Convex Drawings of Intersecting Families of Simple Closed Curves
Bette Bultena,
Department of Computer Science,
University of Victoria, Canada.
Branko Grünbaum,
Department of Mathematics,
University of Washington, USA.
Frank Ruskey,
Department of Computer Science,
University of Victoria, Canada.
Abstract:
A FISC or family of intersecting simple closed curves
is a collection of simple closed curves in the plane
with the properties that there is some open region common to the
interiors of all the curves, and that every two curves intersect
in finitely many points.
Let F be a FISC.
Intersections of the curves represent the vertices of a plane graph,
G(F), whose edges are the curve arcs between the vertices.
The directed dual of G(F),
denoted D(F),
is the dual graph of G(F), but with
edges oriented to indicate inclusion in fewer interiors of the curves.
A convex drawing of G(F) is one in which every curve is convex.
The graph G(F) has a convex drawing if there is some FISC
C whose curves are all convex and where F
can be transformed into C by a continuous transformation
of the plane.
We prove that G(F) has a convex drawing if and only if
D(F) contains only one source and only one sink.

The postscript file;
the dvi file.
Please send me a note
if you download a copy  Thanks!

For more on Venn diagrams see the
Survey of
Venn Diagrams.

Selected Citations:

Johann Linhart and Ronald Ortner,
A Note on Convex Realizability of Arrangements of Pseudocircles
Geombinatorics, 18 (2008) 6671.

P. Hamburger, A. Sali and G. Petruska,
Saturated chain partitions in ranked partially ordered sets,
and nonmonotone symmetric 11Venn diagrams,
Studia Sci. Math. Hungar. 41 (2004) 147191.

Jerrold Griggs, Charles E. Killian and Carla D. Savage,
Venn Diagrams and Symmetric Chain Decompositions in the Boolean Lattice,
Electronic Journal of Combinatorics, Volume 11 (no. 1), #R2, (2004).
Back to list of publications.