MINIMUM AREA POLYOMINO VENN DIAGRAMS

Abstract. 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 n regions, in a diagram of n polyominoes, consists of exactly one unit square. We construct minimum area polyVenn diagrams in bounding rectangles of size 2 r × 2 c whenever r, c ≥ 2. Our construction is inductive, and depends on two “expansion ” results. First, a minimum area polyVenn diagram in a 2 r ×2 c rectangle can be expanded to produce another that fits into a 2 r+1 ×2 c+1 rectangle. Secondly, when r = 2, it can also be expanded to produce a polyVenn diagram in a 2 r × 2 c+3 bounding rectangle. Finally, we construct polyVenn diagrams in bounding rectangles of size (2 n/2 − 1) × (2 n/2 + 1) if n is even, but where the empty set is not represented as a unit square