The Cerny conjecture for automata respecting intervals of a directed   graph
  The \v{C}ern\'y's conjecture states that for every synchronizing automaton with n states there exists a reset word of length not exceeding (n-11)^2. We prove this conjecture for a class of automata preserving certain properties of intervals of a directed graph. Our result unifies and generalizes some earlier results obtained by other authors. 
M. Grech and A. Kisielewicz
cs.FL