Synchronizing automata with finitely many minimal synchronizing words

A synchronizing word for a given synchronizing DFA is called minimal if none of its proper factors is synchronizing. We characterize the class of synchronizing automata having only finitely many minimal synchronizing words (the class of such automata is denoted by FG). Using this characterization we prove that any such automaton possesses a synchronizing word of length at most 3n-5. We also prove that checking whether a given DFA A is in FG is co-NP-hard and provide an algorithm for this problem which is exponential in the number of states A.