Minimum area polyomino Venn diagrams

• Main Authors: Bette Bultena, Matthew Klimesh, Frank Ruskey
• Format: Article
• Language: English
• Published: Carleton University 2012-09-01
• Series: Journal of Computational Geometry
• Online Access: http://jocg.org/index.php/jocg/article/view/91

Polyomino Venn (or polyVenn) diagrams are Venn diagrams whose curves are the perimeters of orthogonal polyominoes drawn on the integer lattice. Minimum area polyVenn diagrams are those in which each of the 2^<em>n</em> intersection regions, in a diagram of <em>n</em> polyominoes, consists of exactly one unit square. We construct minimum area polyVenn diagrams in bounding rectangles of size 2^<em>r</em>×2^<em>c</em> whenever <em>r</em>, <em>c</em> ≥ 2. Our construction is inductive, and depends on two <q>expansion</q>results. First, a minimum area polyVenn diagram in a 2^<em>r</em>×2^<em>c</em> rectangle can be expanded to produce another that fits into a 2^<em>r</em>+1×2^<em>c</em>+1 rectangle. Second, a minimum area polyVenn in a 2²×2^<em>c</em> rectangle can be expanded to produce another that fits into a 2²×2^<em>c</em>+3 bounding rectangle. Finally, for even <em>n</em> we construct <em>n</em>-set polyVenn diagrams in bounding rectangles of size (2^<em>n</em>/2-1)×(2^<em>n</em>/2+1) in which the empty set is <em>not</em> represented as a unit square.