Extremality of the unique translation-invariant Gibbs measure for hard-core models on the Cayley tree of order $k=3$

We study fertile hard-core models with three states and an activity parameter $\lambda>0$ on the Cayley tree of order $k=3$. It is known that there are four types of such models. For two of them, we find the regions where the unique translation-invariant Gibbs measure is (not) extremal. For one of the considered models, we find the conditions under which the extremal measure is not unique.