  We introduce two new invariants of a Noetherian (standard graded) local ring
$(R, \mathfrak m)$ that measure the number of generators of certain kinds of
reductions of $\mathfrak m,$ and we study their properties. Explicitly, we
consider the minimum among the number of generators of ideals $I$ such that
either $I^2 = \mathfrak m^2$ or $I \supseteq \mathfrak m^2$ holds. We
investigate subsequently the case that $R$ is the quotient of a polynomial ring
$k[x_1, \dots, x_n]$ by an ideal $I$ generated by homogeneous quadratic forms,
and we compute these invariants. We devote specific attention to the case that
$R$ is the quotient of a polynomial ring $k[x_1, \dots, x_n]$ by the edge ideal
of a finite simple graph $G.$
