  Any finite simple graph $G = (V,E)$ can be represented by a collection
$\mathscr{C}$ of subsets of $V$ such that $uv\in E$ if and only if $u$ and $v$
appear together in an odd number of sets in $\mathscr{C}$. Let $c_2(G)$ denote
the minimum cardinality of such a collection. This invariant is equivalent to
the minimum dimension of a faithful orthogonal representation of $G$ over
$\mathbb{F}_2$ and is closely connected to the minimum rank of $G$. We show
that $c_2(G) = \operatorname{mr}(G,\mathbb{F}_2)$ when
$\operatorname{mr}(G,\mathbb{F}_2)$ is odd, or when $G$ is a forest. Otherwise,
$\operatorname{mr}(G,\mathbb{F}_2)\leq c_2(G)\leq
\operatorname{mr}(G,\mathbb{F}_2)+1$. Furthermore, we show that the following
are equivalent for any graph $G$ with at least one edge: i.
$c_2(G)=\operatorname{mr}(G,\mathbb{F}_2)+1$; ii. the adjacency matrix of $G$
is the unique matrix of rank $\operatorname{mr}(G,\mathbb{F}_2)$ which fits $G$
over $\mathbb{F}_2$; iii. there is a minimum collection $\mathscr{C}$ as
described in which every vertex appears an even number of times; and iv. for
every component $G'$ of $G$, $c_2(G') = \operatorname{mr}(G',\mathbb{F}_2) +
1$. We also show that, for these graphs, $\operatorname{mr}(G,\mathbb{F}_2)$ is
twice the minimum number of tricliques whose symmetric difference of edge sets
is $E$. Additionally, we provide a set of upper bounds on $c_2(G)$ in terms of
the order, size, and vertex cover number of $G$. Finally, we show that the
class of graphs with $c_2(G)\leq k$ is hereditary and finitely defined. For odd
$k$, the sets of minimal forbidden induced subgraphs are the same as those for
the property $\operatorname{mr}(G,\mathbb{F}_2)\leq k$, and we exhibit this set
for $c_2(G)\leq2$.
