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MINIMUM AREA POLYOMINO VENN DIAGRAMS∗
Bette Bultena,† Matthew Klimesh,‡ and Frank Ruskey†
Abstract. Polyomino Venn (or polyVenn) diagrams are Venn diagrams whose curves are
the perimeters of polyominoes drawn on the integer lattice. Minimum area polyVenn diagrams are those in which each of the 2n intersection regions, in a diagram of n polyominoes,
consists of exactly one unit square.
We construct minimum area polyVenn diagrams in bounding rectangles of size 2r ×2c
whenever r, c ≥ 2. Our construction is inductive, and depends on two “expansion” results.
First, a minimum area polyVenn in a 22 × 2c rectangle can be expanded to produce another
one that fits into a 22 ×2c+3 bounding rectangle. Second, a minimum area polyVenn diagram
in a 2r × 2c rectangle can be expanded to produce another one that fits into a 2r+1 × 2c+1
rectangle. Finally, for even n we construct n-set polyVenn diagrams in bounding rectangles
of size (2n/2 − 1) × (2n/2 + 1) in which the empty set is not represented as a unit square.
1 Introduction
Venn diagrams have been drawn with a greater variety of shapes than John Venn must have
imagined. Much research has been concerned with the nature of the curves, for example,
whether they have rotational symmetry or are convex ([3],[8]). In this paper, the curves are
the perimeters of polyominoes and we further require that each region of the diagram be a
unit square.
Venn diagrams in which the area of each intersection region is in proportion to the
size of the population that it represents are said to be area proportional, and this is now
an active area of research; see, for example, [6] and [11]. It is also of interest to produce
diagrams in which each region has the same area, particularly if the shape of each region
is identical. Such diagrams might be useful because of their visual properties, or because
of the ease of labelling them. For example, the article [4] uses a minimum area polyomino
Venn diagram from [6] for data visualization in the field of microbiology. Our constructions
can be viewed as embeddings of the hypercube into grid graphs (e.g., [1],[2]) and thus may
be of interest to the graph embedding community.
Perhaps the first polyomino Venn diagrams appeared around 2000 on a website of
Mark Thompson [12], which introduced the idea of using the perimeters of polyominoes as
the curves of Venn diagrams, with the additional property that each intersection region is
∗
The work of B. Bultena was supported in part by a Canada Research Scholarship.
The work of F. Ruskey was supported in part by an NSERC Discovery Grant.
†
Dept. of Computer Science, University of Victoria, CANADA
‡
Jet Propulsion Laboratory, California Institue of Technology, USA
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a unit square. Chow and Ruskey [6] further investigated the problem of minimizing the
number of unit squares in an n-set polyVenn, and asked, but did not resolve, the question
of whether minimum area polyVenn diagrams exist for every number of curves.
In this paper, we settle several conjectures/questions from [6]. Most notably we
show how to construct minimum area polyomino diagrams in bounding rectangles of size
2r × 2c whenever r, c ≥ 2.
A
A
A
B
A
B
A
C
B
B
C
C
C
A B A
C
C
B AB
C
C
B
Figure 1: A (1, 2)-polyVenn. Two depictions are shown: first, the three polyominoes and
their positions within the base region, and second, a more compact depiction where each
grid square contains the labels of the pieces that it intersects.
2 Denitions
Consider the set of three L-shaped tetrominoes A, B, C, shown at the top of Figure 1. If
these are overlapped in the obvious way, then all subsets of {A, B, C} occur as a unique unit
square in the result, the 2 × 4 rectangle shown at the bottom of Figure 1. In other words,
the curves that comprise the perimeters of these tetrominoes form a Venn diagram when
overlaid as shown.
Definition 1. A polyomino [7] is set of unit squares whose corners lie on the integer lattice
and whose perimeter forms a single simple closed curve.
For our purposes, unit squares, and therefore polyominoes, are taken to include their
interiors. Note that in our definition of a polyomino no holes are allowed. We use the term
Venn diagram in the sense defined by Grünbaum in [9].
