Most binary matrices have no small defining set

Bodkin, Carly; Liebenau, Anita; Wanless, Ian M.

doi:10.1016/j.disc.2020.112035

Cited by 1 publication

(1 citation statement)

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“…This result is currently being generalized to arbitrary m as part of a more general result in [1]. Given that it is easy to show that any element of Λ m 2m has a defining set (and hence critical set) of size at most 2m 2 (simply take all the 1's), this result is in some sense asympotically optimal.…”

Section: Introductionmentioning

confidence: 94%

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

Cavenagh

Wright

2019

J of Combinatorial Designs

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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), then we say that D is a defining set for M. A critical set is a minimal defining set (the deletion of any entry results in more than one completion). We give a new equivalent definition of critical sets in (0,1)matrices and apply this theory to Λ m m 2 , the set of (0,1)matrices of dimensions m m 2 × 2 with uniform row and column sum m. The smallest possible size for a defining set of a matrix in Λ m m 2 is m 2 [N. Cavenagh, J. Combin. Des. 21 (2013), pp. 253-266], and the infimum (the largest-smallest defining set size for members of Λ m m 2 ) is known asymptotically [N. Cavenagh and R. Ramadurai, J Combin Des. 2019;27:522-536. wileyonlinelibrary.com/journal/jcd

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Section: Introductionmentioning

confidence: 94%