Recognising weakly stable matrices

A max-plus matrix A is called weakly stable if the sequence (orbit) does not reach an eigenvector of A for any starting vector x unless x is an eigenvector. This is in contrast to previously studied strongly stable (robust) matrices for which the orbit reaches an eigenvector with any non-trivial starting vector. Max-plus matrices are used to describe multiprocessor interactive systems for which reachability of a steady regime is equivalent to reachability of an eigenvector by a matrix orbit.
We prove that an irreducible matrix is weakly stable if and only if its critical graph is a Hamiltonian cycle in the associated graph. We extend this condition to reducible matrices. These criteria can be checked in polynomial time.