Defining Sets and Critical Sets in (0,1)‐Matrices

Abstract

If D is a partially filled‐in (0, 1)‐matrix with a unique completion to a (0, 1)‐matrix M (with prescribed row and column sums), we say that D is a defining set for M. If the removal of any entry of D destroys this property (i.e. at least two completions become possible), we say that D is a critical set for M. In this note, we show that the complement of a critical set for a (0, 1)‐matrix M is a defining set for M. We also study the possible sizes (number of filled‐in cells) of defining sets for square matrices M with uniform row and column sums, which are also frequency squares. In particular, we show that when the matrix is of even order 2m and the row and column sums are all equal to m, the smallest possible size of a critical set is precisely m2. We give the exact structure of critical sets with this property. © 2012 Wiley Periodicals, Inc. J. Combin. Designs 21: 253–266, 2013

Critical Sets of 2-Balanced Latin Rectangles

“…Observe that any (0, 1)-matrix m × n is in fact a subset of F m,n . Indeed, any trade in a (0, 1)-matrix is a disjoint union of trade cycles ( [3]; an equivalent result is also shown by Lemma 3.2.1 of [2]). …”

“…This is useful because it is well-known that a (0, 1)-matrix has no trades (and is thus the unique member of its class A(R, S)) if and only if its rows and columns can be rearranged so that a line of non-increasing gradient can be drawn with all the 0's below and the 1's above. This statement of the Gale-Ryser theorem (see [2]) is given as Lemma 3 in [3].…”

The maximum, supremum, and spectrum for critical set sizes in (0,1)‐matrices

“…With R and S as above, we next define A R S ′( , ) to be the set of all m n × ⋆ (0, 1, )-matrices with 1. at most r i 1's in row i, 2. at most n r − i 0's in row i, 3. at most s j 1's in column j, 4.…”

“…Precise bounds for scs(Λ ) n x were obtained in [6] for small values of x and upper bounds for general x, including scs ≤ m (Λ ) m m 2 2 . In [3], it is shown that ≥ x n x scs(Λ ) min{ , ( − ) } n x 2 2 , a corollary of which is:…”

Rank relations between a {0, 1}-matrix and its complement

“…{ , }-matrices are closely related to graph theory and combinatorial mathematics [1][2][3][4]. They also have a wide range of practical applications in statistics and probability [5][6][7][8].…”