dc.contributor.author Cavenagh, Nicholas J. dc.date.accessioned 2012-10-30T22:09:38Z dc.date.available 2012-10-30T22:09:38Z dc.date.copyright 2012-07-24 dc.date.issued 2013 dc.identifier.citation Cavenagh, N. J. (2013). Defining Sets and Critical Sets in (0,1)-Matrices. Journal of Combinatorial Designs, 21(6), 253-266. en_NZ dc.identifier.issn 1520-6610 dc.identifier.uri https://hdl.handle.net/10289/6768 dc.description.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. en_NZ dc.language.iso en dc.publisher Wiley Periodicals, Inc en_NZ dc.relation.ispartof Journal of Combinatorial Designs dc.subject (0,1)-matrix en_NZ dc.subject defining set en_NZ dc.subject critical set en_NZ dc.subject frequency square en_NZ dc.subject F-square en_NZ dc.subject Gale-Ryser Theorem en_NZ dc.title Defining sets and critical sets in (0,1)-matrices. en_NZ dc.type Journal Article en_NZ dc.identifier.doi 10.1002/jcd.21326 en_NZ dc.relation.isPartOf Journal of Combinatorial Designs en_NZ pubs.begin-page 253 en_NZ pubs.elements-id 38394 pubs.end-page 266 en_NZ pubs.issue 6 en_NZ pubs.volume 21 en_NZ
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