  Consider the following decision problem: for a given monotone Boolean
function $f$ decide, whether $f$ is read-once. For this problem, it is
essential how the input function $f$ is represented. Our contribution consists
of the following two results. We show that we can test in polynomial-time
whether a given expression $C\lor D$ computes a read-once function, provided
that $C$ is a read-once monotone CNF and $D$ is a read-once monotone DNF and
all the variables of $C$ occur also in $D$ (recall that due to Gurvich, the
problem is coNP-complete when $C$ is read-2). The second result states that
this is a coNP-complete problem to decide whether the expression $\Psi\land
D_n$ computes a read-once function, where $D_n$ is as above and $\Psi$ is the
$\bigwedge-\bigvee-\bigwedge$ depth-3 read-once monotone Boolean formula (so
that the entire expression $\Psi\land D_n$ is depth-3 read-2). This result
improves the result of \cite{elbassioni2011readability} in the depth and the
result of \cite{gurvich2010it} in the readability of the input formula.
