On quasi-equigenerated and Freiman cover ideals of graphs

A quasi-equigenerated monomial ideal $I$ in the polynomial ring $R= k[x_1, \ldots, x_n]$ is a Freiman ideal if $\mu(I^2) = l(I)\mu(I)- \binom{l(I)}{2}$ where $l(I)$ is the analytic spread of $I$ and $\mu(I)$ is the number of minimal generators of $I$. Freiman ideals are special since there exists an exact formula computing the minimal number of generators of any of their powers. In this work we address the question of characterizing which cover ideals of simple graphs are Freiman.