  A graph $G$ on $n$ vertices is a \emph{threshold graph} if there exist real
numbers $a_1,a_2, \ldots, a_n$ and $b$ such that the zero-one solutions of the
linear inequality $\sum \limits_{i=1}^n a_i x_i \leq b$ are the characteristic
vectors of the cliques of $G$. Introduced in [Chv{\'a}tal and Hammer, Annals of
Discrete Mathematics, 1977], the \emph{threshold dimension} of a graph $G$,
denoted by $\dimth(G)$, is the minimum number of threshold graphs whose
intersection yields $G$. Given a graph $G$ on $n$ vertices, in line with
Chv{\'a}tal and Hammer, $f_G\colon \{0,1\}^n \rightarrow \{0,1\}$ is the
Boolean function that has the property that $f_G(x) = 1$ if and only if $x$ is
the characteristic vector of a clique in $G$. A Boolean function $f$ for which
there exists a graph $G$ such that $f=f_G$ is called a \emph{graphic} Boolean
function. It follows that for a graph $G$, $\dimth(G)$ is precisely the minimum
number of \emph{majority} gates whose AND (or conjunction) realizes the graphic
Boolean function $f_G$. The fact that there exist Boolean functions which can
be realized as the AND of only exponentially many majority gates motivates us
to study threshold dimension of graphs. We give tight or nearly tight upper
bounds for the threshold dimension of a graph in terms of its treewidth,
maximum degree, degeneracy, number of vertices, size of a minimum vertex cover,
etc. We also study threshold dimension of random graphs and graphs with high
girth.