Definition 2. An n-Venn diagram consists of n simple closed curves C1 , C2 , · · · , Cn drawn
in the plane such that each of the 2n intersections
X1 ∩ X2 ∩ · · · ∩ Xn ,
(1)
where Xi is either the open interior or the open exterior of the curve Ci , is both non-empty
and connected.
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Venn diagram definitions sometimes require that the curves intersect each other at
a finite number of points, but for our purposes this requirement is too constraining. Let
[n] = {1, . . . , n} and let P([n]) denote the power set of [n]. Let int(S) denote the interior
of S ⊂ R2 . Given polyominoes P1 , . . . , Pn and a set U that contains the union of the
polyominoes, for each I ∈ P([n]) it is convenient to define
\
\
(U )
int(U \ Pi ).
(2)
XI (P1 , . . . , Pn ) =
int(Pi ) ∩
i∈I
(U )
i∈[n]\I
(U )
We may write XI
instead of XI (P1 , . . . , Pn ) when the polyomino list is clear from
(U )
context. We call each XI an intersection region.
Definition 3. A polyomino Venn diagram, or polyVenn, consists of polyominoes P1 , . . . , Pn
(R2 )
such that for all I ∈ P([n]) the set XI
is non-empty and connected. A minimum area
polyomino Venn diagram is a polyVenn together with a base region G that contains the
(G)
union of all the polyominoes, such that for all I ∈ P([n]) the set XI is a single unit square.
Note that for any set of polyominoes that yield a polyVenn, their perimeters make
up a Venn diagram.
In the minimum area polyVenn definition the base region provides a kind of consistency, in that the empty set is represented by a bounded region in the same way as the
other subsets of [n].
Definition 4. An (r, c)-polyVenn is an n-set minimum area polyVenn with n = r + c in
which the base region G is a 2r × 2c rectangle.
By convention we will only consider (r, c)-polyVenns with r ≤ c, since if this is not
the case the diagram can be rotated 90◦ to make it so.
The polyVenn in Figure 1 is an example of a (1, 2)-polyVenn. Note that in an n-set
polyVenn, each polyomino covers exactly 2n−1 unit squares, and intersects with every other
polyomino in exactly 2n−2 squares.
Proposition 1. A (0, c)-polyVenn does not exist for any c > 2.
Proof. Consider a (0, c)-polyVenn where c ≥ 2. Each of the c polyominoes must consist
of 2c−1 horizontally consecutive squares. Without loss of generality, let P1 be the leftmost
polyomino. All other polyominoes must each intersect P1 in its rightmost 2c−2 squares.
There is only one polyomino consisting of 2c−1 horizontally consecutive squares that does
this. A polyVenn cannot have two totally overlapping polyominoes, so we must have c ≤
2.
3 Expanding an Existing Diagram
It is natural to ask whether n-set minimum area polyVenns exist for all n. If they do, is
there a method to construct them? A well-known question [10] and an open problem [13]
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ask if a simple Venn diagram of n curves can be extended to a simple Venn diagram of n + 1
curves. We ask a similar question. We want to know if it is possible to extend a minimum
area polyVenn by stretching the diagram and adding more polyomino pieces.
Our primary tool for attacking this question is a technique we call expansion. The
idea is to take an existing (r, c)-polyVenn V and expand it into an (r + r0 , c + c0 )-polyVenn
by the addition of r0 + c0 polyominoes. The original polyominoes in V are “expanded” by
0
0
uniformly stretching vertically by a factor of 2r and horizontally by a factor of 2c . We refer
0
0
to this construction as a 2r × 2c expansion. We also use the term “expansion” to refer to
the stretching of the original polyVenn or its components; the meaning will be clear from
0
0
context. The key element of a 2r ×2c expansion is finding the r0 +c0 new polyominoes. The
construction of these new polyominoes will depend on r, r0 , c, c0 , but not otherwise on V .
Figure 2: A (2, 3)-polyVenn created by expansion.
Figure 2 shows an expansion of the (1, 2)-polyVenn of Figure 1 into a (2, 3)-polyVenn.
Each polyomino is positioned on the 4 × 8 grid. This grid is comprised of 8 minigrids, which
are the (in this case 2 × 2) expansions of the unit squares of the original polyVenn. The top
three polyominoes are the expanded originals from Figure 1. The bottom two polyominoes
have the property that they contain exactly one domino for each minigrid in the diagram.
