# Stabbing Rectangles and Stabbing Disks.

December 2011

## Date of Award

Winter 12-12-2012

## Institute Name (Publisher)

Indian Statistical Institute

## Document Type

Master's Dissertation

## Degree Name

Master of Technology

Computer Science

## Department

Advance Computing and Microelectronics Unit (ACMU-Kolkata)

## Supervisor

Bishnu, Arijit (ACMU-Kolkata; ISI)

## Abstract (Summary of the Work)

Proximity graph [JT92; Lio; Tou91] is a graph where the edges between the vertices of the graph depends on the neighborliness of vertices. Proximity graph can be intuitively defined as follows: given a point set P in the plane, the vertices of the graphs, there is an edge between a pair of vertices p, q âˆˆ P if they satisfy some particular notion of neighborliness.Proximity graphs can be used in shape analysis and in data mining [JT92; Tou]. In graph drawing, a problem related to proximity graphs is to find the classes of graphs that admit a proximity drawing for some notion of proximity, and whenever possible to efficiently decide, for a given graph, whether such a drawing exists [BETT99; Lio].In the case of Gabriel graphs, GG(P), the notion of neighborliness of a pair of vertices a, b is the closed disk Dab with diameter ab. An edge ab is in the Gabriel graph of a point set P if and only if P âˆ© Dab = {a, b} (see Figure 1.1(Left)) [ADH10]. Gabriel graphs were introduced by Gabriel and Sokal [GS69] in the context of geographic variation analysis.In the case of Delaunay graphs, DG(P) [ADH11], the region of influence of a pair of vertices a, b is the set of closed disks Dab with chord ab. An edge ab is in the Delaunay graph of a point set P if and only if there exists a disk dab âˆˆ Dab such that P âˆ© dab = {a, b}In this thesis, we consider the problems related to the Witness graphs (gen-Figure 1.1: Gabriel graph. Left: The vertices defining the shaded disk are adjacent because their disk doesnâ€™t contain any other vertex, in contrast to the other vertices defining the unshaded disk. Right: Witness Gabriel graph. Black points are the vertices of the graph, white points are the witnesses. Each pair of vertices defining a shaded disk are adjacent and the pairs defining the unshaded disks are not.eralization of proximity graphs).1.1 The Witness Gabriel Graphs The witness Gabriel graph [ADH10] GGâˆ’(P, W) is defined by two sets of points P and W; P is the set of vertices of the graph and W is the set of witnesses. There is an edge ab in GGâˆ’(P, W) if and only if there is no point of W in Dab\\{a, b} (see Figure 1.1(Right)). The witness Gabriel graphs were introduced by Aronov et al. [ADH10] in 2010.1.2 The Witness Delaunay GraphsThewitness Delaunay graph[ADH11] of a point set P of vertices in the plane, with respect to a point set W of witnesses, denoted DGâˆ’(P, W), is the graph with vertex set P in which two points x, y âˆˆ P are adjacent if and only if thereis an open disk that does not contain any witness w âˆˆ W whose bounding circle passes through x and y.In graph drawing, a problem that is attracting substantial research is to find the number of witness points to remove all the edges of a witness graph. This problem can also be defined independently, as to find the size of the stabbing set for a point set P under some proximity notion. Stabbing set for a point set P is defined as follows: Let S be a family of geometric objects with nonempty interiors, each one associated to a finite subset of P.

ProQuest Collection ID: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:28843223

## Control Number

ISI-DISS-2011-304