  The asynchronous dynamics associated with a Boolean network $f : \{0,1\}^n
\to \{0,1\}^n$ is a finite deterministic automaton considered in many
applications. The set of states is $\{0,1\}^n$, the alphabet is $[n]$, and the
action of letter $i$ on a state $x$ consists in either switching the $i$th
component if $f_i(x)\neq x_i$ or doing nothing otherwise. This action is
extended to words in the natural way. We then say that a word $w$ {\em fixes}
$f$ if, for all states $x$, the result of the action of $w$ on $x$ is a fixed
point of $f$. A whole family of networks is fixable if its members are all
fixed by the same word, and the fixing length of the family is the minimum
length of such a word. In this paper, we are interested in families of Boolean
networks with relatively small fixing lengths. Firstly, we prove that fixing
length of the family of networks with acyclic asynchronous graphs is $\Theta(n
2^n)$. Secondly, it is known that the fixing length of the whole family of
monotone networks is $O(n^3)$. We then exhibit two families of monotone
networks with fixing length $\Theta(n)$ and $\Theta(n^2)$ respectively, namely
monotone networks with tree interaction graphs and conjunctive networks with
symmetric interaction graphs.