The two dominoes within each minigrid make up a single (1, 1)-polyVenn.
0
Although it occurs in this example (and in any 2 × 2 expansion), in a general 2r ×
2 expansion the intersection of a given minigrid with the added polyominoes does not
necessarily need to form an (r0 , c0 )-polyVenn: the intersection of a given polyomino with
the minigrid might not be connected even though the whole polyomino is simply connected.
c0
0
0
Suppose we want to form a 2r × 2c expansion of an (r, c)-polyVenn by adding
0
0
polyominoes E1 , . . . , Er0 +c0 . Let Gij be the 2r × 2c miningrid expanded from the unit
square in row i and column j of the original base rectangle (with the upper left square
having indices (1, 1)). The relevant condition on the new polyominoes is that for every Gij
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and every T ∈ P([r0 + c0 ]), the set
\
int(Gij ∩ Et ) ∩
\
int(Gij \ Et )
(3)
t∈[r0 +c0 ]\T
t∈T
is a single unit square. When this condition holds for a given Gij we will say the added
polyominoes satisfy the expanded minigrid intersection (EMI) property for Gij .
We now can state and prove a straightforward but useful expansion result.
Theorem 2. Let V be an (r, c)-polyVenn in a rectangle G. Suppose r0 , c0 ≥ 0 and let
0
0
X(·) stretch the plane by factors of 2r vertically and 2c horizontally, so that polyominoes
P1 , . . . , Pr+c in G become X(P1 ), . . . , X(Pr+c ) in X(G). Suppose also that E1 , . . . , Er0 +c0
are polyominoes contained within X(G) that satisfy the EMI property for each minigrid of
X(G). Then augmenting X(V ) by E1 , . . . , Er0 +c0 results in an (r + r0 , c + c0 )-polyVenn that
0
0
is a 2r × 2c expansion of V .
Proof. We need to show that the set of polyominoes
S = {X(P1 ), . . . , X(Pr+c )} ∪ {E1 , . . . , Er0 +c0 }
together with the bounding rectangle X(G) make up an (r + r0 , c + c0 )-polyVenn.
Clearly each member of S is a polyomino: each X(Pk ) is an expanded polyomino,
and each Et is a polyomino by hypothesis. It remains to show that the required intersection
regions all consist of a single unit square. Each such region is of the form
!
!
\
\
\
\
int(Et )∩
int(X(G) \ Et ) ,
int(X(Pk ))∩
int(X(G) \ X(Pk )) ∩
k∈K
|
t∈T
k∈[r+c]\K
{z
t∈[r0 +c0 ]\T
}
A
(4)
where K ∈ P([r + c]) and T ∈
P([r0
+
c0 ]).
(G)
Note that the subexpression A is equal to X(XK (P1 , . . . , Pr+c )). However, since V
(G)
is an (r, c)-polyVenn, XK (P1 , . . . , Pr+c ) consists of a single unit square in G. Thus A is
an expanded unit square, which is a minigrid Gij for some 1 ≤ i ≤ 2r and 1 ≤ j ≤ 2c .
Therefore (4) can be written as
!
\
\
int(Gij ) ∩
int(Et ) ∩
int(X(G) \ Et )
t∈T
which, since Gij ⊂ X(G), is equal to
\
int(Gij ∩ Et ) ∩
t∈T
t∈[r0 +c0 ]\T
\
int(Gij \ Et ).
t∈[r0 +c0 ]\T
But this is just (3), which by the EMI property is a single grid square, so the proof is
complete.
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4 Two Expansions
We have two expansion results for (r, c)-polyVenns, Theorem 3 and Theorem 4.
Theorem 3. For c ≥ 0, a (2, c)-polyVenn can be expanded to a (2, c + 3)-polyVenn.
Proof. Let V be a (2, c)-polyVenn. We will exhibit a 1 × 8 expansion by constructing
polyominoes E1 , E2 , and E3 such that the conditions of Theorem 2 are satisfied.
