Acyclic automata with easy-to-find short regular expressions

Computing short regular expressions equivalent to a given finite automaton is a hard task. We present a class of acyclic automata for which it is easy-to-find a regular expression that has linear size. We call those automata UDR. A UDR automaton is characterized by properties of its underlying digraph. We give a characterisation theorem and an efficient algorithm to determine if an acyclic automaton is UDR, that can be adapted to compute an equivalent short regular expression.