  Let $G = (V,w)$ be a weighted undirected graph with $m$ edges. The cut
dimension of $G$ is the dimension of the span of the characteristic vectors of
the minimum cuts of $G$, viewed as vectors in $\{0,1\}^m$. For every $n \ge 2$
we show that the cut dimension of an $n$-vertex graph is at most $2n-3$, and
construct graphs realizing this bound.
  The cut dimension was recently defined by Graur et al.\ \cite{GPRW20}, who
show that the maximum cut dimension of an $n$-vertex graph is a lower bound on
the number of cut queries needed by a deterministic algorithm to solve the
minimum cut problem on $n$-vertex graphs. For every $n\ge 2$, Graur et al.\
exhibit a graph on $n$ vertices with cut dimension at least $3n/2 -2$, giving
the first lower bound larger than $n$ on the deterministic cut query complexity
of computing mincut. We observe that the cut dimension is even a lower bound on
the number of \emph{linear} queries needed by a deterministic algorithm to
solve mincut, where a linear query can ask any vector $x \in
\mathbb{R}^{\binom{n}{2}}$ and receives the answer $w^T x$. Our results thus
show a lower bound of $2n-3$ on the number of linear queries needed by a
deterministic algorithm to solve minimum cut on $n$-vertex graphs, and imply
that one cannot show a lower bound larger than this via the cut dimension.
  We further introduce a generalization of the cut dimension which we call the
$\ell_1$-approximate cut dimension. The $\ell_1$-approximate cut dimension is
also a lower bound on the number of linear queries needed by a deterministic
algorithm to compute minimum cut. It is always at least as large as the cut
dimension, and we construct an infinite family of graphs on $n=3k+1$ vertices
with $\ell_1$-approximate cut dimension $2n-2$, showing that it can be strictly
larger than the cut dimension.
