  Let $I\subset S=K[x_1,\dots,x_n]$ be a squarefree monomial ideal, $K$ a
field. The $k$th squarefree power $I^{[k]}$ of $I$ is the monomial ideal of $S$
generated by all squarefree monomials belonging to $I^k$. The biggest integer
$k$ such that $I^{[k]}\ne(0)$ is called the monomial grade of $I$ and it is
denoted by $\nu(I)$. Let $d_k$ be the minimum degree of the monomials belonging
to $I^{[k]}$. Then, $\text{depth}(S/I^{[k]})\ge d_k-1$ for all $1\le
k\le\nu(I)$. The normalized depth function of $I$ is defined as
$g_{I}(k)=\text{depth}(S/I^{[k]})-(d_k-1)$, $1\le k\le\nu(I)$. It is expected
that $g_I(k)$ is a non-increasing function for any $I$. In this article we
study the behaviour of $g_{I}(k)$ under various operations on monomial ideals.
Our main result characterizes all cochordal graphs $G$ such that for the edge
ideal $I(G)$ of $G$ we have $g_{I(G)}(1)=0$. They are precisely all cochordal
graphs $G$ whose complementary graph $G^c$ is connected and has a cut vertex.
As a far-reaching application, for given integers $1\le s<m$ we construct a
graph $G$ such that $\nu(I(G))=m$ and $g_{I(G)}(k)=0$ if and only if
$k=s+1,\dots,m$. Finally, we show that any non-increasing function of
non-negative integers is the normalized depth function of some squarefree
monomial ideal.
