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.

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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) 66-71.
- P. Hamburger, A. Sali and G. Petruska, Saturated chain partitions in ranked partially ordered sets, and non-monotone symmetric 11-Venn diagrams, Studia Sci. Math. Hungar. 41 (2004) 147-191.
- 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).

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