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      Defining sets and critical sets in (0,1)-matrices.

      Cavenagh, Nicholas J.
      DOI
       10.1002/jcd.21326
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      Cavenagh, N. J. (2013). Defining Sets and Critical Sets in (0,1)-Matrices. Journal of Combinatorial Designs, 21(6), 253-266.
      Permanent Research Commons link: https://hdl.handle.net/10289/6768
      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.
      Date
      2013
      Type
      Journal Article
      Publisher
      Wiley Periodicals, Inc
      Collections
      • Computing and Mathematical Sciences Papers [1454]
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