Over the last decades, several questions related to the synchronization of automata have been studied. Among the ones that remain open, Černý's Conjecture seems to be the most relevant one. In 2008, at the School on Algebraic Theory of Automata in Lisbon, Mikhail V. Volkov suggested several problems related to that conjecture. One of them was the problem of finding universal reset words for totally synchronizing digraphs, that is, those such that every automaton obtained from them can be synchronized.
In this lecture, we will study the problem of complete synchronization for the class of monotonic graphs, that can vaguely be described as those for which the resulting automata preserve some linear order in the state set.