A $\{0,1\}$-matrix $\mathsf{A}$ is balanced if it does not contain a
submatrix of odd order having exactly two 1's per row and per column. A graph
is balanced if its clique-matrix is balanced. No characterization of minimally
unbalanced graphs is known, and even no conjecture on the structure of such
graphs has been posed, contrarily to what happened for perfect graphs. In this
paper, we provide such a characterization for the class of diamond-free graphs
and establish a connection between minimally unbalanced diamond-free graphs and
Dyck-paths