Synchronizing Automata with Large Reset Lengths

We study synchronizing automata with the shortest reset words of relatively large length. First, we refine the Frankl-Pin result on the length of the shortest words of rank $m$, and the B\'eal, Perrin, and Steinberg results on the length of the shortest reset words in one-cluster automata. The obtained results are applied to computation aimed in extending the class of small automata for which the \v{C}ern\'y conjecture is verified and discovering new automata with special properties regarding synchronization. In particular, a new class of slowly synchronizing automata on a ternary alphabet is constructed and a conjecture on $cn$-extendable sets in synchronizing automata is disproved.