In this paper we investigate careful synchronization of one-cluster partial automata. First we prove that in general case the shortest carefully synchronizing word for such automata is of length $2^\frac{ n}{ 2} + 1$, where $n$ is the number of states of an automaton. Additionally we prove that checking whether a given one-cluster partial automaton is carefully synchronizing is NP-hard even in the case of binary alphabet.
Categories: cs.FL cs.CC