Careful Synchronization of One-Cluster Automata

In this paper, we investigate the careful synchronization of one-cluster partial automata. First, we prove that the shortest carefully synchronizing word for such automata can be 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 for the binary alphabet.