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Unit Disk Graph Recognition is NPHard
 Computational Geometry. Theory and Applications
, 1993
"... Unit disk graphs are the intersection graphs of unit diameter closed disks in the plane. This paper reduces SATISFIABILITY to the problem of recognizing unit disk graphs. Equivalently, it shows that determining if a graph has sphericity 2 or less, even if the graph is planar or is known to have s ..."
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Unit disk graphs are the intersection graphs of unit diameter closed disks in the plane. This paper reduces SATISFIABILITY to the problem of recognizing unit disk graphs. Equivalently, it shows that determining if a graph has sphericity 2 or less, even if the graph is planar or is known to have sphericity at most 3, is NPhard. We show how this reduction can be extended to 3 dimensions, thereby showing that unit sphere graph recognition, or determining if a graph has sphericity 3 or less, is also NPhard. We conjecture that Ksphericity is NPhard for all fixed K greater than 1. 1 Introduction A unit disk graph is the intersection graph of a set of unit diameter closed disks in the plane. That is, each vertex corresponds to a disk in the plane, and two vertices are adjacent in the graph if the corresponding disks intersect. The set of disks is said to realize the graph. Of course, the unit of distance is not critical, since the disks realize the same graph even if the coordina...
Map Labeling Problems
"... I hereby declare that I am the sole author of this thesis. I authorize the University of Waterloo to lend this thesis to other institutions or individuals for the purpose of scholarly research. I further authorize the University of Waterloo to reproduce this thesis by photocopying or by other means, ..."
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I hereby declare that I am the sole author of this thesis. I authorize the University of Waterloo to lend this thesis to other institutions or individuals for the purpose of scholarly research. I further authorize the University of Waterloo to reproduce this thesis by photocopying or by other means, in total or in part, at the request of other institutions or individuals for the purpose of scholarly research. ii The University of Waterloo requires the signatures of all persons using or photocopying this thesis. Please sign below, and give address and date. iii Abstract One of the most challenging tasks in cartography is the labeling of maps attaching text to geographic features. Many of the issues become simpler when the features are points, for example cities on a largescale map, because we expect the text to be placed horizontally and close to the associated point. We want labels that do not overlap and are large enough to be readable. Even simple formulations of this problem are NPcomplete. One such formulation is the pointfeature label placement problem: given a set of points in the plane, and an axisparallel rectangular label associated with each point, place each label with one corner at the associated point such that no two labels overlap. This problem is known to be NPcomplete. Modeling each label as a fixed rectangle in this way is quite limiting. Researchers have considered approximation algorithms where each label can be scaled. In this thesis, we propose an alternative formulation to the map labeling problem. We introduce the use of elastic labels, where each label is a rectangle with fixed area, but varying in height and width. Then, we define the elastic labeling problem as determining the choice of height and width of each label, and the corner of the label to place at the associated point, so that no two labels overlap. This problem is useful when the goal of placing a label at a given point is to associate some text, consisting of more than one word, with the point. In this case we can write the specified text inside the label using one, two, or more rows, as long as the label is placed at the specified point.