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# Face-balanced, Venn and polyVenn diagrams

 dc.contributor.author Bultena, Bette dc.date.accessioned 2013-08-29T23:20:15Z dc.date.available 2013-08-29T23:20:15Z dc.date.copyright 2013 en_US dc.date.issued 2013-08-29 dc.identifier.uri http://hdl.handle.net/1828/4873 dc.description.abstract A \emph{simple} $n$-\emph{Venn diagram} is a collection of $n$ simple intersecting closed curves in the plane where exactly two curves meet at any intersection point; en_US the curves divide the plane into $2^n$ distinct open regions, each defined by its intersection of the interior or exterior of each of the curves. A Venn diagram is \emph{reducible} if there is a curve that, when removed, leaves a Venn diagram with one less curve and \emph{irreducible} if no such curve exists. A Venn diagram is \emph{extendible} if another curve can be added, producing a Venn diagram with one more curve. Currently it is not known whether every simple Venn diagram is extendible by the addition of another curve. We show that all simple Venn diagrams with $5$ curves or less are extendible to another simple Venn diagram. We also show that for certain Venn diagrams, a new extending curve is relatively easy to produce. We define a new type of diagram of simple closed curves where each curve divides the plane into an equal number of regions; we call such a diagram a \emph{face-balanced} diagram. We generate and exhibit all face-balanced diagrams up to and including those with $32$ regions; these include all the Venn diagrams. Venn diagrams exist where the curves are the perimeters of polyominoes drawn on the integer lattice. When each of the $2^n$ intersection regions is a single unit square, we call these \emph{minimum area polyomino Venn diagrams}, or \emph{polyVenns}. We show that polyVenns can be constructed and confined in bounding rectangles of size $2^r \times 2^c$ whenever $r, c \ge 2$ and $n=r+c$. We show this using two constructive proofs that extend existing diagrams. Finally, for even $n$, we construct polyVenns with $n$ polyominoes in $(2^{n/2} - 1) \times (2^{n/2} + 1)$ bounding rectangles in which the empty set is not represented as a unit square. dc.language English eng dc.language.iso en en_US dc.subject Venn diagram en_US dc.subject graph theory en_US dc.subject computational geometry en_US dc.subject minimum area Venn diagram en_US dc.subject Winkler's conjecture en_US dc.title Face-balanced, Venn and polyVenn diagrams en_US dc.type Thesis en_US dc.contributor.supervisor Ruskey, Frank dc.degree.department Dept. of Computer Science en_US dc.degree.level Doctor of Philosophy Ph.D. en_US dc.rights.temp Available to the World Wide Web en_US dc.identifier.bibliographicCitation Bette Bultena, Branko Gr\"{u}nbaum, Frank Ruskey. Convex Drawings of Intersecting Families of Simple Closed Curves.'' 11th Canadian Conference of Computational Geometry, (1999): 33-54. en_US dc.identifier.bibliographicCitation Bette Bultena, Frank Ruskey. Minimum Area Polyomino Venn Diagrams.'' Journal of Computation Geometry, 3(2012): 154-167. en_US dc.identifier.bibliographicCitation Bette Bultena, Frank Ruskey. Venn Diagrams with Few Vertices.'' Electronic Journal of Combinatorics, 5(1998): R44. en_US dc.description.scholarlevel Graduate en_US dc.description.proquestcode 0405 en_US dc.description.proquestcode 0984 en_US
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