Symmetric Monotone Venn Diagrams with Seven Curves

Tao Cao, Department of Computer Science, University of Victoria, Canada.
Khalegh Mamakani, Department of Computer Science, University of Victoria, Canada.
Frank Ruskey, Department of Computer Science, University of Victoria, Canada.

Abstract:

An n-Venn diagram consists of n curves drawn in the plane in such a way that each of the 2n possible intersections of the interiors and exteriors of the curves forms a connected non-empty region. A k-region in a diagram is a region that is in the interior of precisely k curves. A n-Venn diagram is symmetric if it has a point of rotation about which rotations of the plane by 2\pi/n radians leaves the diagram fixed; it is polar symmetric if it is symmetric and its stereographic projection about the infinite outer face is isomorphic to the projection about the innermost face. A Venn diagram is monotone if every k-region is adjacent to both some (k-1)-region (if k > 0) and also to some k+1 region (if k < n). A Venn diagram is simple if at most two curves intersect at any point. We prove that the so-called Grunbaum encoding uniquely identifies monotone symmetric n-Venn diagrams and describe an algorithm that produces an exhaustive list of all of the monotone symmetric n-Venn diagrams. That algorithm is used to prove that there are exactly 23 simple monotone symmetric 7-Venn diagrams, of which 6 are polar symmetric.

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