It is convenient to describe E1 , E2 , and E3 by their intersections with the expanded
unit squares of V . Toward that end consider the following triples of subsets of a 1 × 8
minigrid:
KKJJ , MLML , KJJK ),
M21 = ( JMKL , KJKJ , MLML ),
M31 = ( JKJK , MKLJ , MLML ),
M41 = ( MLML , JJKK , JKKJ ),
M11 = (
JJKK , MLML , KJJK ),
M22 = ( MKLJ , JKJK , MLML ),
M32 = ( KJKJ . JMKL , MLML ),
M42 = ( MLML , KKJJ , JKKJ ).
M12 = (
We remark that each component of each Mi1 is a horizontal reflection of the same component
of Mi2 . For a given minigrid Gij (with 1 ≤ i ≤ 22 and 1 ≤ j ≤ 2c ), set the intersections
(E1 ∩ Gij , E2 ∩ Gij , E3 ∩ Gij ) to be Mi1 when j is odd and Mi2 when j is even.
The EMI property for E1 , E2 , and E3 with all of the 1 × 8 minigrids is readily
verified by inspection; for each Mij , each of the needed 23 = 8 intersections (complement
or not for each component) is a unit square.
It must also be shown that E1 , E2 , and E3 are polyominoes (specifically, that they
are simply connected). This can be verified by inspection for c = 0, 1. For larger c, the
resulting polyominoes are formed by repeatedly concatenating the c = 1 case polyominoes
with themselves. Again by inspection, this pattern produces simply-connected shapes (e.g.,
see Figure 3). Thus Theorem 2 produces the desired conclusion.
Theorem 4. For r, c ≥ 1, an (r, c)-polyVenn can be expanded to an (r + 1, c + 1)-polyVenn.
Proof. Let V be an (r, c)-polyVenn. We will exhibit a 2 × 2 expansion by constructing
polyominoes E1 and E2 such that the conditions of Theorem 2 are satisfied.
We describe E1 and E2 by their intersections with the expanded grid squares of V .
Each such intersection will correspond to one of the following pairs of subsets of a 2 × 2
minigrid:
Mnw = (
E , G ),
Mne = (
G , D ),
Msw = (
F , E ),
Mse = (
D , F ).
For a given 2 × 2 minigrid square Gij in the expanded bounding rectangle, set the intersections (E1 ∩ Gij , E2 ∩ Gij ) to be Mnw , Mne , Msw , or Mse depending on whether Gij is in
the upper left, upper right, lower left, or lower right quadrant of the expanded bounding
rectangle.
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P1
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P3
P2
P4
X(P1)
X(P2)
X(P3)
X(P4)
E1
E2
E3
Figure 3: Example of the 1 × 8 expansion used in the proof of Theorem 3, applied to the
(2, 2)-polyVenn with polyominoes P1 , P2 , P3 , P4 .
As in the proof of Theorem 3, the EMI property is easily verified by inspection of
the minigrid intersections with the added polyominoes. In this case these intersections are
Mnw , Mne , Msw , and Mse , which are all (1, 1)-polyVenns.
It is also required to show that E1 and E2 are polyominoes. It is clear from Figure 4
that E1 and E2 are simply connected for square grids up to 16 × 16. One can imagine the
pattern for all rectangles with height greater than four. We do not pursue a more rigorous
argument, but we note that one could be produced by showing, for k = 1, 2, that: (1) the
construction does not produce any diagonal-only connections in Ek , (2) the exterior of Ek
is connected, since from any square that is not a member of Ek , one can exit the bounding
rectangle with a straight path that does not intersect Ek , and (3) Ek is connected since,
from any square in Ek , one can reach the center 2 × 2 region of Ek with no more than a
two line-segment path with Ek . From these facts, one could conclude that the perimeters
of E1 and E2 are simple closed curves and thus E1 and E2 are polyominoes.
Applying Theorem 2 completes the proof.
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Figure 4: The constructed E1 and E2 described in the proof of Theorem 4, shown for
(r, c) = (1, 1), (2, 2), (3, 3), and (4, 4). Note that the construction is valid for any r, c ≥ 1.
