in which connected component of which cell it lies. We also show that the arcs of the trisector are monotonic in some direction. These properties imply that points on the trisector of three lines can be sorted along each branch using only linear semi-algebraic tests.

AN OPTIMAL ALGORITHM FOR COMPUTING (≤K)-LEVELS, WITH APPLICATIONS

This paper gives an optimal O(n log n+nk) time algorithm for constructing the levels 1,…, k in an arrangement of n lines in the plane. This algorithm is extended to compute these levels in an arrangement of n unbounded x-monotone polygonal convex chains, of which each pair intersects at most a constant number of times. We then show how these results can be used to solve several geometric optimization problems including the weak separation problem for sets of red and blue points or polygons, the maximum line transversal problem for sets of line segments, the densest hemisphere problem for sets of points on a sphere and the optimal corridor problem for sets of points in the plane. All of the algorithms are quality-sensitive; they run faster if the optimal solution is a good one.

Transversals to Line Segments in Three-Dimensional Space

AbstractWe completely describe the structure of the connected components of transversalsnto a collection of n line segments in ℝ3. Generically, the set of transversals to four segments consists of zero or two lines. We catalog the non-generic cases and show that n≥ 3 arbitrary line segments in ℝ3 admit at most n connected components of line transversals, and that this bound can be achieved in certain configurations when the segments are coplanar, or they all lie on a hyperboloid of one sheet. This implies a tight upper bound of n on the number of geometric permutations of line segments in ℝ3.

Universal Sets of n Points for One-bend Drawings of Planar Graphs with n Vertices

This paper shows that any planar graph with n vertices can be point-set embedded with at most one bend per edge on a universal set of n points in the plane. An implication of this result is that any number of planar graphs admit a simultaneous embedding without mapping with at most one bend per edge.

Slicing an ear using prune-and-search

Abstract It is well known that a diagonal of a simple polygon P can be found in linear time with a simple and practically efficient algorithm. An ear of P is a triangle such that one of its edges is a diagonal of P and the remaining two edges are edges of P . An ear of P can easily be found by first triangulating P and subsequently searching the triangulation. However, although a polygon can be triangulated in linear time, such a procedure is conceptually difficult and not practically efficient. In this note we show that an ear of P can be found in linear time with a simple, practically efficient algorithm that does not require pre-triangulating P .

The Graham scan triangulates simple polygons

Abstract The Graham scan is a fundamental backtracking technique in computational geometry which was originally designed to compute the convex hull of a set of point in the plane [9] and has since found application in several different contexts. In this note we show how to use the Graham scan to triangulate a simple polygon. The resulting algorithm triangulates an n-vertex polygon P in O(kn) time where k −1 is the number of concave vertices in P. Although the worst case running time of the algorithm is O(n2), it is easy to implement and is therefore of practical interest.

The Expected Number of 3D Visibility Events Is Linear

In this paper, we show that, amongst

Lines and Free Line Segments Tangent to Arbitrary Three-Dimensional Convex Polyhedra

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