Symmetric chain decompositions and independent families of curves.
No Thumbnail Available
Files
Date
2003-07-08
Authors
Journal Title
Series/Report No.
Journal ISSN
Volume Title
Publisher
Abstract
This thesis shows that symmetric independent families of n curves with the minimum possible number of regions exist for all n less than or equal to 16. Recent research has shown that such an independent family of curves exists for all prime n [Griggs, Killian, and Savage 2002]. For composite n, before this thesis, such a symmetric independent family of curves was known to exist only for n = 2,4,6,8,9, and 10.
An independent family of curves is a collection of simple closed curves intersecting at finite number of points. If we label the curves with 1,2,...,n, then each region is labeled with the set of labels of the curves containing that region. If every subset of [1,2,...,n] is a label for at least one region, then the collection of curves is an independent family of curves. If we rotate any curve around a point by an angle of (2π/n) radians (n-1) times and each time it coincides with one of the other curves, then the collection of curves is a symmetric independent family of curves.
We solve this geometric problem by first solving a combinatorial problem of looking for symmetric chain decompositions (SCD's) of necklace-representative posets with the chain cover property (CCP). We then show the way of constructing a symmetric independent family of curves with the minimum possible number of regions from an SCD of necklace-representative poset with the CCP.
Description
Keywords
graph drawing, Venn diagrams, periodic necklaces, planar embedding, geometric duals
Citation
Degree
MS
Discipline
Computer Science
