  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
{\em defining set} for $M$. A {\em critical set} is a minimal defining set (the
deletion of any entry results in more than one completion). We give a new
classification of critical sets in $(0,1)$-matrices and apply this theory to
$\Lambda_{2m}^m$, the set of $(0,1)$-matrices of dimensions $2m\times 2m$ with
uniform row and column sum $m$.
  The smallest possible size for a defining set of a matrix in $\Lambda_{2m}^m$
is $m^2$
  \cite{Cav}, and the infimum (the largest smallest defining set size for
members of $\Lambda_{2m}^m$) is known asymptotically \cite{CR}.
  We show that no critical set of size larger than $3m^2-2m$ exists in an
element of $\Lambda_{2m}^m$ and that there exists a critical set of size $k$ in
an element of $\Lambda_{2m}^m$ for each $k$ such that $m^2\leq k\leq
3m^2-4m+2$. We also bound the supremum (the smallest largest critical set size
for members of $\Lambda_{2m}^m$) between $\lceil (3m^2-2m+1)/2\rceil$ and
$2m^2-m$.
