Computing short regular expressions equivalent to a given finite
automaton is a hard task. In this work we present a class of acyclic automata
for which it is possible to obtain in time O(n
$^{2}$
log n) an
equivalent regular expression of size O(n). A characterisation of this class is
made using properties of the underlying digraphs that correspond to the
series-parallel digraphs class. Using this characterisation we present an
algorithm for the generation of automata of this class and an enumerative
formula for the underlying digraphs with a given number of vertices.