Figure 5: Alternative E1 and E2 polyominoes for producing 2 × 2 expansions, shown for
(r, c) = (1, 1), (2, 2), (3, 3), and (4, 4). The general construction is valid for any r, c ≥ 1.
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Alternative constructions of E1 and E2 are possible in the proof of Theorem 4.
Figure 5 illustrates an interesting construction that is also defined on a 2r+1 ×2c+1 rectangle
when r, c ≥ 1. Let E10 and E20 be the polyominoes in this case, while E1 and E2 are the
polyominoes described in the proof of Theorem 4. This construction is very similar: The
center 4 × 4 regions of E10 and E20 are identical to the same regions of E1 and E2 . Also the
same 2 × 2 pairs of subsets are applied to the same quadrants. However, instead of using
only the first element of each subset for E1 and the second for E2 , the pairs are alternating
in E10 and E20 , so adjacent minigrids in a quadrant always contain both the first and second
element of the pair.
Although the simple connectivity is clear to the eye from the examples in Figure 5,
the argument that E10 and E20 are polyominoes would be laid out in the same manner as
the description provided in the proof of Theorem 4.
5 Summary of Existence Results
A
A
A
A
B
B
A
B
A
B
Figure 6: The trivial polyVenns.
Theorems 3 and 4 provide a way of inductively determining ranges of (r, c) pairs for
which an (r, c)-polyVenn exists. But in order to begin we must establish some base cases.
It turns out to be sufficient to consider base cases with r = 0 and r = 1. Figure 6
exhibits (trivial) (0, c)-polyVenns for c = 0, 1, 2 and the (1, 1)-polyVenn. The trivial (1, 2)polyVenn was shown in Figure 1.
In Figures 7 and 8, we show a (1, 3)-polyVenn and a (1, 4)-polyVenn, respectively.
For c > 4 we do not know of any (1, c)-polyVenns and we make the following conjecture:
Conjecture 5. A (1, c)-polyVenn does not exist for c > 4.
A motivation for this conjecture comes from the observation that having only two
rows is very constraining in the construction of polyVenns; for example no (0, 1) expansion
seems possible on an existing (1, c)-polyVenn. Also, a partial computer search has not
produced a (1, 5)-polyVenn.
However, given the base cases and the expansion theorems, we summarize the known
polyVenns in the following theorem:
Theorem 6. For every n > 0 and for every r, c > 2 where r+c = n, there exists a minimum
polyomino Venn diagram bound in a 2r × 2c bounding rectangle.
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D C D C
B
B
C
B
A
A
C
C D
A
A
D
B A B A B A B A B
C D
D
D C D C
A
B
C
D
Figure 7: A (1, 3)-polyVenn with nice symmetry.
D
C
D
B
D
B
BC
AB
D
D
BC A BC A BC A BC A C A BC A C
C
E D
D
D E D E D E
A
D
A
B
D E D E D E
B
B C A B AB
AB
E D E D E
E
A
B
C
D
E
Figure 8: A (1, 4)-polyVenn.
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E
163
C A C A
E
E
A
A
E
C
E
C
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Proof. Applying Theorems 3 and 4 with the base cases completes the proof.
Table 1 shows how the base cases are extended and summarizes our state of knowledge about for which r and c an (r, c)-polyVenn exists.
20
20
F6
21
22
23
24
25
26
27
28
..
.
21
F6
F6
22
F6
F6
T4
23
X
F7
T4
T4
24
X
F8
T4
T4
T4
25
X
?
T3
T4
T4
T4
26
X
?
T3
T4
T4
T4
T4
27
X
?
T3
T4
T4
T4
T4
T4
28
X
?
T3
T4
T4
T4
T4
T4
T4
···
···
···
···
···
···
···
···
···
···
..
.
Table 1: Our knowledge of existence of (r, c)-polyVenns. Rows and columns are labelled
with bounding box dimensions (2r and 2c ). Fn indicates existence by example given in
Figure n; ‘X’ indicates nonexistence following from Proposition 1; T3 and T4 indicate
existence implied by Theorems 3 and 4, respectively; and question marks are currently
open cases.
