Extremality criteria for the supereigenvector space in max-plus algebra

We present necessary and sufficient criteria for a max-algebraic supereigenvector, i.e., solution of the system $A\otimes\textbf{x}\geq\textbf{x}$ with $A\in\overline{\mathbb{R}}^{n\times n}$ in max-plus algebra, to be an extremal. also show that suggested extremality can verified $O(n^2)$ time any given $\textbf{x}$.