Study on Stabbing Rectangles and Circles Induced by Point Sets.

Date of Submission

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

Subject Name

Computer Science


Advance Computing and Microelectronics Unit (ACMU-Kolkata)


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 ab‾. An edge ab is in the Gabriel graph of a point set P if and only if P∩ 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 ab‾. An edge ab is in the Delaunay graph of a point set P if and only if there exists a disk dab ∈ Dab such that P ∩ 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] GG─(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 GG─(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. │ADH10│ in 2010.1.2 The Witness Delaunay GraphsThe witress Delaunay groph │JADHI1│ of a point set Pof vertices in the plaar, with respect to a puint set W of witaes, denoted DG─(P.W), is the graph with verte set Pis which two points , y ∈ P are adjacent if and only if there is an open disk that dos ot contain say winess w ∈ 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.


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


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