Minimally unbalanced diamond-free graphs and Dyck-paths

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.