# Study on Stabbing Rectangles and Circles Induced by Point Sets.

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)

Prorimity graph [JT92; Lio; Tou91] is a graph where the edges between the vertices of the graph depends on the neighborliness of vertices. Prozimity 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 P if they satisfy some particular notion of neighborliness.Proximity graphs can be used in shape analysis and in data mining |JT92; Toul. 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 abâ€¾. An edge ab is in the Gabriel graph of a point set P if and only if Pâˆ© Dab = {a, b} (see Figure %3D 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 abâ€¾. 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© das = {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 doesn't contain any other vertex, in contrast to the other verfices 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 GraphsThe witness Gabriel graph (ADH10] GGâ”€(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 GGâ”€(P,W) if and only if there is no point of W in Dab\\{a, b} (see Figure I.1(Right)). The witness Gabriel graphs were introduced by Aronov et al. â”‚ADH10â”‚ in 2010.1.2 The Witness Delaunay GraphsThe witress Delaunay groph â”‚JADHI1â”‚ of a point set Pof vertices in the plaar, with respect to a puint set W of witaes, denoted DGâ”€(P.W), is the graph with verte set Pis which two points , y âˆˆ P are adjacent if and only if there is an open disk that dos ot contain say winess w âˆˆ W whose bounding cricle passes through x and y.Ia graph drawing, a peobkem that is attracting salstantial revoech is to fiad the mamber of witneo poitta to remove all the edges of a witna graph.

## Comments

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:28843286

## Control Number

ISI-DISS-2011-380

## Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

## DOI

http://dspace.isical.ac.in:8080/jspui/handle/10263/6913

COinS