6 Rectangles That Omit the Empty Set
In this section we discuss the problem of finding n-set polyVenns in which each polyomino
fits into an h × w rectangle where 2n − 1 = hw, so that the region for the empty set (the
intersection of the exterior of all of the polyominoes) is the exterior of the rectangle. We
refer to such polyVenns as minimum area empty-set-omitting polyVenns or ESO-polyVenns.
For n > 2 it is only possible for such polyVenns to exist when Mn = 2n −1 is not a Mersenne
prime, since otherwise the resulting diagram must have dimensions 1 × Mn and would imply
the existence of a polyVenn that violates Proposition 1. For n > 2, the first 8 such numbers
are 24 − 1 = 15 = 3 · 5, 26 − 1 = 63 = 32 · 7, 28 − 1 = 255 = 3 · 5 · 17, 29 − 1 = 511 = 7 · 73,
210 − 1 = 1023 = 3 · 11 · 31, 211 − 1 = 2047 = 23 · 89, and 212 − 1 = 4095 = 32 · 5 · 7 · 13.
This problem was first mentioned in [6].
Theorem 7. For all r ≥ 0 there exists a minimum area ESO-polyVenn with a (2r − 1) ×
(2r + 1) base rectangle.
Proof. We use a simple construction of a polyVenn of dimension (2r − 1) × (2r + 1), where
r = n/2, based on the successive 2 × 2 expansions of the (1, 1)-polyVenn of Figure 6. We
can use either the construction detailed in the proof of Theorem 4 (epitomized by Figure 4)
or the alternative construction (epitomized by Figure 5). In either case the resulting (r, r)polyVenns have two easily verified properties. First, the empty set is represented by the
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Figure 9: Converting the base case (1, 1)-polyVenn to a 1 × 3 polyVenn that omits the
empty set.
Figure 10: Omiting the empty set: a polyVenn on the (2n/2 − 1) × (2n/2 + 1) grid for n = 8.
upper left square of the bounding rectangle. Second, if the top row of the diagram is
removed, rotated 90◦ clockwise, and attached to the right side of the diagram (with the
formerly rightmost square even with the bottom of the diagram), then the rearranged
polyominoes continue to be simply connected (i.e., they are still polyominoes). Performing
the rearrangement described in this second property results in a minimum area polyVenn
whose base region is a (2r − 1) × (2r + 1) rectangle with an extra square attached at the
top of the rightmost column. But by the first property this extra square represents the
empty set, so the resulting polyominoes actually form a minimum area ESO-polyVenn in
the (2r − 1) × (2r + 1) rectangle.
Figure 9 demonstrates how this construction converts the (1, 1)-polyVenn to a polyVenn
on a 1 × 3 grid. Figure 10 shows the result of the construction when n = 8 and the (4, 4)polyVenn was constructed using repeated 2×2 expansions of the type illustrated in Figure 5.
We do not know of any minimum area ESO-polyVenns for grid dimensions not
covered by this construction. The next few such dimensions are
63 = 3 × 21, 255 = 3 × 85 = 5 × 51, and 511 = 7 × 73.
Given the seemingly mysterious nature of the Mersenne primes, we feel that there
will not be any general construction possible when n is odd and Mn is not prime.
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7 Final Remarks and Open Problems
If a (1, 5)-polyVenn does not exist, this might be established through either an exhaustive
computer search or through a more mathematical proof. We have expended some effort
in both of these directions. A computer search can easily find a large number of (1, 4)polyVenns but exhaustively searching for a (1, 5)-polyVenn may be beyond our current
technique. Since our results show that (2, c)-polyVenns exist for all c, we do know that
minimum area polyVenns exist with very long and thin bounding rectangles.
The facts that (r, c)-polyVenns exist for all r, c ≥ 2 and that minimum area emptyset-omitting polyVenns exist for bounding rectangles of size (2n/2 −1)×(2n/2 +1) for even n
suggest that additional interesting constructions of minimum area polyVenns may exist. In
particular it would be interesting to see a construction of minimum area empty-set-omitting
polyVenns for another infinite family of bounding rectangles.
Acknowledgements
We thank the referees for their helpful comments, which have improved the presentation of
this paper.
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